Multiple Correct Answers MCQs for Sub-Topics of Topic 4: Geometry

Question 1. Which of the following statements correctly describe a point in geometry?

(A) It has a definite size.

(B) It represents a location.

(C) It has no dimension.

(D) It can be represented by a dot.

Question 2. Identify the correct descriptions of a line in Euclidean geometry:

(A) It has a definite length.

(B) It extends infinitely in one direction.

(C) It has no breadth.

(D) It is determined by two distinct points.

Question 3. Which of the following are types of lines based on their relationship in a plane?

(A) Straight lines

(B) Parallel lines

(C) Intersecting lines

(D) Curved lines

Question 4. A plane in geometry can be described as:

(A) A flat surface.

(B) Extending infinitely in all directions.

(C) Having two dimensions.

(D) Containing a finite number of points.

Question 5. Which of the following have definite endpoints?

(A) A line

(B) A ray

(C) A line segment

(D) A curve (if closed)

Question 6. Consider a ray $\overrightarrow{AB}$. Which of the following statements are true?

(A) Point A is the endpoint.

(B) It extends infinitely from A through B.

(C) It has two endpoints.

(D) It is a part of a line.

Question 7. Which of the following are examples of closed curves?

(A) A circle

(B) A square boundary

(C) A parabola

(D) A line segment

Question 8. If two distinct points P and Q are given, which of the following geometric elements can be uniquely determined?

(A) A line segment PQ

(B) A ray starting at P and passing through Q

(C) A line passing through P and Q

(D) A circle with diameter PQ

Question 9. Consider three distinct points A, B, and C. If they are collinear, it means:

(A) They lie on the same plane.

(B) They lie on the same line.

(C) They form a triangle.

(D) A line can be drawn through all three points.

Question 10. Which of the following can be modelled by parts of a ruled paper?

(A) Parallel lines

(B) Intersecting lines

(C) Line segments

(D) Rays

Question 11. Identify the statements that are true about geometric planes.

(A) A plane has thickness.

(B) Three non-collinear points define a unique plane.

(C) If two distinct lines intersect, they lie in the same plane.

(D) A plane has a boundary.

Question 12. A geometric ray is defined by:

(A) An endpoint.

(B) Direction.

(C) A definite length.

(D) Extending infinitely in both directions.

Question 1. Which of the following instruments are typically used for measuring lengths of line segments?

(A) Protractor

(B) Ruler

(C) Measuring tape

(D) Compass (for copying or comparing lengths)

Question 2. The standard units for measuring length in the SI system include:

(A) Degrees

(B) Metres

(C) Centimetres

(D) Kilograms

Question 3. When comparing the lengths of two line segments $\overline{AB}$ and $\overline{CD}$, we can:

(A) Visually estimate which is longer.

(B) Measure both lengths and compare the numerical values.

(C) Use a compass to transfer the length of one segment onto the other.

(D) Assume they are equal if they are parallel.

Question 4. An angle is formed by:

(A) Two intersecting lines.

(B) Two rays with a common endpoint.

(C) A line segment and a ray.

(D) Three non-collinear points.

Question 5. In the angle $\angle \text{PQR}$, the vertex is:

(A) Point P

(B) Point Q

(C) Point R

(D) The common endpoint of the rays.

Question 6. The arms of $\angle \text{XYZ}$ are:

(A) The line segments XY and YZ.

(B) The rays $\overrightarrow{YX}$ and $\overrightarrow{YZ}$.

(C) The point Y.

(D) The lines containing the rays.

Question 7. A point is in the interior of an angle if it lies:

(A) On the vertex.

(B) Between the arms of the angle.

(C) On one of the arms.

(D) In the region bounded by the two rays and a segment connecting points on the rays.

Question 8. The standard unit for measuring angles is the degree ($\circ$). Which of the following are also units for measuring angles?

(A) Radians

(B) Gradians

(C) Minutes ($'$)

(D) Litres

Question 9. The measure of an angle depends on:

(A) The length of the arms.

(B) The distance of the vertex from other points.

(C) The opening between the arms.

(D) The unit used for measurement.

Question 10. Which of the following statements about angle measurement are true?

(A) A full rotation is $360^\circ$.

(B) A straight angle is half of a full rotation.

(C) A right angle is a quarter of a full rotation.

(D) Angles are always positive measures.

Question 11. When measuring an angle using a protractor, the vertex of the angle should be placed at the centre of the protractor, and one arm should align with the $0^\circ$ mark. True or False?

(A) True

(B) False

(C) Only for acute angles

(D) Only for obtuse angles

Question 12. If two line segments have the same length, which notation is used to express this?

(A) AB $\sim$ CD

(B) AB $\cong$ CD

(C) length(AB) = length(CD)

(D) AB || CD

Question 1. Which of the following angle measures represent an acute angle?

(A) $30^\circ$

(B) $89^\circ$

(C) $90^\circ$

(D) $105^\circ$

Question 2. Which of the following angle measures represent an obtuse angle?

(A) $85^\circ$

(B) $90^\circ$

(C) $95^\circ$

(D) $179^\circ$

Question 3. A right angle measures:

(A) Exactly $90^\circ$

(B) Half of a straight angle

(C) A quarter of a complete angle

(D) Less than an acute angle

Question 4. A reflex angle is an angle that measures:

(A) More than $90^\circ$

(B) More than $180^\circ$

(C) Less than $360^\circ$

(D) Exactly $360^\circ$

Question 5. When two lines are perpendicular, they:

(A) Intersect at exactly one point.

(B) Form four right angles at the intersection.

(C) Are parallel to each other.

(D) Their symbol is $\perp$.

Question 6. Which of the following figures can have a perpendicular bisector?

(A) A line

(B) A ray

(C) A line segment

(D) A complete line

Question 7. A perpendicular bisector of a line segment:

(A) Passes through the midpoint of the segment.

(B) Is perpendicular to the segment.

(C) Is parallel to the segment.

(D) Contains all points equidistant from the endpoints of the segment.

Question 8. Which angle types can be formed by two rays originating from the same point?

(A) Zero angle

(B) Complete angle

(C) Acute angle

(D) Straight line

Question 9. The hands of a clock can form various angles. At 9 o'clock, the hands form a:

(A) Right angle

(B) $90^\circ$ angle

(C) Obtuse angle (reflex angle is greater than 180)

(D) Acute angle

Question 10. If a line $p$ is perpendicular to a line $q$, and line $r$ is parallel to line $q$, then:

(A) $p$ is parallel to $r$.

(B) $p$ is perpendicular to $r$.

(C) The angle between $p$ and $r$ is $90^\circ$.

(D) $q$ is perpendicular to $r$.

Question 11. A zero angle and a complete angle:

(A) Have the same measure numerically ($0^\circ$ vs $360^\circ$ is different).

(B) Represent zero rotation or full rotation, respectively.

(C) Have their arms coinciding.

(D) Are formed by the same set of rays.

Question 12. The symbol for perpendicularity between two lines $l$ and $m$ is:

(A) $l || m$

(B) $l \cong m$

(C) $l \perp m$

(D) The angle between them is $90^\circ$.

Question 1. Which of the following pairs of angle measures could be complementary angles?

(A) $30^\circ, 60^\circ$

(B) $45^\circ, 45^\circ$

(C) $50^\circ, 40^\circ$

(D) $70^\circ, 110^\circ$

Question 2. Which of the following pairs of angle measures could be supplementary angles?

(A) $60^\circ, 120^\circ$

(B) $90^\circ, 90^\circ$

(C) $100^\circ, 80^\circ$

(D) $45^\circ, 45^\circ$

Question 3. Two angles are adjacent if they share:

(A) A common vertex.

(B) A common side (arm).

(C) No interior points in common.

(D) Their sum is $90^\circ$.

Question 4. A linear pair of angles:

(A) Are always adjacent.

(B) Have non-common arms forming a straight line.

(C) Are always supplementary.

(D) Are always equal.

Question 5. When two lines intersect, which pairs of angles are equal?

(A) Adjacent angles

(B) Vertically opposite angles

(C) Angles forming a linear pair

(D) Angles opposite the point of intersection

Question 6. If two angles are supplementary, they can be:

(A) Both acute angles.

(B) Both right angles.

(C) One acute and one obtuse angle.

(D) A straight angle and a zero angle.

Question 7. If two angles are complementary, they must be:

(A) Both acute angles.

(B) Both equal angles.

(C) Adjacent angles.

(D) Supplementary to each other's complements.

Question 8. If two angles form a linear pair, then:

(A) Their sum is $180^\circ$.

(B) They are supplementary.

(C) They are adjacent.

(D) They are vertically opposite.

