Triangles: Introduction and Types

Triangles: Introduction and Types - Triangle: Definition and Basic Elements

Among the vast family of polygons, the triangle holds a special and fundamental place. As the polygon with the minimum possible number of sides (three), it is the simplest closed straight-sided figure in Euclidean geometry. Triangles are the building blocks of many other polygons and complex shapes, and their properties are extensively studied.

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Definition

A triangle is defined as a polygon with exactly three sides and three vertices. It is the simplest type of polygon.

More precisely, if A, B, and C are three points that do not lie on the same straight line (i.e., they are non-collinear points), then the figure formed by the three line segments connecting these points pairwise – $\overline{\text{AB}}$, $\overline{\text{BC}}$, and $\overline{\text{CA}}$ – is called a triangle. The points A, B, and C are the vertices of the triangle.

A triangle with vertices A, B, and C is symbolically denoted by $\triangle \text{ABC}$. The order of the vertices in the notation can be any permutation (e.g., $\triangle \text{ABC}$, $\triangle \text{BCA}$, $\triangle \text{CAB}$, etc., all refer to the same triangle).

The condition that the vertices must be non-collinear is important. If A, B, and C were collinear, the segments would simply lie on a single line and would not form a closed figure.

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Basic Elements of a Triangle

Every triangle has exactly six fundamental elements:

1. Three Sides: These are the line segments that form the boundary of the triangle. They connect the vertices. In $\triangle \text{ABC}$, the sides are the line segment $\overline{\text{AB}}$, the line segment $\overline{\text{BC}}$, and the line segment $\overline{\text{CA}}$.
2. Three Vertices: These are the three non-collinear points where the sides of the triangle meet. In $\triangle \text{ABC}$, the vertices are the points A, B, and C.
3. Three Interior Angles: These are the angles formed inside the triangle at each vertex by the two adjacent sides. At vertex A, the interior angle is formed by sides $\overline{\text{AB}}$ and $\overline{\text{AC}}$; it can be denoted as $\angle \text{BAC}$ or simply $\angle \text{A}$. Similarly, at vertex B, the angle is $\angle \text{ABC}$ (or $\angle \text{B}$), formed by sides $\overline{\text{BA}}$ and $\overline{\text{BC}}$. At vertex C, the angle is $\angle \text{BCA}$ (or $\angle \text{C}$), formed by sides $\overline{\text{CB}}$ and $\overline{\text{CA}}$.

Here is a diagram illustrating these basic elements:

Sometimes, the term "elements" is used more broadly to include other specific points, lines, or segments associated with a triangle, such as medians, altitudes, angle bisectors, circumcenter, incenter, etc. However, the sides, vertices, and interior angles are the defining basic elements.

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Interior and Exterior of a Triangle

Just like other closed curves, a triangle divides the plane in which it lies into three distinct sets of points:

• The interior of the triangle: This is the region of the plane that is strictly inside the triangle's boundary.
• The boundary of the triangle: This consists of the triangle itself – the set of points that lie on the three sides (including the vertices).
• The exterior of the triangle: This is the region of the plane that is outside the triangle's boundary and is not part of the interior or the boundary.

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Fundamental Property: Angle Sum Property (Introduction)

One of the most fundamental and frequently used properties of triangles in Euclidean geometry is the relationship between the measures of their three interior angles. This property is a direct consequence of Euclid's Fifth Postulate (the Parallel Postulate) or one of its equivalents.

Angle Sum Property of a Triangle: The sum of the measures of the three interior angles of any triangle in Euclidean geometry is always equal to $180^\circ$ (or two right angles).

For any triangle $\triangle \text{ABC}$:

We will explore the proof and implications of this crucial property in a later section.

Triangles: Types of Triangles based on Sides (Scalene, Isosceles, Equilateral)

Triangles are diverse in shape, and one common way to classify them is by comparing the lengths of their three sides. This classification divides triangles into three distinct categories, each with its own set of properties.

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Classification by Side Lengths

1. Scalene Triangle

A scalene triangle is a triangle in which all three sides have different lengths. No two sides are equal in length.

