Chapter 3 Understanding Quadrilaterals (Concepts)

Introduction to Curves and Polygons

Geometry deals with shapes, sizes, and properties of space. In previous classes, you have encountered various plane figures like triangles, squares, rectangles, and circles. Before we dive deeper into the study of quadrilaterals, let's refresh our understanding of some fundamental concepts, starting with different types of curves and introducing polygons.

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Understanding Curves

In mathematics, a curve is a continuous and connected set of points. When you draw a shape on a piece of paper without lifting your pencil, you are tracing a curve. Curves can be straight, bent, or a combination of both.

Simple Curve:

A curve is called a simple curve if it does not cross itself at any point between its endpoints. Think of drawing a shape without having your pencil line intersect itself.

Examples include a circle, an oval, a simple wavy line, or a triangle drawn in one stroke.

Non-Simple Curve:

A curve that crosses itself at one or more points is called a non-simple curve.

Examples include the shape of the alphabet 'X' or a figure-eight shape drawn without lifting the pencil.

Open Curve:

A curve is called an open curve if its starting and ending points are not the same. The curve does not close itself.

Examples include a straight line segment, a ray, a simple wavy line, or the shape of the alphabet 'C'.

Closed Curve:

A curve is called a closed curve if its starting and ending points are the same. The curve forms a closed loop.

Examples include a circle, a square, a triangle, or an oval. Closed curves divide the plane into three parts: the interior (inside the curve), the exterior (outside the curve), and the boundary (the curve itself).

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Polygons

Now we come to a very important type of plane figure built from straight lines.

A polygon is a simple closed curve that is made up entirely of line segments.

Let's break down this definition:

• Simple: The curve does not cross itself.
• Closed: The curve forms a complete loop, with no open ends.
• Made up of line segments: The boundary of the figure consists only of straight line segments, not curved parts.

Examples of polygons: Triangles, Squares, Rectangles, Pentagons, Hexagons, etc.

Examples of figures that are NOT polygons:

Let's understand why the non-examples are not polygons:

• Figure (a) is an open curve (not closed).
• Figure (b) is a simple closed curve, but it is not made up entirely of line segments (it has curved parts).
• Figure (c) is made up of line segments, but it is not a simple curve (it crosses itself).
• Figure (d) is a simple curve made of line segments, but it is not closed.

Sides, Vertices, and Diagonals

Every polygon is defined by its fundamental components:

• Sides: The line segments that make up the polygon's boundary are called its sides. For an n-sided polygon, there are n sides.
• Vertices: The points where two adjacent sides meet are called vertices (singular: vertex). An n-sided polygon also has n vertices.
• Diagonals: A diagonal is a line segment that connects two non-adjacent (non-consecutive) vertices. For example, in a quadrilateral, you can draw a line from one vertex to the opposite vertex; this line is a diagonal.

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Classification of Polygons

Polygons are classified based on the number of sides they have. Since the number of sides is equal to the number of vertices, they can also be classified by the number of vertices. A polygon with $n$ sides is often called an $n$-gon.

Here is a table showing the classification of polygons based on the number of sides:

Number of Sides ($n$)	Classification
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon (or Septagon)
8	Octagon
9	Nonagon (or Enneagon)
10	Decagon
12	Dodecagon
...	...
$n$	$n$-gon

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Convex and Concave Polygons

Polygons can be categorised into two main types based on their shape or the measure of their interior angles: convex and concave polygons.

Convex Polygon:

A polygon is called a convex polygon if for every side of the polygon, the entire polygon lies on the same side of the line containing that side. Another way to identify a convex polygon is that all its interior angles are less than $180^\circ$. A key property is that all the diagonals of a convex polygon lie entirely inside the polygon.

Examples: Triangle, Square, Rectangle, Regular pentagon, Regular hexagon.

Concave Polygon:

A polygon is called a concave polygon if at least one of its interior angles is greater than $180^\circ$ (this is called a reflex angle). In a concave polygon, it is possible to draw a line containing a side such that part of the polygon lies on both sides of the line. Also, at least one diagonal of a concave polygon lies partly or entirely outside the polygon. Concave polygons appear to "dent inwards".

Examples: A dart shape (a concave quadrilateral), a star shape made from line segments.