Question 9. Consider two intersecting lines forming four angles. Which of the following are true?

(A) Adjacent angles form linear pairs.

(B) Vertically opposite angles are equal.

(C) The sum of all four angles is $360^\circ$.

(D) Any two non-adjacent angles are vertically opposite.

Question 10. If angle A and angle B are complementary, and angle B and angle C are supplementary, which conclusion(s) can be drawn?

(A) $\angle A + \angle B = 90^\circ$

(B) $\angle B + \angle C = 180^\circ$

(C) $\angle C = 90^\circ + \angle A$

(D) $\angle C$ is an obtuse angle if $\angle A$ is acute.

Question 11. If two lines intersect and one of the angles formed is $90^\circ$, then the lines are perpendicular. This means:

(A) All four angles formed are $90^\circ$.

(B) Each adjacent angle forms a linear pair summing to $180^\circ$.

(C) Vertically opposite angles are equal to $90^\circ$.

(D) The lines are parallel.

Question 12. Two angles are adjacent and their sum is $90^\circ$. They are:

(A) Complementary angles.

(B) Adjacent complementary angles.

(C) Angles forming a right angle.

(D) A linear pair.

Question 1. When a transversal intersects two lines, it creates various angle pairs. Which of the following are types of angle pairs formed?

(A) Corresponding angles

(B) Alternate interior angles

(C) Vertically opposite angles

(D) Adjacent angles

Question 2. Consider a transversal intersecting two lines. Alternate interior angles are located:

(A) On the same side of the transversal.

(B) On opposite sides of the transversal.

(C) Between the two lines.

(D) Outside the two lines.

Question 3. Consider a transversal intersecting two lines. Corresponding angles are located:

(A) On the same side of the transversal.

(B) On opposite sides of the transversal.

(C) One interior and one exterior to the two lines.

(D) At corresponding positions at each intersection.

Question 4. If a transversal intersects two parallel lines, which of the following angle pairs are equal?

(A) Corresponding angles

(B) Alternate interior angles

(C) Alternate exterior angles

(D) Consecutive interior angles

Question 5. If a transversal intersects two parallel lines, which of the following angle pairs are supplementary?

(A) Corresponding angles

(B) Alternate interior angles

(C) Consecutive interior angles

(D) Angles forming a linear pair

Question 6. Which of the following are criteria to prove that two lines are parallel when intersected by a transversal?

(A) A pair of corresponding angles are equal.

(B) A pair of alternate interior angles are equal.

(C) A pair of consecutive interior angles are complementary.

(D) A pair of consecutive interior angles are supplementary.

Question 7. If lines $l$ and $m$ are intersected by a transversal $t$, and a pair of corresponding angles are equal, then:

(A) Line $l$ is parallel to line $m$.

(B) Alternate interior angles are equal.

(C) Consecutive interior angles are supplementary.

(D) The lines are perpendicular.

Question 8. If two lines are parallel, then any transversal intersecting them will create angles such that:

(A) All acute angles formed are equal.

(B) All obtuse angles formed are equal.

(C) An acute angle and an obtuse angle on the same side of the transversal are supplementary.

(D) Vertically opposite angles are supplementary.

Question 9. Consider two lines and a transversal. If a pair of alternate interior angles are not equal, then:

(A) The lines are perpendicular.

(B) The lines are not parallel.

(C) Corresponding angles are not equal.

(D) Consecutive interior angles are not supplementary.

Question 10. If line A is parallel to line B, and line B is parallel to line C, then:

(A) Line A is parallel to line C.

(B) This is related to Euclid's Fifth Postulate (or an equivalent axiom).

(C) The lines are all in the same plane.

(D) Any transversal will form equal corresponding angles with all three lines.

Question 11. In the context of parallel lines and a transversal, which type(s) of angles are always equal?

(A) Alternate Interior Angles

(B) Corresponding Angles

(C) Vertically Opposite Angles (regardless of parallelism)

(D) Consecutive Interior Angles

Question 12. If a transversal intersects two lines and one consecutive interior angle is $90^\circ$, then:

(A) The other consecutive interior angle is also $90^\circ$.

(B) The lines are parallel.

(C) The transversal is perpendicular to both lines.

(D) All eight angles formed are $90^\circ$.

Question 1. Euclidean geometry is based on a deductive system starting from:

(A) Undefined terms

(B) Definitions

(C) Axioms (Common Notions)

(D) Theorems

Question 2. Which of the following are considered undefined terms in classical Euclidean geometry?

(A) Point

(B) Line

(C) Plane

(D) Triangle

Question 3. Axioms (or Common Notions) in Euclidean geometry are statements that are:

(A) Proven logically.

(B) Assumed to be universally true.

(C) Applicable only to geometry.

(D) Self-evident truths.

Question 4. Postulates in Euclidean geometry are statements that are:

(A) Assumed to be true.

(B) Specific to geometry.

(C) Proven from axioms.

(D) Definitions of terms.

Question 5. Theorems in Euclidean geometry are statements that are:

(A) Assumed to be true without proof.

(B) Proven using logical reasoning.

(C) Derived from definitions, axioms, and postulates.

(D) Always obvious.

Question 6. Euclid's Postulate 1 states that a straight line can be drawn between any two points. This implies that:

(A) Given two points, a line exists passing through them.

(B) The line drawn is unique.

(C) The line has a definite length.

(D) Any two points are collinear.

Question 7. Which of the following statements are equivalent to Euclid's Fifth Postulate?

(A) Through a point not on a given line, there is exactly one line parallel to the given line (Playfair's Axiom).

(B) The sum of angles in a triangle is $180^\circ$.

(C) The sum of interior angles on the same side of a transversal intersecting two parallel lines is $180^\circ$.

(D) Any two straight lines can intersect at most at one point.

Question 8. Non-Euclidean geometries differ from Euclidean geometry primarily by changing or replacing:

(A) The definitions.

(B) Euclid's fifth postulate.

(C) The axioms that are common notions.

(D) The other four postulates.

Question 9. Consider the axiom "If equals be added to equals, the wholes are equal." This is an example of a Common Notion, which:

(A) Applies to quantities of any kind, not just geometric.

(B) Is assumed to be true.

(C) Needs a formal proof.

(D) Is the basis for angle measurement.

Question 10. The phrase "Things which coincide with one another are equal to one another" implies:

(A) Congruent figures are equal in measure.

(B) If two segments have the same length, they can be made to coincide.

(C) This is a definition.

(D) This is a postulate.

Question 11. According to Euclidean postulates, which geometric construction is always possible?

(A) Drawing a straight line between any two points.

(B) Extending a line segment indefinitely.

(C) Drawing a circle with any centre and radius.

(D) Drawing a line parallel to a given line through a point not on it.

Question 12. A proof in Euclidean geometry is a sequence of statements where each statement is justified by:

(A) A definition.

(B) An axiom or postulate.

(C) A previously proven theorem.

(D) An opinion or observation.

Question 1. Which of the following are characteristics of a polygon?

(A) It is a simple closed curve.

(B) It is made up entirely of line segments.

(C) The line segments intersect only at endpoints.

(D) It has at least 2 sides.

Question 2. The line segments forming a polygon are called its:

(A) Diagonals

(B) Edges (in 3D context)

(C) Sides

(D) Boundary segments

Question 3. The common endpoints of the sides of a polygon are called its:

(A) Angles

(B) Vertices

(C) Corners

(D) Nodes

Question 4. A diagonal of a polygon connects:

(A) Two adjacent vertices.

(B) Two non-adjacent vertices.

(C) A vertex to the midpoint of the opposite side.

(D) Two points on the boundary that are not endpoints of the same side.

Question 5. Based on the number of sides, a polygon with 5 sides is called a:

(A) Quadrilateral

(B) Hexagon

(C) Pentagon

(D) Quinquagon

Question 6. A convex polygon is characterized by:

(A) All interior angles being less than $180^\circ$.

(B) All diagonals lying entirely inside the polygon.

(C) At least one interior angle being greater than $180^\circ$.

(D) All vertices pointing outwards.

Question 7. A regular polygon is a polygon that is:

(A) Equilateral (all sides equal).

(B) Equiangular (all angles equal).

(C) Convex.

(D) Always a square or a rectangle.

Question 8. Which of the following are examples of irregular polygons?

(A) A rectangle (not a square)

(B) A rhombus (not a square)

(C) An isosceles triangle (not equilateral)

(D) A square

Question 9. The sum of the interior angles of a polygon with $n$ sides is given by:

(A) $(n-2) \times 180^\circ$

(B) $(2n-4) \times 90^\circ$

(C) $360^\circ$ (for any polygon, this is exterior angles sum)

(D) $n \times (\text{measure of each interior angle})$ (only for regular polygons)

Question 10. Which of the following figures are polygons?

