# Class 7 Properties of Triangle

Sum of Lengths of Two Sides of a Triangle

Properties of Triangle Worksheet

## Types of Triangles

Triangles are named on the basis of the lengths of their sides and the measures of their angles. Different types of triangles are given below.

- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle
- Acute Triangle
- Right Triangle
- Obtuse Triangle

## Scalene Triangle

A triangle whose all three sides length are not equal is known as scalene triangle. Figure below shows an example of scalene triangle.

PQ ≠ PR ≠ QR

## Isosceles Triangle

A triangle whose two sides are equal is known as an isosceles triangle. Figure below shows an example of isosceles triangle.

Here, AB = AC

## Equilateral Triangle

A triangle whose all sides are equal to one another is known as equilateral triangle. Figure below shows an example of equilateral triangle.

Here, AB = BC = AC and ∠ABC = ∠BAC = ∠ACB = 60°

## Acute Triangle

A triangle whose all the angles are acute is known as acute angled triangle or acute triangle.

Equilateral triangle is also an acute angled triangle.

## Right-Angled Triangle

A triangle whose one angle is a right angle is known as a right-angled triangle.

Triangle ABC is a right-angled triangle, where ∠ABC = 90°

## Obtuse Triangle

A triangle whose one angle is more than 90° is known as obtuse-angled triangle.

Here, triangle PQR is an obtuse angled triangle, where ∠PQR is an obtuse angle.

## Perimeter of a Triangle

The sum of the lengths of the sides of a triangle is called its perimeter. Let's assume a, b and c are the three sides of a triangle, then its perimeter = a + b + c

## Properties of Triangle

- The sum of the angles of a triangle is 180°
- A triangle can not have more than one right angle
- A triangle can not have more than one obtuse angle
- In a right-angled triangle, the other two angles are acute, and their sum is 90°

Let's see some examples to understand these properties.

**Example 1.** If two angles of a triangle are 65° and 45°, then find the third angle.

**Solution.** As we know sum of all the angles of a triangle is 180°. Two angles are provided, they are 65° and 45°. Let's assume the third angle is 'x'.

x + 65° + 45° = 180°

⇒ x = 180° − 110°

⇒ x = 70°

Hence, the third angle is 70°.

**Example 2.** Find all the angles of right-angled isosceles triangle.

**Solution.** Here, ∠ABC = 90° and AB = BC

∠BAC = ∠BCA = x°

As we know ∠ABC + ∠BAC + ∠BCA = 180°

⇒ 90° + x° + x° = 180°

⇒ 90° + 2x° = 180°

⇒ 2x° = 180° − 90°

⇒ 2x° = 90°

⇒ x° = 45°

So, ∠ABC = 90°, ∠BAC = 45° and ∠BCA = 45°

**Example 3.** If the angles of a triangle are in the ratio 2 : 3 : 4, then find the three angles of the triangle.

**Solution.** Let's assume all the three angles are 2x°, 3x°, and 4x°.

As we know sum of all the angles are equal to 180°

2x° + 3x° + 4x° = 180°

⇒ 9x° = 180°

⇒ x° = ^{180°}⁄_{9}

⇒ x° = 20°

So, the angles of the triangles are 40°, 60° and 80°.

**Example 4.** The sum of two angles of an isosceles triangle is equal to it's third angle. Determine the measures of all the angles.

**Solution.** Let's assume PQR is the triangle and ∠P, ∠Q, and ∠R it's angles.

∠P + ∠Q = ∠R

As we know ∠P + ∠Q + ∠R = 180°

⇒ ∠R + ∠R = 180°

⇒ 2∠R = 180°

⇒ ∠R = 90°

As it is a right-angled isosceles triangle, it's ∠P = ∠Q

So, ∠P = ∠Q = 45° and ∠R = 90°

**Example 5.** One of the acute angles of a right-angled triangle is 50°. Find the other acute angle.

**Solution.** Let's assume the other acute angle is p°.

As we know, sum of all the angles of a triangle is 180°.

90° + 50° + p° = 180°

⇒ p° = 180° − 140°

⇒ p° = 40°

So, the third acute angle is 40°.

**Example 6.** In below given figure DE || BC, ∠A = 30°, and ∠C = 40°. Find the value of x, y and z.

**Solution.** In triangle ABC, we have ∠A = 30°, and ∠C = 40°

∠A + ∠B + ∠C = 180°

⇒ 30° + y° + 40° = 180°

⇒ y° + 70° = 180°

⇒ y° = 110°

Here, DE || BC and transversal AB intersect them at point B and D. So corresponding angles will be equal.

∠B = ∠ADE

⇒ y° = z°

⇒ z° = 110°

Similarly, transversal AC intersect them at point E and C. So corresponding angles will be equal.

