# Triangles

A closed figure made with 3 line segments forms the shape of a triangle.

In this mini-lesson, we will explore everything about triangles, which are commonly seen around us.

You can observe the shape of a triangle in signboards, your favorite sandwiches, or birthday banners.

So let's get started!

## What is a Triangle?

A triangle is a closed figure or shape with 3 sides, 3 angles and 3 vertices.

A triangle with vertices P, Q, and R is denoted as △PQR.

The elements of a triangle are its 3 angles and its 3 sides.

Let's look at triangle PQR.

The elements of this triangle are:

• $$\angle PQR$$
• $$\angle QRP$$
• $$\angle RPQ$$
• Side PQ
• Side QR
• Side RP

You can explore the relationship between the elements of a triangle by using the triangle calculator and dragging the vertices of the triangle shown below.

## Different Types of Triangles

Triangles can be classified based on their angles or the length of their sides.

### Classification of Triangles by Angles

Type of Triangle  Example Explanation
Right-angled triangle

A right triangle has one of its vertex angles as 90°.

In the above triangle, $$\angle \text{ACB}$$ = 90°.

Acute-angled triangle

An acute triangle has all its internal angles as acute (less than 90°).

Obtuse-angled triangle

An obtuse triangle has one of its internal angles as obtuse (greater than 90°).

In the given triangle, $$\angle \text{DFE}$$= 120$$^\circ$$

### Classification of Triangles by Sides

Type of Triangle  Example Explanation
Equilateral triangle

If all three sides of a triangle are equal, it is called an equilateral triangle.

In the given triangle, all three sides are equal in length.

AB = BC = AC

Isosceles triangle

If any two sides of a triangle are equal, it is called an isosceles triangle.

In the given triangle, two sides are equal.

RS = RT

Scalene triangle

If a triangle has three unequal sides, it is called a scalene triangle.

In the given triangle, all three sides are of unequal lengths.

XZ ≠ XZ ≠ YZ

## Properties of a Triangle

Some of the important properties of a triangle are listed below.

### Property 1 - Angle Sum Property

The angle sum property states that the sum of the three interior angles of a triangle is 180°

In the given triangle,

$$\angle \text{D} + \angle \text{E} + \angle \text{F} = 180^\circ$$

### Property 2 - Triangle Inequality Property

The triangle inequality property states that the sum of the length of the two sides of a triangle is greater than the third side.

in $$\Delta PQR$$

• p + q > r  ( 6 + 4 > 3)
• r + p > q (3 + 4  > 6)
• r + q > p ( 6 + 3 > 4)

### Property 3 - Pythagorean Theorem

The pythagorean theorem states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.

 Hypotenuse² = Base² + Altitude²

### Property 4 - Side opposite the greater angle is the longest side

To understand the side opposite the greater angle is the longest side property, let's consider the triangle given below.

In this triangle, $$\angle \text{A}$$ is the greatest angle.

Thus, side BC is the longest side.

### Property 5 - Exterior Angle Property

The exterior angle property of a triangle is always equal to the sum of the interior opposite angles.

In the given triangle, Exterior angle E1  = $$\angle \text{ABC} + \angle \text{BCA}$$

A triangle has 3 exterior angles and these exterior angles add up to 360° for any polygon.

### Property 6 - Congruence Property

In the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are equal.

• $$\angle\text{ ABC} = \angle\text{PQR}$$
• $$\angle\text{ BAC} = \angle\text{QPR}$$
• $$\angle\text{ BCA} = \angle\text{QRP}$$
• AB = PQ
• AC = PR
• BC = QR

Challenging Questions

 1. Emma said a triangle has at least one set of parallel lines. Is she right?  Why? 2. Can a triangle be formed with dimensions 17 inches, 9 inches, and 30 inches?

## Formulas of a Triangle

The formulas of a triangle help us determine its area and perimeter.

### Perimeter of a Triangle

Consider △XYZ.

The lengths of the sides of the triangle are represented as ‘a’, ‘b’ and ‘c’ respectively.

 Perimeter = a + b + c

### Area of a Triangle

In the given triangle $$ABC$$,

\begin{align}\text{Area of }\Delta ABC = \frac{1}{2} h\times \text{BC}\end{align}

$$\text{BC}$$ is the base and $$h$$ is the height of the triangle.

 \begin{align}\frac{1}{2} \times \text{Base} \times \text{Height}\end{align}

Tips and Tricks
1. A triangle cannot have more than one right angle.
2. A triangle cannot have all angles less than 60°
3. The angles opposite the equal sides of an isosceles triangle are equal.