(A) A figure formed by joining 4 points with 4 line segments that cross in the middle.

(B) A circle.

(C) A triangle.

(D) A figure made of 5 line segments forming a closed boundary without crossing.

Question 11. A concave polygon:

(A) Has at least one interior angle greater than $180^\circ$.

(B) Is sometimes called a re-entrant polygon.

(C) Has at least one diagonal outside the polygon.

(D) Has all interior angles less than $180^\circ$.

Question 12. The minimum number of vertices a polygon can have is:

(A) 2

(B) 3

(C) Equal to the minimum number of sides.

(D) Equal to the minimum number of interior angles.

Question 1. A triangle is a polygon defined by:

(A) Two non-collinear points.

(B) Three collinear points.

(C) Three non-collinear points.

(D) Three line segments joining three non-collinear points.

Question 2. The basic elements of a triangle include:

(A) Sides

(B) Vertices

(C) Angles

(D) Diagonals

Question 3. A triangle is classified as isosceles if:

(A) All three sides are equal.

(B) Exactly two sides are equal.

(C) At least two sides are equal.

(D) All three angles are equal.

Question 4. Which of the following correctly describe a triangle with all sides of different lengths?

(A) Isosceles triangle

(B) Scalene triangle

(C) Irregular polygon

(D) Equilateral triangle

Question 5. An equilateral triangle is a triangle where:

(A) All sides are equal.

(B) All angles are equal.

(C) Each angle measures $60^\circ$.

(D) It is also an isosceles triangle.

Question 6. A triangle is classified as acute-angled if:

(A) One angle is acute.

(B) All three angles are acute.

(C) No angle is $90^\circ$ or more.

(D) One angle is less than $90^\circ$.

Question 7. A triangle is classified as obtuse-angled if:

(A) One angle is greater than $90^\circ$.

(B) All three angles are obtuse.

(C) It cannot have a right angle.

(D) The sum of two angles is less than the third angle.

Question 8. A right-angled triangle is a triangle where:

(A) One angle is exactly $90^\circ$.

(B) The other two angles are complementary.

(C) It cannot have an obtuse angle.

(D) All three angles are right angles.

Question 9. Which of the following combinations of angles can form a triangle?

(A) $50^\circ, 60^\circ, 70^\circ$

(B) $90^\circ, 90^\circ, 0^\circ$

(C) $30^\circ, 40^\circ, 110^\circ$

(D) $100^\circ, 40^\circ, 30^\circ$

Question 10. An equilateral triangle is also a(n):

(A) Isosceles triangle

(B) Acute-angled triangle

(C) Equiangular triangle

(D) Right-angled triangle

Question 11. Can a triangle be both isosceles and right-angled?

(A) Yes

(B) Yes, if its angles are $45^\circ, 45^\circ, 90^\circ$

(C) No

(D) Yes, if its sides are proportional to $1, 1, \sqrt{2}$

Question 12. A triangle divides the plane into:

(A) The interior region.

(B) The exterior region.

(C) The boundary (the triangle itself).

(D) Only two regions.

Question 1. The Angle Sum Property of a triangle states that the sum of the interior angles is:

(A) $180^\circ$

(B) A constant value regardless of the triangle type

(C) Equal to a straight angle

(D) $90^\circ$ for a right triangle

Question 2. The Exterior Angle Property of a triangle states that the measure of an exterior angle is equal to:

(A) The sum of the two adjacent interior angles.

(B) The sum of the two opposite interior angles.

(C) $180^\circ$ minus the adjacent interior angle.

(D) Greater than either of the opposite interior angles.

Question 3. In an isosceles triangle, if two sides are equal, then:

(A) The angles opposite these sides are equal.

(B) The median to the unequal side is also an altitude and an angle bisector.

(C) All three angles are equal (only if it's equilateral).

(D) The angle between the equal sides is equal to the base angles.

Question 4. The Triangle Inequality Theorem states that in any triangle:

(A) The sum of the lengths of any two sides is greater than the length of the third side.

(B) The difference between the lengths of any two sides is less than the length of the third side.

(C) The longest side is opposite the largest angle.

(D) The sum of all side lengths is $180^\circ$.

Question 5. In $\triangle$ABC, if $\angle A > \angle B$, then based on the side-angle relationship:

(A) BC > AC

(B) AC > BC

(C) The side opposite $\angle A$ is longer than the side opposite $\angle B$.

(D) AB is the smallest side.

Question 6. Which of the following sets of lengths can form the sides of a triangle?

(A) 2 cm, 3 cm, 5 cm

(B) 4 cm, 5 cm, 6 cm

(C) 7 cm, 7 cm, 10 cm

(D) 1 cm, 1 cm, 1 cm

Question 7. In $\triangle$PQR, if PQ = QR, then:

(A) $\angle P = \angle R$

(B) $\angle Q = \angle R$

(C) The triangle is isosceles.

(D) The altitude from Q to PR bisects $\angle Q$.

Question 8. If the angles of a triangle are in the ratio $2:3:4$, then:

(A) The sum of the ratios is 9.

(B) The angles are $40^\circ, 60^\circ, 80^\circ$.

(C) The triangle is acute-angled.

(D) The triangle is right-angled.

Question 9. The angle opposite the shortest side in a triangle is always the:

(A) Largest angle.

(B) Smallest angle.

(C) A right angle.

(D) Can be acute or obtuse.

Question 10. If in $\triangle$XYZ, $\angle X = 90^\circ$, then:

(A) $\angle Y + \angle Z = 90^\circ$.

(B) $\triangle$XYZ is a right-angled triangle.

(C) YZ is the longest side (hypotenuse).

(D) $\angle Y$ and $\angle Z$ are both acute.

Question 11. The sum of an interior angle and its corresponding exterior angle at a vertex of a triangle is always:

(A) $90^\circ$

(B) $180^\circ$

(C) A straight angle

(D) Equal to the sum of the other two interior angles

Question 12. An exterior angle of a triangle can be:

(A) Acute (e.g., if the opposite interior angles are obtuse).

(B) Right (e.g., if the opposite interior angles are $45^\circ$ and $45^\circ$).

(C) Obtuse (e.g., if the opposite interior angles are $30^\circ$ and $40^\circ$).

(D) Equal to the sum of the two remote interior angles.

Question 1. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If the sides are $a$, $b$, and $c$ (hypotenuse), this means:

(A) $a^2 + b^2 = c^2$

(B) $c = \sqrt{a^2 + b^2}$

(C) The theorem applies to acute triangles.

(D) The theorem applies to triangles with a $90^\circ$ angle.

Question 2. In a right triangle ABC, with the right angle at B, the hypotenuse is:

(A) Side AB

(B) Side BC

(C) Side AC

(D) The longest side

Question 3. Which of the following sets of numbers are Pythagorean triplets?

(A) (3, 4, 5)

(B) (5, 12, 13)

(C) (8, 15, 17)

(D) (6, 8, 10)

Question 4. The converse of the Pythagorean theorem is used to:

(A) Find the length of the hypotenuse.

(B) Prove that a triangle is right-angled.

(C) Find the length of a leg.

(D) Determine the type of angle opposite the longest side.

Question 5. If the sides of a triangle are $a, b, c$ and $a^2 + b^2 = c^2$, then:

(A) The angle opposite side $c$ is a right angle.

(B) The triangle is a right-angled triangle.

(C) $c$ is the hypotenuse.

(D) The angle opposite side $a$ is acute.

Question 6. A triangle has side lengths 7 cm, 24 cm, and 25 cm. Which of the following are true?

(A) $7^2 + 24^2 = 49 + 576 = 625$.

(B) $25^2 = 625$.

(C) The triangle is right-angled.

(D) The right angle is opposite the side of length 24 cm.

Question 7. Applications of the Pythagorean theorem include finding:

(A) The diagonal of a rectangle.

(B) The height of a pole leaning against a wall (assuming it forms a right triangle).

(C) The distance between two points in a coordinate plane.

(D) The angles of an equilateral triangle.

Question 8. If the sides of a triangle are $a, b, c$ and $a^2 + b^2 < c^2$, then the triangle is:

(A) Right-angled.

(B) Obtuse-angled (angle opposite $c$).

(C) Acute-angled.

(D) Cannot form a triangle.

Question 9. If the sides of a triangle are $a, b, c$ and $a^2 + b^2 > c^2$, then the triangle is:

(A) Right-angled.

(B) Obtuse-angled.

(C) Acute-angled (angle opposite $c$).

(D) Necessarily equilateral.