If $\triangle \text{ABC}$ is a scalene triangle with side lengths represented by $a, b, c$ opposite to vertices A, B, C respectively, then $a \neq b$, $b \neq c$, and $a \neq c$. That is, $\text{BC} \neq \text{AC}$, $\text{AC} \neq \text{AB}$, and $\text{BC} \neq \text{AB}$.

A direct consequence of having unequal sides is that in a scalene triangle, all three interior angles also have different measures. The smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.

In the diagram, the sides are marked with 1, 2, and 3 dashes, indicating that their lengths are all different.

2. Isosceles Triangle

An isosceles triangle is a triangle in which at least two sides are equal in length. Note the phrasing "at least two", which is important for the relationship with equilateral triangles.

If in $\triangle \text{ABC}$, the length of side $\overline{\text{AB}}$ is equal to the length of side $\overline{\text{AC}}$ ($\text{AB} = \text{AC}$), then $\triangle \text{ABC}$ is an isosceles triangle. The two equal sides ($\overline{\text{AB}}$ and $\overline{\text{AC}}$ in this case) are often called the legs, and the third unequal side ($\overline{\text{BC}}$) is called the base of the isosceles triangle.

A key property of an isosceles triangle, known as the Isosceles Triangle Theorem (or Base Angles Theorem), states that the angles opposite the equal sides are also equal. These equal angles are called the base angles (the angles at the base).

If $\text{AB} = \text{AC}$ in $\triangle \text{ABC}$, then the angle opposite side $\overline{\text{AC}}$ (which is $\angle \text{B}$) is equal to the angle opposite side $\overline{\text{AB}}$ (which is $\angle \text{C}$). So, $\text{m}\angle \text{B} = \text{m}\angle \text{C}$.

In the diagram, sides $\overline{\text{AB}}$ and $\overline{\text{AC}}$ are marked with a single dash, indicating $\text{AB} = \text{AC}$. Angles $\angle \text{B}$ and $\angle \text{C}$ are marked with matching arcs, indicating $\text{m}\angle \text{B} = \text{m}\angle \text{C}$.

3. Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal in length.

If $\text{AB} = \text{BC} = \text{CA}$ in $\triangle \text{ABC}$, then it is an equilateral triangle.

A significant property of an equilateral triangle is that it is also equiangular, meaning all three interior angles are equal in measure. Since the sum of the interior angles of any triangle is $180^\circ$ (by the Angle Sum Property), the measure of each angle in an equilateral triangle is $180^\circ / 3 = 60^\circ$.

In the diagram, all sides are marked with a single dash, and all angles are marked with matching arcs and the measure $60^\circ$.

An equilateral triangle is a special type of isosceles triangle because if all three sides are equal, then any two sides are also equal. However, the converse is not true; an isosceles triangle is not necessarily equilateral (unless its third angle is also $60^\circ$).

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Summary of Triangle Classification by Sides

Type of Triangle	Side Lengths	Angle Measures
Scalene	All 3 sides are different lengths	All 3 angles are different measures
Isosceles	At least 2 sides are equal lengths	At least 2 angles (opposite the equal sides) are equal measures
Equilateral	All 3 sides are equal lengths	All 3 angles are equal measures ($60^\circ$ each)

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Triangles: Types of Triangles based on Angles (Acute-angled, Obtuse-angled, Right-angled)

Besides classifying triangles based on the lengths of their sides, we can also categorise them according to the measures of their interior angles. This classification is equally important and leads to three distinct types of triangles.

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Classification by Angle Measures

The sum of the interior angles of any Euclidean triangle is always $180^\circ$. Based on how this $180^\circ$ is distributed among the three angles, we classify triangles as follows:

1. Acute-angled Triangle (or Acute Triangle)

An acute-angled triangle is a triangle in which all three interior angles are acute angles. An acute angle is an angle that measures greater than $0^\circ$ and less than $90^\circ$.

If $\triangle \text{ABC}$ is acute-angled, then $\text{m}\angle \text{A} < 90^\circ$, $\text{m}\angle \text{B} < 90^\circ$, and $\text{m}\angle \text{C} < 90^\circ$.