All the polygons we will primarily study in this chapter, especially quadrilaterals like parallelograms, rectangles, squares, rhombuses, and trapeziums, are convex polygons.

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Regular and Irregular Polygons

Polygons can also be classified based on the equality of their sides and angles.

Regular Polygon:

A polygon is called a regular polygon if it is both equilateral (all its sides have equal length) and equiangular (all its interior angles have equal measure).

Examples: An equilateral triangle (all 3 sides and all 3 angles are equal), a square (all 4 sides and all 4 angles are equal), a regular pentagon (all 5 sides and all 5 angles are equal), a regular hexagon, a regular octagon, etc.

Irregular Polygon:

A polygon that is not regular is called an irregular polygon. In an irregular polygon, either the sides are not all equal, or the angles are not all equal, or both.

Examples: A scalene triangle (sides and angles are generally unequal), an isosceles triangle (sides are not all equal unless it's equilateral), a rectangle (all angles are equal ($90^\circ$), but sides are not necessarily equal unless it's a square), a rhombus (all sides are equal, but angles are not necessarily equal unless it's a square), a general trapezium.

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Angle Sum Property of a Polygon (Interior Angles)

The sum of the measures of the interior angles of a polygon is a very useful property that depends only on the number of sides of the polygon.

Consider a convex polygon with $n$ sides. We can divide this polygon into a number of triangles by drawing all possible diagonals from a single vertex. If the polygon has $n$ vertices, choosing one vertex leaves $(n-1)$ other vertices. Two of these are adjacent to the chosen vertex, so $(n-1) - 2 = (n-3)$ vertices are non-adjacent. Drawing diagonals from the chosen vertex to each of these $(n-3)$ non-adjacent vertices divides the polygon into $(n-2)$ triangles.

We know that the sum of the interior angles of a triangle is $180^\circ$. The sum of the interior angles of the polygon is equal to the sum of the interior angles of all the triangles formed by this triangulation.

Since there are $(n-2)$ triangles, and the sum of angles in each is $180^\circ$, the sum of the interior angles of a polygon with $n$ sides is:

Let's verify this formula for shapes we know:

• For a triangle ($n=3$ sides): Sum of interior angles $= (3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ$. This matches the well-known property of triangles.
• For a quadrilateral ($n=4$ sides): Sum of interior angles $= (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$. Let's see this visually:

A diagonal divides a quadrilateral into two triangles. The sum of angles of the quadrilateral is the sum of angles of the two triangles, which is $180^\circ + 180^\circ = 360^\circ$. The formula works.

• For a pentagon ($n=5$ sides): Sum of interior angles $= (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$.

If the polygon is regular, all its $n$ interior angles are equal. So, the measure of each interior angle of a regular polygon with $n$ sides is:

Sum of Exterior Angles of a Polygon:

An exterior angle of a polygon is formed by one side and extending the adjacent side. At each vertex, the interior angle and the corresponding exterior angle form a linear pair, so their sum is $180^\circ$.

A remarkable property is that the sum of the measures of the exterior angles (one at each vertex) of any convex polygon is always $360^\circ$. This is true regardless of the number of sides.

For a regular polygon with $n$ sides, all exterior angles are equal. So, the measure of each exterior angle of a regular polygon with $n$ sides is:

Note that for a regular polygon, the sum of an interior angle and its corresponding exterior angle is $180^\circ$. So, $\frac{(n-2)180^\circ}{n} + \frac{360^\circ}{n} = \frac{180n - 360 + 360}{n} = \frac{180n}{n} = 180^\circ$, which is consistent.

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Answer:

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Answer:

Quadrilateral and Kinds of Quadrilaterals

Building upon our understanding of polygons, we now turn our attention to a specific type of polygon that has four sides: the quadrilateral. Quadrilaterals are fundamental geometric shapes that appear frequently in architecture, design, and everyday objects.

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What is a Quadrilateral?

A quadrilateral is a polygon with exactly four sides. It is a simple closed curve made up of four line segments. Because it has four sides, it also has four vertices and four interior angles.