Question 10. In an isosceles right-angled triangle, if the hypotenuse is $h$, the length of each equal leg is:

(A) $h/2$

(B) $h/\sqrt{2}$

(C) $\sqrt{2}h$

(D) $\sqrt{h^2/2}$

Question 11. The Pythagorean theorem relates the areas of squares built on the sides of a right-angled triangle. The area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. True or False?

(A) True

(B) False

(C) Only for specific side lengths

(D) This is a common visual proof of the theorem.

Question 12. If a triangle has side lengths $3k, 4k, 5k$ where $k$ is a positive constant, then:

(A) It is a right-angled triangle.

(B) The angle opposite the side $5k$ is $90^\circ$.

(C) The sides form a Pythagorean triplet scaled by $k$.

(D) It is an isosceles triangle.

Question 1. Two geometric figures are congruent if:

(A) They have the same shape.

(B) They have the same size.

(C) They can be made to coincide exactly by rigid transformations (translation, rotation, reflection).

(D) They are mirror images of each other.

Question 2. Which of the following pairs of figures are necessarily congruent?

(A) Two circles with the same radius.

(B) Two squares with the same side length.

(C) Two equilateral triangles with the same side length.

(D) Two triangles with the same angles.

Question 3. Two line segments $\overline{AB}$ and $\overline{CD}$ are congruent if:

(A) They are parallel.

(B) They have the same length.

(C) $AB = CD$ (referring to length).

(D) Their midpoints coincide.

Question 4. Two angles $\angle A$ and $\angle B$ are congruent if:

(A) They are adjacent.

(B) They are vertically opposite.

(C) Their measures are equal ($m\angle A = m\angle B$).

(D) They are formed by congruent rays.

Question 5. The SAS congruence criterion for triangles states that two triangles are congruent if:

(A) Two angles and the included side are equal.

(B) Two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other.

(C) The sides are proportional.

(D) All three angles are equal.

Question 6. Which of the following are valid congruence criteria for triangles?

(A) SSS

(B) AAA

(C) ASA

(D) AAS

Question 7. The RHS congruence criterion is specifically for right-angled triangles and involves:

(A) The right angle.

(B) The hypotenuse.

(C) Any two sides.

(D) One side.

Question 8. CPCTC is a principle used after proving triangles are congruent. It means:

(A) Corresponding Parts of Congruent Triangles are Congruent.

(B) If two triangles are congruent, their corresponding sides are equal.

(C) If two triangles are congruent, their corresponding angles are equal.

(D) It is a criterion for proving congruence.

Question 9. If $\triangle \text{ABC} \cong \triangle \text{XYZ}$, then by CPCTC:

(A) $\angle A = \angle X$

(B) AB = XY

(C) BC = XZ

(D) $\angle C = \angle Z$

Question 10. Congruence of figures is an equivalence relation because it is:

(A) Reflexive (A $\cong$ A)

(B) Symmetric (If A $\cong$ B, then B $\cong$ A)

(C) Transitive (If A $\cong$ B and B $\cong$ C, then A $\cong$ C)

(D) Commutative

Question 11. If two triangles are congruent by the ASA criterion, it means we have shown equality of:

(A) Two sides and an angle.

(B) Two angles and the included side.

(C) Two angles and a non-included side.

(D) The sides adjacent to the equal angles and the included angle.

Question 12. Which of the following pairs of figures are NOT necessarily congruent?

(A) Two copies of the same Aadhaar card.

(B) Two circles with different radii.

(C) Two squares with different side lengths.

(D) A triangle and its reflection in a mirror.

Question 1. Two geometric figures are similar if:

(A) They have the same shape.

(B) They have the same size.

(C) Their corresponding angles are equal.

(D) Their corresponding sides are proportional.

Question 2. For two triangles to be similar ($\triangle \text{ABC} \sim \triangle \text{PQR}$), it is necessary that:

(A) $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$.

(B) $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$.

(C) They are congruent.

(D) They have the same area.

Question 3. Which of the following are valid criteria for proving similarity of triangles?

(A) AA (Angle-Angle)

(B) SSS (Side-Side-Side Proportionality)

(C) SAS (Side-Angle-Side Proportionality)

(D) AAS (Angle-Angle-Side)

Question 4. The Basic Proportionality Theorem (BPT), also known as Thales Theorem, deals with:

(A) A line parallel to one side of a triangle.

(B) The ratio of sides when a parallel line intersects the other two sides.

(C) The angles of a triangle.

(D) Proving triangles are congruent.

Question 5. In $\triangle$ABC, if a line DE is drawn parallel to BC, intersecting AB at D and AC at E, then:

(A) $\triangle \text{ADE} \sim \triangle \text{ABC}$.

(B) $\frac{AD}{DB} = \frac{AE}{EC}$.

(C) $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$.

(D) DE = $\frac{1}{2}$ BC (only if D and E are midpoints).

Question 6. The converse of the Basic Proportionality Theorem is used to prove that:

(A) A line segment is parallel to a side of a triangle.

(B) A line divides two sides of a triangle proportionally.

(C) Two triangles are congruent.

(D) A point is the midpoint of a side.

Question 7. If two triangles are congruent, then they are also similar. True or False?

(A) True

(B) False

(C) Yes, with a similarity ratio of 1:1.

(D) Yes, because corresponding angles are equal and corresponding sides are proportional (ratio 1:1).

Question 8. If two triangles are similar, then:

(A) They are necessarily congruent.

(B) Their corresponding angles are equal.

(C) Their corresponding sides are proportional.

(D) Their corresponding medians and altitudes are also proportional to the sides.

Question 9. All squares are similar. True or False?

(A) True

(B) False

(C) Yes, because all angles are $90^\circ$ and sides are in proportion (equal ratios).

(D) No, because their sizes can be different.

Question 10. If $\triangle \text{ABC} \sim \triangle \text{DEF}$, and AB:DE = 2:1, then:

(A) $\angle A = \angle D$

(B) AC:DF = 2:1

(C) BC:EF = 2:1

(D) Area($\triangle \text{ABC}$):Area($\triangle \text{DEF}$) = 2:1

Question 11. The AA similarity criterion is derived from the fact that if two pairs of angles are equal, the third pair must also be equal due to the angle sum property of a triangle. True or False?

(A) True

(B) False

(C) Yes, because $180^\circ - (\angle A + \angle B) = 180^\circ - (\angle D + \angle E)$ if $\angle A = \angle D$ and $\angle B = \angle E$.

(D) No, it's an independent axiom.

Question 12. Which of the following pairs of figures are always similar?

(A) Two equilateral triangles

(B) Two circles

(C) Two isosceles triangles

(D) Two rhombuses

Question 1. If two triangles are similar, the ratio of their areas is equal to:

(A) The ratio of their corresponding sides.

(B) The square of the ratio of their corresponding sides.

(C) The ratio of the squares of their corresponding altitudes.

(D) The ratio of the squares of their corresponding medians.

Question 2. If $\triangle \text{ABC} \sim \triangle \text{DEF}$ and AB/DE = 3/4, then:

(A) Area($\triangle \text{ABC}$)/Area($\triangle \text{DEF}$) = 9/16

(B) Perimeter($\triangle \text{ABC}$)/Perimeter($\triangle \text{DEF}$) = 3/4

(C) The ratio of corresponding altitudes is 3:4.

(D) $\angle A / \angle D = 3/4$ (Angle measures are equal, not proportional).

Question 3. The areas of two similar triangles are $49 \text{ cm}^2$ and $64 \text{ cm}^2$. What is the ratio of their corresponding sides?

(A) $\sqrt{49} : \sqrt{64}$

(B) 7:8

(C) 49:64

(D) 8:7

Question 4. In a right-angled triangle, drawing an altitude from the right angle to the hypotenuse creates two smaller triangles. These smaller triangles are:

(A) Similar to each other.

(B) Similar to the original triangle.

(C) Necessarily congruent to each other.

(D) Have angles equal to the original triangle's acute angles.

Question 5. In right $\triangle$ABC, right-angled at B, BD is the altitude to AC. Then:

(A) $\triangle \text{ADB} \sim \triangle \text{BDC}$.

(B) $\triangle \text{ADB} \sim \triangle \text{ABC}$.

(C) $\triangle \text{BDC} \sim \triangle \text{ABC}$.

(D) $BD^2 = AD \times DC$.

Question 6. Similarity of triangles is used in real-world applications like:

(A) Calculating the height of tall objects using shadows.

(B) Creating scale models.

(C) Map making.

(D) Designing structures like bridges or buildings.