## Solved Examples

 Example 1

The perimeter of a triangular garden is given as 26 feet.

If two of its sides measure 7 feet and 11 feet respectively, what is the measure of the third side?

Solution

We know that the perimeter of a triangle is the sum of all three sides

$$\implies$$ 26 = 7 + 11 + unknown side

Therefore, the unknown side is given by:

\begin{align} &=26 \!-\! (7 \!+\! 11) \\ &= 8 \;\text{feet} \end{align}

 $$\therefore$$ The third side measures 8 feet.
 Example 2

Emma is building a triangular, wooden birdhouse as shown.

If two of the angles measure 46° and 62°, what is the measure of the third angle?

Solution

We know that the sum of the angles of a triangle adds up to 180°

Therefore, the unknown angle is:

\begin{align} &=180^\circ \!-\! (46^\circ \!+\! 62^\circ) \\ &= 72^\circ\end{align}

 $$\therefore$$ The third angle measures 72°
 Example 3

In a right triangle, two of the sides measure 3 inches and 4 inches.

What is the measure of the hypotenuse?

Solution

By Pythagoras theorem, we know that, hypotenuse2 = base2 + height2

Substituting the values, we get:

\begin{align} \text{hypoteneuse}^2 &=3^2 +4^2 \\ &= 9 + 16\\ &= 25\: \text{inches} \\ \implies \text{hypoteneuse} &= 5 \:\text{inches} \end{align}

 $$\therefore$$ Hypotenuse = 5 inches
 Example 4

One of the acute angles of a right-angled triangle is 45°

Find the other angle.

Identify the type of triangle thus formed.

Solution

Given, angle1 =  90° and angle2 = 45°

We know that the sum of the angles of a triangle adds up to 180°

Therefore,

\begin{align} \text{angle}_3 &=180^\circ \!-\! (90^\circ \!+\! 45^\circ) \\ &= 45^\circ\end{align}

Since two angles measure the same, it is an isosceles triangle.

 $$\therefore$$ △ABC is an isosceles triangle
 Example 5

The sides of a right triangular park are in the ratio 3: 4: 5 and its perimeter is 1080 feet.

The right angle is formed between the sides $$3x$$ and $$4x$$ in the figure.

Find the area of the right triangular park.

Solution

Given:

• Perimeter = 1080 feet
• Sides of the triangle are in the ratio 3:4:5

Therefore, the sides of triangle are

\begin{align} \text{side}_1 &= \frac{3}{12}\times 1080\\\ &= 270 \text{ feet}\\\\ \text{side}_2 &= \frac{4}{12}\times 1080\\\ &= 360\text{ feet} \\\\ \text{side}_3 &= \frac{5}{12}\times 1080\\\ &= 450\text{ feet} \end{align}

Since the right angle is formed between the sides $$3x$$ and $$4x$$, the height of the triangle = 270 feet and base = 360 feet

Area of a triangle is $$\dfrac{1}{2}\times \text{Base}\times \text{Height}$$

\begin{align} \text{Area of the park} &= \frac{1}{2} \times 270 \times 360 \\ &= 48600\: \text{feet}^2\end{align}

 $$\therefore$$ Area of the park = 48,600 feet2

Important Notes
1. A triangle is a 3-sided closed shape.
2. The perimeter of a triangle is the sum of all three sides.
3. The area of a triangle is half the product of its height and base.
4. Herons formula and Pythagoras theorem are important formulas related to triangles.
5. The sum of the angles of a triangle adds up to 180°.

## Interactive Questions

Here are few questions for you to practice.

## Let's Summarize

We hope you enjoyed learning about the Triangle with the simulations and practice questions. Now, you will be able to easily solve problems related to the different kinds of triangles.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.

## 1. Is a straight line a triangle?

No, a straight line cannot be a triangle, because it does not enclose an area, which a triangle does.

## 2. Which triangle cannot be drawn?

According to the triangle inequality theorem, the sum of two sides of a triangle must be greater than the third side. Therefore, a triangle which contradicts the triangle inequality theorem cannot be drawn.

## 3. How many degrees is a triangle?

The sum of all the interior angles of a triangle is $$180^{\circ}$$.

Therefore, a triangle adds up to $$180^{\circ}$$.

grade 9 | Questions Set 1