Consider a quadrilateral named ABCD:

Let's identify the different parts of this quadrilateral:

• Sides: The four line segments forming the boundary are AB, BC, CD, and DA.
• Vertices: The four points where the sides meet are A, B, C, and D.
• Angles: The four interior angles are $\angle$DAB (or $\angle$A), $\angle$ABC (or $\angle$B), $\angle$BCD (or $\angle$C), and $\angle$CDA (or $\angle$D).
• Diagonals: Line segments connecting opposite vertices are AC and BD.
• Adjacent Sides: Pairs of sides that share a common vertex. These are (AB, BC), (BC, CD), (CD, DA), and (DA, AB).
• Opposite Sides: Pairs of sides that do not share a common vertex. These are (AB, DC) and (BC, AD).
• Adjacent Angles: Pairs of angles that share a common side (i.e., are at adjacent vertices). These are ($\angle$A, $\angle$B), ($\angle$B, $\angle$C), ($\angle$C, $\angle$D), and ($\angle$D, $\angle$A).
• Opposite Angles: Pairs of angles that are at opposite vertices. These are ($\angle$A, $\angle$C) and ($\angle$B, $\angle$D).

Angle Sum Property of a Quadrilateral:

From our study of the angle sum property of polygons, we know that the sum of the interior angles of any polygon with $n$ sides is $(n-2) \times 180^\circ$. For a quadrilateral, $n=4$.

Sum of interior angles of a quadrilateral $= (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$.

This property holds for any quadrilateral, whether it is convex or concave.

Answer:

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Kinds of Quadrilaterals

Quadrilaterals are further classified into different types based on the relationships between their sides, angles, and diagonals. Let's study some common types.

1. Parallelogram:

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

In parallelogram ABCD, AB is parallel to DC (AB || DC), and AD is parallel to BC (AD || BC).

Properties of a Parallelogram:

Parallelograms have several distinct properties:

1. Opposite sides are equal in length: The lengths of opposite sides are equal. In parallelogram ABCD, AB = DC and AD = BC.
2. Opposite angles are equal in measure: The measures of opposite angles are equal. In parallelogram ABCD, $\angle$A = $\angle$C and $\angle$B = $\angle$D.
3. Adjacent angles are supplementary: The sum of the measures of any two adjacent angles is $180^\circ$. This is because consecutive angles are interior angles on the same side of a transversal intersecting parallel lines.

   $\angle A + \angle B = 180^\circ$

   (Considering parallel lines AD || BC and transversal AB)

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

   (Considering parallel lines AB || DC and transversal BC)

   $\angle C + \angle D = 180^\circ$

   (Considering parallel lines BC || AD and transversal CD)

   $\angle D + \angle A = 180^\circ$

   (Considering parallel lines DC || AB and transversal DA)

4. Diagonals bisect each other: The two diagonals of a parallelogram intersect each other at their midpoints. If diagonals AC and BD intersect at point O, then AO = OC and BO = OD.
   

A quadrilateral is a parallelogram if any one of the following conditions is met:

• Opposite sides are parallel.
• Opposite sides are equal.
• Opposite angles are equal.
• Diagonals bisect each other.
• A pair of opposite sides is parallel and equal.

Answer:

Some Special Parallelograms

In the previous section, we introduced the parallelogram as a quadrilateral with both pairs of opposite sides parallel. Parallelograms have specific properties related to their sides, angles, and diagonals. Now, let's look at some quadrilaterals that are actually special types of parallelograms because they have all the properties of a parallelogram plus some additional unique properties.

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Rhombus

A rhombus is a parallelogram in which all four sides are equal in length.

Since a rhombus is a type of parallelogram, it automatically possesses all the properties of a parallelogram. These include:

• Opposite sides are parallel.
• Opposite sides are equal (this is reinforced by the definition that *all* sides are equal).
• Opposite angles are equal.
• Adjacent angles are supplementary (sum is $180^\circ$).
• Diagonals bisect each other.

In rhombus ABCD, AB || DC, AD || BC, and AB = BC = CD = DA.

Additional Properties of a Rhombus:

Besides the parallelogram properties, a rhombus has these special characteristics:

1. All sides are equal:

   By definition, AB = BC = CD = DA.

2. Diagonals bisect each other at right angles:

   The diagonals of a rhombus not only bisect each other (like in any parallelogram), but they also intersect at a $90^\circ$ angle. If diagonals AC and BD intersect at point O, then AO = OC, BO = OD, and $\angle$AOB = $\angle$BOC = $\angle$COD = $\angle$DOA = $90^\circ$.