Question 7. If the ratio of perimeters of two similar triangles is $p:q$, then the ratio of their areas is:

(A) $p:q$

(B) $p^2:q^2$

(C) $\sqrt{p}:\sqrt{q}$

(D) $(p/q)^2$

Question 8. If $\triangle \text{LMN} \sim \triangle \text{RST}$ and Area($\triangle \text{LMN}$):Area($\triangle \text{RST}$) = 25:36. If LM = 10 cm, then RS is:

(A) 12 cm

(B) 15 cm

(C) $\frac{10 \times 6}{5}$ cm

(D) $\sqrt{36/25} \times 10$ cm

Question 9. If a line segment joins the midpoints of two sides of a triangle, the smaller triangle formed is similar to the original triangle with a similarity ratio of:

(A) 1:1 (if congruent)

(B) 1:2

(C) 2:1 (original to smaller)

(D) Side of smaller : corresponding side of original = 1:2

Question 10. If the similarity ratio of two similar triangles is $1:k$, then the ratio of their areas is:

(A) $1:k$

(B) $1:k^2$

(C) $k^2:1$

(D) $\sqrt{1}:\sqrt{k}$

Question 1. Which of the following are types of quadrilaterals?

(A) Triangle

(B) Parallelogram

(C) Trapezium

(D) Kite

Question 2. The sum of the interior angles of any convex quadrilateral is:

(A) $180^\circ$

(B) $360^\circ$

(C) Equal to the sum of angles around a point.

(D) $(4-2) \times 180^\circ$.

Question 3. A parallelogram has the following properties:

(A) Opposite sides are parallel.

(B) Opposite sides are equal.

(C) Opposite angles are equal.

(D) Diagonals are perpendicular.

Question 4. The diagonals of a parallelogram:

(A) Bisect each other.

(B) Are always equal.

(C) Are always perpendicular.

(D) Divide the parallelogram into four triangles of equal area.

Question 5. Which of the following are special types of parallelograms?

(A) Rectangle

(B) Rhombus

(C) Square

(D) Trapezium

Question 6. A rectangle is a parallelogram where:

(A) All sides are equal.

(B) All angles are $90^\circ$.

(C) Diagonals are equal.

(D) Adjacent angles are supplementary.

Question 7. A rhombus is a parallelogram where:

(A) All sides are equal.

(B) Diagonals are perpendicular.

(C) Diagonals bisect the angles.

(D) All angles are equal (only if it's a square).

Question 8. A square is a quadrilateral that is simultaneously:

(A) A rectangle

(B) A rhombus

(C) A parallelogram

(D) A kite

Question 9. Which of the following properties are true for a square?

(A) All sides are equal.

(B) All angles are $90^\circ$.

(C) Diagonals are equal and perpendicular bisectors of each other.

(D) Opposite sides are parallel.

Question 10. A trapezium (or trapezoid) is a quadrilateral with:

(A) At least one pair of parallel sides.

(B) Exactly one pair of parallel sides.

(C) All sides equal.

(D) Diagonals bisecting each other.

Question 11. In a kite, which of the following is true?

(A) All four sides are equal.

(B) Two pairs of adjacent sides are equal.

(C) Diagonals are perpendicular.

(D) One diagonal is the perpendicular bisector of the other.

Question 12. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. This statement is:

(A) A theorem.

(B) The converse of a property of parallelograms.

(C) True.

(D) False.

Question 1. The Mid-Point Theorem applies to triangles and involves:

(A) The midpoints of two sides.

(B) A line segment connecting the midpoints of two sides.

(C) The third side of the triangle.

(D) The angles of the triangle.

Question 2. In $\triangle$ABC, if D is the midpoint of AB and E is the midpoint of AC, then according to the Mid-Point Theorem:

(A) DE || BC

(B) DE = BC

(C) DE = $\frac{1}{2}$ BC

(D) $\triangle \text{ADE} \sim \triangle \text{ABC}$ with ratio 1:2.

Question 3. The converse of the Mid-Point Theorem involves:

(A) A line through the midpoint of one side of a triangle.

(B) A line parallel to another side of the triangle.

(C) The third side of the triangle.

(D) Proving that the line bisects the third side.

Question 4. In $\triangle$PQR, S is the midpoint of PQ. A line through S parallel to QR intersects PR at T. According to the converse of the Mid-Point Theorem:

(A) T is the midpoint of PR.

(B) ST = $\frac{1}{2}$ QR.

(C) $\triangle \text{PST} \sim \triangle \text{PQR}$.

(D) $\frac{PS}{PQ} = \frac{PT}{PR} = \frac{ST}{QR}$.

Question 5. An application of the Mid-Point Theorem is proving that the figure formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram. This proof involves:

(A) Drawing the diagonals of the quadrilateral.

(B) Applying the Mid-Point Theorem in the triangles formed by the diagonals.

(C) Showing that opposite sides of the inner figure are parallel.

(D) Showing that opposite sides of the inner figure are equal.

Question 6. In $\triangle$ABC, D, E, F are midpoints of AB, BC, CA respectively. Then $\triangle$DEF is formed. Which of the following are true?

(A) DE || AC

(B) EF || AB

(C) FD || BC

(D) DE = $\frac{1}{2}$ AC

Question 7. In $\triangle$ABC, D, E, F are midpoints of AB, BC, CA respectively. The figure ADEF is a:

(A) Parallelogram

(B) Rectangle

(C) Rhombus

(D) Square

Question 8. The Mid-Point Theorem can be derived using concepts from:

(A) Coordinate geometry.

(B) Vector algebra.

(C) Similarity of triangles.

(D) Trigonometry.

Question 9. In $\triangle$ABC, D, E are midpoints of AB, AC. If perimeter of $\triangle$ABC is 20 cm, the perimeter of $\triangle$ADE is:

(A) 10 cm

(B) 15 cm

(C) $\frac{1}{2}$ of perimeter of $\triangle$ABC

(D) 5 cm

Question 10. If the figure formed by joining the midpoints of a quadrilateral is a rhombus, then the diagonals of the original quadrilateral are:

(A) Equal.

(B) Perpendicular.

(C) Parallel.

(D) Bisected by each other.

Question 11. If the figure formed by joining the midpoints of a quadrilateral is a rectangle, then the diagonals of the original quadrilateral are:

(A) Equal.

(B) Perpendicular.

(C) Parallel.

(D) Bisected by each other.

Question 12. The segment joining the midpoints of two sides of a triangle divides the triangle into a smaller triangle and a trapezium. The smaller triangle is similar to the original triangle. True or False?

(A) True

(B) False

(C) Yes, the smaller triangle is congruent to the other three triangles formed by joining all midpoints.

(D) Yes, the ratio of similarity is 1:2.

Question 1. Area of a plane figure is the measure of the surface enclosed by its boundary. It is measured in:

(A) Linear units (like cm)

(B) Square units (like $\text{cm}^2$)

(C) Cubic units (like $\text{cm}^3$)

(D) Units of area.

Question 2. Two figures are said to be equal in area if:

(A) They are congruent.

(B) They have the same boundary.

(C) They enclose the same amount of surface.

(D) They can be decomposed into a finite number of congruent parts.

Question 3. If two figures are congruent, then:

(A) They are similar.

(B) They are equal in area.

(C) They have the same perimeter.

(D) They have the same shape but different size.

Question 4. Consider two parallelograms on the same base and between the same parallels. Which of the following is true?

(A) They have the same area.

(B) They are congruent.

(C) Their heights are equal.

(D) Their perimeters are equal.

Question 5. Consider two triangles on the same base and between the same parallels. Which of the following is true?

(A) They have the same area.

(B) They are congruent.

(C) Their heights are equal.

(D) Their vertices opposite the base lie on the same line parallel to the base.

Question 6. If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram. This means:

(A) Area(Triangle) = $\frac{1}{2} \times$ Area(Parallelogram)

(B) The height of the triangle and parallelogram (relative to the common base) are equal.

(C) The base is common to both figures.

(D) This is a theorem based on areas.

Question 7. If two triangles have the same base and equal areas, then:

(A) They are congruent.

(B) They lie between the same parallels.

(C) Their corresponding heights are equal.

(D) Their vertices opposite the base are collinear with the base.

Question 8. In parallelogram ABCD, diagonal AC divides it into two triangles $\triangle$ABC and $\triangle$ADC. These two triangles:

(A) Are congruent.

(B) Have equal areas.

(C) Are on the same base.

(D) Have the same height (from B and D to AC).

Question 9. A median of a triangle divides it into two triangles which are:

(A) Congruent.

(B) Similar.

(C) Equal in area.

(D) On equal bases and have the same vertex (hence same height).

Question 10. If the area of parallelogram ABCD is $50 \text{ cm}^2$, and E is any point on side AB, what is the area of $\triangle$CDE?

(A) $50 \text{ cm}^2$

(B) $25 \text{ cm}^2$

(C) Half the area of the parallelogram.