Example: A triangle with angle measures $60^\circ, 70^\circ, 50^\circ$ is acute-angled because $60^\circ < 90^\circ$, $70^\circ < 90^\circ$, and $50^\circ < 90^\circ$, and their sum is $60^\circ + 70^\circ + 50^\circ = 180^\circ$. An equilateral triangle, having all angles equal to $60^\circ$, is always an acute-angled triangle.

2. Obtuse-angled Triangle (or Obtuse Triangle)

An obtuse-angled triangle is a triangle in which one of the interior angles is an obtuse angle. An obtuse angle is an angle that measures greater than $90^\circ$ and less than $180^\circ$.

If $\triangle \text{ABC}$ is obtuse-angled, then one of its angles (say $\angle \text{A}$) has $\text{m}\angle \text{A} > 90^\circ$. The other two angles ($\angle \text{B}$ and $\angle \text{C}$) must necessarily be acute, because if either were $90^\circ$ or greater, the sum of the angles would exceed $180^\circ$. Thus, a triangle can have at most one obtuse angle.

Example: A triangle with angle measures $30^\circ, 40^\circ, 110^\circ$. It is obtuse-angled because $110^\circ > 90^\circ$, and $30^\circ + 40^\circ + 110^\circ = 180^\circ$.

3. Right-angled Triangle (or Right Triangle)

A right-angled triangle is a triangle in which one of the interior angles is a right angle. A right angle measures exactly $90^\circ$.

If $\triangle \text{ABC}$ is right-angled, then one of its angles (say $\angle \text{A}$) has $\text{m}\angle \text{A} = 90^\circ$. The other two angles ($\angle \text{B}$ and $\angle \text{C}$) must be acute and complementary (their sum is $90^\circ$), because $\text{m}\angle \text{B} + \text{m}\angle \text{C} = 180^\circ - 90^\circ = 90^\circ$. A triangle can have at most one right angle.

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side of the triangle. The other two sides that form the right angle are called the legs or arms of the right triangle.

Example: A triangle with angle measures $90^\circ, 60^\circ, 30^\circ$. It is right-angled because one angle is $90^\circ$. The $60^\circ$ and $30^\circ$ angles are acute and sum to $90^\circ$.

Right-angled triangles are of particular importance in geometry and trigonometry due to the Pythagorean Theorem, which relates the lengths of the sides ($leg_1^2 + leg_2^2 = hypotenuse^2$).

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Summary of Triangle Classification by Angles

Type of Triangle	Angle Properties
Acute-angled	All 3 angles are acute ($< 90^\circ$)
Obtuse-angled	Exactly 1 angle is obtuse ($> 90^\circ$)
Right-angled	Exactly 1 angle is a right angle ($= 90^\circ$)

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Combining Classifications (Sides and Angles)

We can combine the classifications based on sides and angles to describe triangles more fully. For example:

• Right Isosceles Triangle: Has one right angle ($90^\circ$) and the two sides forming the right angle (the legs) are equal in length. The angles opposite the equal sides (the base angles) are equal, and since the other angle is $90^\circ$, each base angle must be $(180^\circ - 90^\circ)/2 = 45^\circ$.
• Acute Isosceles Triangle: Has two equal sides and all three angles are acute. (e.g., angles $70^\circ, 70^\circ, 40^\circ$)
• Obtuse Isosceles Triangle: Has two equal sides and one obtuse angle. The obtuse angle must be between the two equal sides. (e.g., angles $20^\circ, 20^\circ, 140^\circ$)
• Right Scalene Triangle: Has one right angle ($90^\circ$) and all three sides are of different lengths. Consequently, all three angles are also of different measures (one is $90^\circ$, the other two are acute and unequal, summing to $90^\circ$). (e.g., angles $90^\circ, 30^\circ, 60^\circ$)
• Acute Scalene Triangle: All three angles are acute, and all three sides are of different lengths.
• Obtuse Scalene Triangle: Has one obtuse angle, and all three sides are of different lengths.
• Equilateral Triangle: Always an acute-angled triangle because all its angles are $60^\circ$.

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