3. Diagonals bisect the angles of the rhombus:

   Each diagonal of a rhombus bisects the two interior angles it connects. For example, diagonal AC bisects $\angle$A and $\angle$C (meaning it divides each angle into two equal angles), and diagonal BD bisects $\angle$B and $\angle$D.

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Rectangle

A rectangle is a parallelogram in which all four interior angles are right angles ($90^\circ$).

Since a rectangle is a type of parallelogram, it has all the properties of a parallelogram:

• Opposite sides are parallel.
• Opposite sides are equal.
• Opposite angles are equal (this is reinforced by the definition that *all* angles are $90^\circ$).
• Adjacent angles are supplementary (since all are $90^\circ$, $90^\circ + 90^\circ = 180^\circ$).
• Diagonals bisect each other.

In rectangle ABCD, AB || DC, AD || BC, and $\angle$A = $\angle$B = $\angle$C = $\angle$D = $90^\circ$.

Additional Properties of a Rectangle:

In addition to the parallelogram properties, a rectangle has these special characteristics:

1. All angles are $90^\circ$:

   By definition, $\angle$A = $\angle$B = $\angle$C = $\angle$D = $90^\circ$.

2. Diagonals are equal in length:

   The lengths of the two diagonals of a rectangle are equal. AC = BD. Since the diagonals also bisect each other (from the parallelogram property), if they intersect at O, then AO = OC and BO = OD. Combining this with AC = BD, we get AO = OC = BO = OD. This means the point of intersection of the diagonals is equidistant from all four vertices.

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Square

A square is a parallelogram that is both a rhombus and a rectangle. This means it has the properties of both these special parallelograms.

Equivalently, a square can be defined as:

• A rectangle with all sides equal.
• A rhombus with all angles equal to $90^\circ$.

In square ABCD, AB || DC, AD || BC, AB = BC = CD = DA, and $\angle$A = $\angle$B = $\angle$C = $\angle$D = $90^\circ$.

Properties of a Square:

A square inherits all properties from parallelograms, rhombuses, and rectangles. Its key properties are:

1. All sides are equal:

   AB = BC = CD = DA (from rhombus property).

2. All angles are $90^\circ$:

   $\angle$A = $\angle$B = $\angle$C = $\angle$D = $90^\circ$ (from rectangle property).

3. Opposite sides are parallel:

   AB || DC and AD || BC (from parallelogram property).

4. Diagonals bisect each other:

   They intersect at their midpoint (from parallelogram property).

5. Diagonals are equal:

   AC = BD (from rectangle property). Thus, the point of intersection is equidistant from all vertices.

6. Diagonals bisect each other at right angles:

   Diagonals AC and BD intersect at O such that $\angle$AOB = $90^\circ$ (from rhombus property).

7. Diagonals bisect the angles:

   Diagonals bisect the vertices' angles (from rhombus property). Since the angles are $90^\circ$, the diagonals divide each vertex angle into two $45^\circ$ angles.

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Summary of Properties of Special Parallelograms

This table summarises the properties of parallelograms, rhombuses, rectangles, and squares, highlighting how the special parallelograms inherit and add properties.

Property	Parallelogram	Rhombus	Rectangle	Square
Opposite sides parallel	Yes	Yes	Yes	Yes
Opposite sides equal	Yes	Yes	Yes	Yes
All sides equal	No (Only opposite sides are equal)	Yes	No (Only opposite sides are equal)	Yes
Opposite angles equal	Yes	Yes	Yes	Yes
All angles $90^\circ$	No (Only opposite angles are equal)	No (Only opposite angles are equal)	Yes	Yes
Diagonals bisect each other	Yes	Yes	Yes	Yes
Diagonals are equal	No (Only bisect each other)	No (Only bisect each other at $90^\circ$)	Yes	Yes
Diagonals bisect at right angles ($90^\circ$)	No (Only bisect each other)	Yes	No (Only bisect each other)	Yes
Diagonals bisect vertex angles	No (Only bisect each other)	Yes	No (Only equal and bisect each other)	Yes

Understanding these properties helps in identifying these figures and solving problems related to them.