(D) Depends on the position of E.

Question 11. Consider $\triangle$ABC and $\triangle$DBC on the same base BC. If Area($\triangle \text{ABC}$) = Area($\triangle \text{DBC}$), then:

(A) AD || BC.

(B) A, B, C, D form a parallelogram.

(C) A and D lie on a line parallel to BC.

(D) The heights of the two triangles from A and D to BC are equal.

Question 12. The formula for the area of a triangle is related to the area of a parallelogram. Area(Triangle) = $\frac{1}{2} \times$ base $\times$ height. This comes from the fact that:

(A) A parallelogram can be divided into two congruent triangles by a diagonal.

(B) A triangle is half of a parallelogram on the same base and between the same parallels.

(C) The area of a rectangle is base times height.

(D) Congruent figures have equal areas.

Question 1. A circle is defined as:

(A) A closed curve.

(B) The set of all points in a plane equidistant from a fixed point.

(C) A curve with constant curvature.

(D) The boundary of a circular region.

Question 2. Basic terms associated with a circle include:

(A) Centre

(B) Radius

(C) Vertex

(D) Diameter

Question 3. A chord of a circle is:

(A) A line segment joining the centre to a point on the circle.

(B) A line segment joining any two points on the circle.

(C) Always the longest line segment inside the circle.

(D) A part of the circumference.

Question 4. The diameter of a circle is:

(A) A special type of chord.

(B) The longest chord.

(C) Twice the length of the radius.

(D) A line segment passing through the centre with endpoints on the circle.

Question 5. The circumference of a circle is:

(A) The area of the circle.

(B) The perimeter of the circle.

(C) Given by the formula $2\pi r$, where $r$ is the radius.

(D) Given by the formula $\pi d$, where $d$ is the diameter.

Question 6. An arc of a circle is:

(A) A segment of the circumference.

(B) Defined by two points on the circle.

(C) Can be a minor arc or a major arc.

(D) A line segment.

Question 7. A sector of a circle is the region bounded by:

(A) A chord and an arc.

(B) Two radii and an arc.

(C) The centre and an arc.

(D) Two points on the circle and the centre.

Question 8. A segment of a circle is the region bounded by:

(A) Two radii and an arc.

(B) A chord and an arc.

(C) Can be a minor segment or a major segment.

(D) The centre and a chord.

Question 9. Two circles are congruent if:

(A) They have the same centre.

(B) They have the same radius.

(C) They have the same diameter.

(D) They have the same circumference.

Question 10. The interior of a circle includes points:

(A) Whose distance from the centre is less than the radius.

(B) On the boundary of the circle.

(C) Whose distance from the centre is greater than the radius.

(D) Inside the circle.

Question 11. All circles are similar. True or False?

(A) True

(B) False

(C) Yes, because they have the same shape.

(D) Yes, any circle can be scaled to match any other circle.

Question 12. A semicircle is:

(A) Half of a circle.

(B) An arc whose endpoints are the ends of a diameter.

(C) A segment bounded by a diameter and an arc.

(D) A sector bounded by two radii forming a $180^\circ$ angle.

Question 1. The angle subtended by a chord at the centre of a circle is:

(A) Always $90^\circ$.

(B) Twice the angle subtended by the same chord at any point on the remaining part of the circle.

(C) Equal to the angle subtended by the same chord at any point on the remaining part of the circle.

(D) Equal for equal chords in the same circle.

Question 2. Equal chords of a circle:

(A) Subtend equal angles at the centre.

(B) Are equidistant from the centre.

(C) Are always parallel.

(D) Divide the circle into equal segments.

Question 3. The perpendicular from the centre of a circle to a chord:

(A) Bisects the chord.

(B) Bisects the angle subtended by the chord at the centre.

(C) Passes through the midpoint of the chord.

(D) Is parallel to the chord.

Question 4. The angle in a semicircle is always a right angle. This implies:

(A) The angle subtended by a diameter at any point on the circumference is $90^\circ$.

(B) If the angle subtended by a chord at a point on the circumference is $90^\circ$, the chord is a diameter.

(C) This is a special case of the theorem relating angle at the centre and angle at the circumference.

(D) The chord forming the semicircle is perpendicular to the tangent at that point.

Question 5. Angles in the same segment of a circle are equal. This means:

(A) Any two angles subtended by the same arc in the same part of the circle are equal.

(B) $\angle APB = \angle AQB$ if P and Q are points on the same arc and AB is the chord.

(C) This applies to angles in both major and minor segments (within their own segment).

(D) Angles subtended by a diameter are equal.

Question 6. If two chords of a circle are equal, then:

(A) They subtend equal angles at the circumference.

(B) They are equidistant from the centre.

(C) The arcs corresponding to the chords are congruent.

(D) The segments corresponding to the chords are congruent.

Question 7. The line joining the centre of a circle to the midpoint of a chord is:

(A) Perpendicular to the chord.

(B) The shortest distance from the centre to the chord.

(C) An angle bisector of the angle subtended by the chord at the centre (if extended).

(D) Parallel to the chord.

Question 8. If the angle subtended by a chord at the centre is $60^\circ$, and the radius is $r$, then the length of the chord is:

(A) $r$ (The triangle formed is equilateral)

(B) $\sqrt{3}r$

(C) $2r$ (if it's a diameter)

(D) Depends on whether it's a minor or major arc (chord length is the same).

Question 9. If two chords are equidistant from the centre of a circle, then:

(A) They are parallel.

(B) They are equal in length.

(C) They subtend equal angles at the centre.

(D) They are perpendicular bisectors of each other.

Question 10. The angle subtended by a major arc at the centre is:

(A) Less than $180^\circ$.

(B) Greater than $180^\circ$.

(C) The reflex angle subtended by the corresponding minor arc.

(D) Twice the angle subtended by the arc at any point on the corresponding minor segment.

Question 11. If the angle in a segment of a circle is obtuse, then the segment is a:

(A) Major segment.

(B) Minor segment.

(C) Semicircle.

(D) Segment corresponding to a major arc.

Question 12. The locus of points equidistant from two points on a circle's circumference is the perpendicular bisector of the chord joining those points. This line also passes through the centre of the circle. True or False?

(A) True

(B) False

(C) Yes, the perpendicular bisector of a chord always passes through the centre.

(D) No, the locus is a circle.

Question 1. A cyclic quadrilateral is a quadrilateral whose vertices:

(A) Lie on a line.

(B) Lie on a circle.

(C) Are collinear.

(D) Can be inscribed in a circle.

Question 2. The property of opposite angles of a cyclic quadrilateral is that they are:

(A) Equal.

(B) Complementary.

(C) Supplementary.

(D) Their sum is $180^\circ$.

Question 3. If ABCD is a cyclic quadrilateral, then:

(A) $\angle A + \angle C = 180^\circ$.

(B) $\angle B + \angle D = 180^\circ$.

(C) The exterior angle at any vertex is equal to the interior opposite angle.

(D) All four angles are equal (only if it's a cyclic rectangle/square).

Question 4. A quadrilateral is cyclic if and only if the sum of any pair of opposite angles is $180^\circ$. This statement is:

(A) The definition of a cyclic quadrilateral.

(B) A theorem about cyclic quadrilaterals.

(C) The converse property of cyclic quadrilaterals.

(D) Always true for any quadrilateral.

Question 5. Which of the following quadrilaterals are always cyclic?

(A) Rectangle

(B) Square

(C) Isosceles trapezium

(D) Parallelogram

Question 6. If a parallelogram is cyclic, then it must be:

(A) A rhombus.

(B) A rectangle.

(C) A square.

(D) A kite.

Question 7. In cyclic quadrilateral PQRS, if $\angle P = 70^\circ$ and $\angle Q = 100^\circ$, then:

(A) $\angle R = 110^\circ$

(B) $\angle S = 80^\circ$

(C) $\angle R + \angle S = 180^\circ$

(D) The exterior angle at P is $110^\circ$.

Question 8. If the exterior angle of a quadrilateral at vertex A is equal to the interior angle at vertex C, then the quadrilateral is:

(A) A parallelogram.

(B) A cyclic quadrilateral.

(C) An isosceles trapezium.

(D) A rhombus.

Question 9. Which of the following statements about cyclic quadrilaterals are true?

(A) All squares are cyclic.

(B) All rectangles are cyclic.

(C) All rhombuses are cyclic.

(D) All parallelograms are cyclic.

Question 10. If the diagonals of a cyclic quadrilateral intersect at right angles, which other properties might hold?

(A) The quadrilateral is a kite.

(B) The sums of the products of opposite sides are equal (Ptolemy's Theorem).

(C) It might be a square or a rhombus.

(D) The center of the circumscribed circle lies at the intersection of the diagonals.

Question 11. A cyclic trapezium must be an isosceles trapezium. True or False?

(A) True

(B) False

(C) Yes, unless the parallel sides are equal (then it's a rectangle/square).

(D) No, any trapezium can be cyclic.

Question 12. If four points A, B, C, D are concyclic (lie on the same circle), then the quadrilateral ABCD is cyclic. The order of vertices matters for defining opposite angles. True or False?

(A) True

(B) False

(C) Yes, ABCD is cyclic, but ACBD is not necessarily.

(D) No, any quadrilateral formed by concyclic points is cyclic regardless of vertex order.

Question 1. A tangent to a circle is a line that intersects the circle at:

(A) Two points.

(B) Exactly one point.

(C) Infinitely many points.

(D) The point of contact.

Question 2. A secant to a circle is a line that intersects the circle at:

(A) Exactly one point.

(B) Two distinct points.

(C) Passes through the centre.

(D) Contains a chord of the circle.

Question 3. The radius drawn to the point of contact of a tangent is:

(A) Parallel to the tangent.

(B) Perpendicular to the tangent.

(C) Forms a $90^\circ$ angle with the tangent.

(D) Bisects the tangent segment from an external point.

Question 4. From a point P outside a circle, how many tangents can be drawn to the circle?

(A) Zero

(B) One

(C) Two

(D) PA and PB, where A and B are points of contact.

Question 5. The lengths of tangents drawn from an external point to a circle are equal. If P is the external point and PA and PB are tangents, then:

(A) PA = PB

(B) $\triangle \text{PAB}$ is an isosceles triangle.

(C) PO bisects $\angle APB$, where O is the centre.

(D) $\triangle \text{PAO} \cong \triangle \text{PBO}$, where O is the centre.

Question 6. If a point lies inside a circle, how many tangents can be drawn from that point to the circle?

(A) Zero

(B) One

(C) Two

(D) The point itself is on the interior, so no tangents can originate from there and touch the circle.

Question 7. If a point lies on the circle, how many tangents can be drawn from that point to the circle?

(A) Zero

(B) Exactly one

(C) Two

(D) The tangent is perpendicular to the radius at that point.

Question 8. Two parallel tangents can be drawn to a circle. The line segment joining the points of contact of two parallel tangents is always a:

(A) Radius

(B) Chord

(C) Diameter

(D) Line segment passing through the centre.

Question 9. A line segment from the centre O to a point of contact A on the tangent is OA. The angle between OA and the tangent at A is:

(A) $0^\circ$

(B) $90^\circ$

(C) A right angle

(D) $180^\circ$

Question 10. If two circles touch each other externally, the distance between their centres is equal to the sum of their radii. If their radii are $r_1$ and $r_2$ and distance between centres is $d$, then:

(A) $d = r_1 + r_2$

(B) The circles have a common tangent at the point of contact.

(C) The point of contact lies on the line joining the centres.

(D) The circles intersect at two points.

Question 11. If two circles touch each other internally, the distance between their centres is equal to the difference of their radii. If $r_1 > r_2$, then:

(A) $d = r_1 - r_2$

(B) The circles have a common tangent at the point of contact.

(C) The point of contact lies on the line joining the centres.

(D) One circle is completely inside the other, except for the point of contact.

Question 12. The segment of a secant inside the circle is a chord. True or False?

(A) True

(B) False

(C) Yes, a chord is a line segment with endpoints on the circle.

(D) No, a secant is a line, not a segment.

Question 1. A figure has line symmetry if there exists a line such that the figure is its own image when reflected across this line. This line is called the axis of symmetry. True or False?

(A) True

(B) False

(C) Yes, the line divides the figure into two congruent halves.

(D) No, line symmetry requires rotation.

Question 2. Which of the following shapes have at least one line of symmetry?

(A) Square

(B) Rectangle

(C) Equilateral triangle

(D) Parallelogram (that is not a rhombus or rectangle)

Question 3. How many lines of symmetry do the following figures have? (Select the correct combination)

(A) Square: 4

(B) Circle: Infinite

(C) Equilateral triangle: 3

(D) Isosceles triangle (not equilateral): 0

Question 4. The reflection of a point (x, y) across the x-axis results in the point:

(A) (-x, y)

(B) (x, -y)

(C) Keeping the x-coordinate the same.

(D) Changing the sign of the y-coordinate.

Question 5. The reflection of a point (x, y) across the y-axis results in the point:

(A) (-x, y)

(B) (x, -y)

(C) Changing the sign of the x-coordinate.

(D) Keeping the y-coordinate the same.

Question 6. Reflectional symmetry is also known as line symmetry. It is a type of:

(A) Transformation.

(B) Isometry (preserves distance).

(C) Dilation.

(D) Rigid motion.

Question 7. Which of the following letters of the English alphabet have horizontal line symmetry?

(A) A

(B) B

(C) C

(D) E

Question 8. Which of the following letters of the English alphabet have vertical line symmetry?

(A) A

(B) H

(C) M

(D) S

Question 9. If a figure has line symmetry about a line $l$, and a point P is on the figure but not on line $l$, then its reflection P' across $l$ must also be:

(A) On the figure.

(B) The same distance from $l$ as P.

(C) Such that $l$ is the perpendicular bisector of PP'.

(D) On line $l$.

Question 10. A line segment has how many lines of symmetry?

(A) One (the line containing the segment)

(B) One (the perpendicular bisector)

(C) Two

(D) Infinite

Question 11. The image of a geometric figure under reflection is congruent to the original figure. True or False?

(A) True

(B) False

(C) Yes, reflection is an isometry.

(D) No, reflection changes orientation.

Question 12. Which of the following shapes have exactly two lines of symmetry?

(A) Square

(B) Rectangle (not a square)

(C) Rhombus (not a square)

(D) Kite (not a rhombus)

Question 1. A figure has rotational symmetry if, after rotating it by an angle less than $360^\circ$ about a fixed point, it coincides with its original position. The fixed point is the:

(A) Axis of symmetry

(B) Centre of rotation

(C) Centroid

(D) Origin

Question 2. The angle of rotational symmetry is the smallest angle of rotation (greater than $0^\circ$) that maps the figure onto itself. For a square, the angles of rotational symmetry are:

(A) $90^\circ$

(B) $180^\circ$

(C) $270^\circ$

(D) $360^\circ$ (this is always rotational symmetry of order 1)

Question 3. The order of rotational symmetry of a figure is the number of times it coincides with itself during a full $360^\circ$ rotation. For a regular hexagon (6 sides), the order of rotational symmetry is:

(A) 3

(B) 6

(C) Equal to the number of sides.

(D) $360^\circ / (\text{angle of rotational symmetry}) = 360^\circ / 60^\circ$.

Question 4. Which of the following figures have rotational symmetry of order greater than 1?

(A) Equilateral triangle

(B) Rectangle (not a square)

(C) Circle

(D) Scalene triangle

Question 5. The angle of rotational symmetry is $360^\circ$ divided by the order of rotational symmetry. If the angle of rotational symmetry is $72^\circ$, the order of rotation is:

(A) 4

(B) 5

(C) $360/72$

(D) Related to a regular pentagon.

Question 6. A figure has rotational symmetry of order 1 if:

(A) It has no rotational symmetry other than the trivial $360^\circ$ rotation.

(B) Its angle of rotational symmetry is $360^\circ$.

(C) It has no line symmetry.

(D) It is asymmetric.

Question 7. Which of the following figures have rotational symmetry of order 2?

(A) Square

(B) Rectangle (not a square)

(C) Rhombus (not a square)

(D) Letter S

Question 8. The centre of rotation for a geometric figure with rotational symmetry can be:

(A) Inside the figure.

(B) On the boundary of the figure.

(C) Outside the figure.

(D) The intersection of its lines of symmetry (if any).

Question 9. A figure can have both line symmetry and rotational symmetry. Which of the following figures have both?

(A) Square

(B) Equilateral triangle

(C) Circle

(D) Parallelogram (not a rhombus or rectangle)

Question 10. A figure has rotational symmetry of order $n$. The distinct angles of rotation (excluding $0^\circ$) that map the figure onto itself are:

(A) $360^\circ/n, 2 \times 360^\circ/n, ..., (n-1) \times 360^\circ/n$.

(B) Multiples of the smallest angle of rotational symmetry.

(C) All angles less than $360^\circ/n$.

(D) $360^\circ$ only.

Question 11. Which of the following letters have rotational symmetry?

(A) O

(B) I

(C) N

(D) Z

Question 12. A shape has rotational symmetry of order $n$. The angle of rotation is $\frac{360^\circ}{n}$. This relationship implies that:

(A) A higher order of rotation means a smaller angle of rotation.