Trapezium and Kite

So far, we have focused on parallelograms and their special types (rhombus, rectangle, square), all of which have both pairs of opposite sides parallel. Now, we will explore two other types of quadrilaterals that do not necessarily have this property across both pairs of sides, but have their own unique characteristics: the Trapezium and the Kite.

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Trapezium (or Trapezoid)

A trapezium is a quadrilateral that has at least one pair of opposite sides parallel.

In the trapezium ABCD shown above, the side AB is parallel to the side DC (AB || DC). These parallel sides are called the bases of the trapezium. The other two sides, AD and BC, are called the non-parallel sides or legs.

It's important to note that some definitions state "exactly one pair of parallel sides". However, the broader definition "at least one pair" is also common and includes parallelograms as special cases of trapeziums. For Class 8, typically, a trapezium is understood to mean a quadrilateral with just one pair of parallel sides.

Properties of a Trapezium:

Based on the definition, a trapezium has the following basic properties:

1. One pair of opposite sides is parallel.
2. Adjacent angles between the parallel sides are supplementary: If the parallel sides are AB and DC, then the angles formed by a non-parallel side acting as a transversal are supplementary. That is, $\angle$A and $\angle$D are supplementary (since AD is a transversal intersecting AB || DC), and $\angle$B and $\angle$C are supplementary (since BC is a transversal intersecting AB || DC).

   If AB || DC, then $\angle$A + $\angle$D = $180^\circ$

   If AB || DC, then $\angle$B + $\angle$C = $180^\circ$

Isosceles Trapezium:

A special type of trapezium is an isosceles trapezium. An isosceles trapezium is a trapezium in which the non-parallel sides are equal in length.

In isosceles trapezium ABCD, where AB || DC, the non-parallel sides AD and BC are equal (AD = BC).

Additional properties of an isosceles trapezium:

1. The non-parallel sides are equal (AD = BC).
2. The angles on each base are equal. That is, the angles formed by the parallel side and each non-parallel side are equal.

   $\angle$DAB = $\angle$CBA

   (Angles on base AB, sometimes called base angles)

   $\angle$ADC = $\angle$BCD

   (Angles on base DC)

3. The diagonals are equal in length (AC = BD).

Answer:

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Kite

A kite is a quadrilateral that has two distinct pairs of consecutive sides that are equal in length.

Note the term "distinct pairs". This means the lengths of the four sides are of the form $a, a, b, b$ where $a \neq b$, and the equal sides are adjacent to each other, forming pairs.

In the kite ABCD shown, the consecutive sides AB and CB are equal (AB = CB), and the consecutive sides AD and CD are equal (AD = CD). These pairs share a common vertex (B and D respectively).

Properties of a Kite:

A kite has the following properties:

1. Two distinct pairs of consecutive sides are equal:

   By definition, AB = CB and AD = CD. The vertices B and D are the endpoints of the diagonal that is the axis of symmetry.

2. The diagonals are perpendicular to each other:

   The diagonal connecting the vertices where the equal sides meet (BD in the figure) is perpendicular to the other diagonal (AC). If diagonals AC and BD intersect at O, then AC $\perp$ BD, meaning $\angle$AOB = $\angle$BOC = $\angle$COD = $\angle$DOA = $90^\circ$.

3. One of the diagonals bisects the other diagonal:

   The diagonal connecting the vertices where the equal sides meet (BD) bisects the other diagonal (AC). If diagonals AC and BD intersect at O, then AO = OC. The diagonal BD is generally not bisected by AC (unless the kite is a rhombus).

4. One pair of opposite angles is equal:

   The angles between the unequal sides are equal. In kite ABCD, $\angle$ABC = $\angle$ADC. The other pair of opposite angles ($\angle$DAB and $\angle$BCD) are generally not equal (unless the kite is a rhombus).

5. One diagonal is an axis of symmetry:

   The diagonal connecting the vertices where the equal sides meet (BD) is an axis of symmetry for the kite. Folding the kite along this diagonal would make the two halves coincide perfectly.

Note: A rhombus is a special case of a kite where all four sides are equal. If all four sides are equal, then any two consecutive sides are equal, satisfying the definition of a kite (even if the pairs are not "distinct" in side length). In a rhombus, both pairs of opposite angles are equal, both diagonals bisect angles, and both diagonals bisect each other at right angles. So, a rhombus is a kite with extra symmetry.

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