(B) $n$ must be an integer greater than or equal to 1.

(C) If $n=1$, the angle is $360^\circ$, which means no rotational symmetry other than the full turn.

(D) The angle of rotation can be any real number.

Question 1. 2-dimensional shapes have length and breadth, while 3-dimensional shapes have:

(A) Length, breadth, and height.

(B) Volume.

(C) Depth.

(D) Area.

Question 2. Which of the following are examples of solid shapes (3D shapes)?

(A) Cube

(B) Sphere

(C) Cylinder

(D) Triangle

Question 3. The flat surfaces of a solid shape are called its faces. These faces can be:

(A) Polygonal.

(B) Curved.

(C) Circular.

(D) Points.

Question 4. The line segments where faces of a solid shape meet are called its edges. The edges can be:

(A) Straight.

(B) Curved.

(C) Points.

(D) Vertices.

Question 5. The points where edges of a solid shape meet are called its vertices. Vertices are typically:

(A) Corners of the solid.

(B) Points where at least three edges meet (for polyhedra).

(C) Always sharp.

(D) Curves.

Question 6. A cuboid is a solid shape with:

(A) 6 rectangular faces.

(B) 12 edges.

(C) 8 vertices.

(D) All faces are squares.

Question 7. A cylinder is a solid shape with:

(A) Two circular bases.

(B) One curved surface.

(C) A total of 3 faces.

(D) Vertices and edges.

Question 8. A cone is a solid shape with:

(A) A circular base.

(B) A curved surface tapering to an apex.

(C) One vertex.

(D) Two faces (base and curved surface).

Question 9. A sphere is a solid shape with:

(A) A curved surface.

(B) No edges.

(C) No vertices.

(D) A definite volume.

Question 10. Which of the following are polyhedra?

(A) Cube

(B) Pyramid

(C) Prism

(D) Sphere

Question 11. A square pyramid has:

(A) A square base.

(B) Four triangular faces.

(C) An apex (vertex opposite the base).

(D) 4 vertices in the base and 1 apex vertex (total 5).

Question 12. The faces of a solid shape can be congruent. For example, in a cube:

(A) All faces are congruent squares.

(B) Opposite faces are congruent rectangles (in a cuboid).

(C) Adjacent faces are congruent.

(D) All faces are triangles (in a triangular prism).

Question 1. Oblique sketches and isometric sketches are methods used for:

(A) Drawing 2D shapes.

(B) Representing 3D solid shapes on a flat surface.

(C) Creating perspective drawings.

(D) Visualizing solids.

Question 2. In an oblique sketch of a cube:

(A) The front face is drawn accurately (true shape and size).

(B) Parallel lines are shown as parallel.

(C) The depth is often shown at a receding angle (e.g., $45^\circ$).

(D) All edge lengths are drawn to scale (true length).

Question 3. In an isometric sketch of a cube:

(A) All edge lengths are drawn to scale (true length).

(B) The angles between the axes are typically $120^\circ$.

(C) Parallel lines are shown as parallel.

(D) The front face is shown in true shape.

Question 4. A cross-section of a solid is the 2D shape obtained by slicing the solid with a plane. Which of the following shapes can be obtained as a cross-section of a cube?

(A) Square

(B) Triangle

(C) Rectangle

(D) Pentagon

Question 5. Which of the following shapes can be obtained as a cross-section of a cylinder?

(A) Circle (slicing parallel to base)

(B) Rectangle (slicing perpendicular to base through axis)

(C) Ellipse (slicing diagonally)

(D) Triangle

Question 6. The different "views" of a 3D shape (Front View, Side View, Top View) are examples of orthographic projections. These views show the shape from different perspectives and are useful for:

(A) Understanding the shape's dimensions.

(B) Representing the shape in 2D.

(C) Calculating the volume.

(D) Engineering and architectural drawings.

Question 7. When you look at a cuboid from the front, side, or top, the view is typically a:

(A) Triangle

(B) Rectangle

(C) Square (if it's a cube or a square face)

(D) Point

Question 8. A net of a solid is a 2D shape that can be folded to form the 3D solid. Different nets can form the same solid. True or False?

(A) True

(B) False

(C) Yes, for a cube, there are multiple distinct nets.

(D) No, each solid has only one net.

Question 9. Which of the following statements are true about visualising solid shapes?

(A) Cross-sections help understand the internal structure.

(B) Shadows are always accurate representations of the shape.

(C) Building 3D models helps in visualization.

(D) Isometric sketches provide a more realistic view than oblique sketches.

Question 10. If you slice a cone vertically through its apex, the cross-section will be a:

(A) Circle

(B) Triangle

(C) Isosceles triangle (for a right cone)

(D) Right triangle (for a right cone, if the base is a point on the plane)

Question 11. The top view of a square pyramid is usually a square with diagonals from the centre to the vertices. This is because:

(A) The base is a square.

(B) The apex is directly above the centre of the base (for a right pyramid).

(C) The triangular faces are projected as triangles in the top view.

(D) The edges from the apex to the base vertices are projected as lines from the centre to the base vertices.

Question 12. When drawing views of a solid using orthographic projection, the lines of projection are parallel to each other and perpendicular to the projection plane. True or False?

(A) True

(B) False

(C) Yes, this is how front, side, and top views are generated.

(D) No, perspective projection uses parallel lines of projection.

Question 1. A polyhedron is a solid whose surface is made up of a finite number of polygonal regions called faces. The intersection of two faces is an edge, and the intersection of edges is a vertex. True or False?

(A) True

(B) False

(C) Yes, this is the definition of a polyhedron.

(D) No, some solids with curved surfaces are also polyhedra.

Question 2. Which of the following solids are polyhedra?

(A) Cube

(B) Sphere

(C) Triangular prism

(D) Cone

Question 3. A convex polyhedron has the property that for every face, the rest of the polyhedron lies entirely on one side of the plane containing that face. Which of the following are convex polyhedra?

(A) Cube

(B) Pyramid

(C) A star polygon extruded into 3D (like a star prism)

(D) Prism

Question 4. A regular polyhedron (Platonic solid) has faces which are congruent regular polygons, and the same number of faces meet at each vertex. Examples of Platonic solids include:

(A) Tetrahedron (4 faces, triangles)

(B) Cube (6 faces, squares)

(C) Octahedron (8 faces, triangles)

(D) Pyramid (base is a square, but side faces are triangles, might not be regular polygons)

Question 5. Euler's formula for convex polyhedra states V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This formula:

(A) Holds for cubes.

(B) Holds for pyramids.

(C) Holds for prisms.

(D) Applies to spheres.

Question 6. Let's verify Euler's formula for a triangular prism. A triangular prism has V=6, E=9, F=5. Is V - E + F = 2 satisfied?

(A) $6 - 9 + 5 = 2$.

(B) Yes.

(C) No.

(D) The formula is $V+E-F=2$.

Question 7. Let's verify Euler's formula for a square pyramid. A square pyramid has V=5, E=8, F=5. Is V - E + F = 2 satisfied?

(A) $5 - 8 + 5 = 2$.

(B) Yes.

(C) No.

(D) It satisfies $V+F-E=2$.

Question 8. If a convex polyhedron has 12 faces and 30 edges, how many vertices does it have?

(A) Use V - E + F = 2.

(B) V - 30 + 12 = 2.

(C) V = 30 - 12 + 2.

(D) V = 20.

Question 9. The faces of an icosahedron are equilateral triangles. It has 20 faces and 12 vertices. How many edges does it have?

(A) Use V - E + F = 2.

(B) 12 - E + 20 = 2.

(C) E = 12 + 20 - 2.

(D) E = 30.

Question 10. Euler's formula can be used to:

(A) Find the number of vertices, edges, or faces of a convex polyhedron if the other two are known.

(B) Check if a given combination of V, E, and F is possible for a convex polyhedron.

(C) Calculate the surface area of a polyhedron.

(D) Determine if a solid is a polyhedron.

Question 11. The dual of a polyhedron has the vertices of the original polyhedron as faces, and vice versa. The dual of a cube is an octahedron. The number of vertices of a cube equals the number of faces of an octahedron, and vice versa. True or False?

(A) True

(B) False

(C) Yes, Cube (8V, 12E, 6F) and Octahedron (6V, 12E, 8F).

(D) No, this is incorrect.

Question 12. A solid has V=10, E=18, F=10. Does this combination satisfy Euler's formula for convex polyhedra? V - E + F = 10 - 18 + 10 = 2.

(A) Yes

(B) No

(C) This could be a convex polyhedron.

(D) This could be a triangular prism joined end-to-end with another triangular prism.