∠AED = ∠C

⇒ x° = 40°

Hence x° = 40°, y° = 110° and z° = 110°

## Exterior Angles of a Triangle

In the above figure ABC is a triangle and its side BC is extended to D. Here, ∠ACD is known as exterior angle and ∠ACB is known as interior angle. Here, ∠ACD and ∠ACB are adjacent supplementary angles.
Interior Angle + Exterior Angle = 180°
∠BAC and ∠ABC is known as interior opposite angles of ∠ACD
∠BAC + ∠ABC = ∠ACD
Let's see some examples to understand it better.

**Example 1.** Find the value of 'x' in the below given figure.

**Solution.** As we know, exterior angle of a triangle is equal to its two interior opposite angles.

x = 55° + 52°

⇒ x = 107°

**Example 2.** Find the measures of ∠ABC and ∠BAC in the below given figure.

**Solution.** Exterior angle of a triangle is equal to its two interior opposite angles.

2x + 3x = 150°

⇒ 5x = 150°

⇒ x = ^{150°}⁄_{5}

⇒ x = 30°

Hence, ∠ABC = 60° and ∠BAC = 90°

**Example 3.** Find the value of 'p' and 'q' in the below given figure.

**Solution.** From the above figure we can see two triangles ABC and ADC.

Let's consider triangle ADC first.

q° + 48° + 62° = 180°

⇒ q° = 180° − 110°

⇒ q° = 70°

Now consider, triangle ABC.

∠ABC + ∠BAC + ∠ACB = 180°

⇒ 50° + p° + 48° + 62° = 180°

⇒ p° + 160° = 180°

⇒ p° = 180° − 160°

⇒ p° = 20°

Hence, p° = 20° and q° = 70°.

**Example 4.** Find the value of 'x', 'y', and 'z' in the below given figure.

**Solution.** Let's find out angle 'x' and 'z' using linear pair rule.

x + 120° = 180°

⇒ x = 60°

z + 122° = 180°

⇒ z = 58°

x + y + z = 180°

⇒ 60° + y + 58° = 180°

⇒ y = 180° − 60° − 58°

⇒ y = 62°

Hence, x = 60°, y = 62° and z = 58°

## Sum of Lengths of Two Sides of a Triangle

Sum of length of any two sides of a triangle is greater than the length of its third side. This rule is applicable for any kind of triangle.

Here, PQ + QR > PR

PQ + PR > QR

PR + QR > PQ

Let's see some examples to understand it better.

**Example 1.** Is it possible to have a triangle with the following sides?

3 cm, 5 cm, 7cm

**Solution.** 3 cm + 5 cm > 7 cm

3 cm + 7 cm > 5 cm

5 cm + 7 cm > 3 cm

Hence, it is proved that sum of lengths of any two sides is greater than the length of third side. Therefore, the triangle is possible.

**Example 2.** Is it possible to have a triangle with the following sides?

10 cm, 4 cm, 5 cm

**Solution.** 5 cm + 4 cm < 10 cm

Here, sum of lengths of any two sides is less than the third sides. So triangle formation can not be possible.

## Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse equals to the sum of the squares of its remaining two sides.

(Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2}

Let's assume, Hypotenuse = c

Base = a

Perpendicular = b

c^{2} = a^{2} + b^{2}

**Example 1.** The hypotenuse of a right-angled triangle is 15 cm long and length of the base is 5 cm. Find the height of the triangle.

Solution. Let's assume ABC is a right-angled triangle as shown in the below figure.

Here, AC = 13 cm and BC = 5 cm

According to Pythagoras theorem AB^{2} + BC^{2} = AC^{2}

⇒ AB^{2} + 5^{2} = 13^{2}

⇒ AB^{2} = 13^{2} − 5^{2}

⇒ AB^{2} = 169 − 25

⇒ AB^{2} = 144

⇒ AB = √ 144

⇒ AB = 12 cm

Hence, height of the triangle is 12 cm.

**Example 2.** Find the value of 'x' in the below given figure.

**Solution.** As we know, AB^{2} + BC^{2} = AC^{2}

AC^{2} = 4^{2} + 3^{2}

⇒ AC^{2} = 16 + 9

⇒ AC = √ 25

⇒ AC = 5 cm

Hence, x is equal to 5 cm.

**Example 3.** A 25 m long ladder riches a roof of 20 m high from the ground on placing it against the wall. How far is the foot of the ladder from the wall?

**Solution.** Let's assume AB is the ladder, B is the roof and CB is the height of the wall and AC is the distance between wall and ladder.

AC^{2} + BC^{2} = AB^{2}

⇒ BC^{2} = AB^{2} − AC^{2}

⇒ BC^{2} = 25^{2} − 20^{2}

⇒ BC^{2} = 625 − 400

⇒ BC = √ 225

⇒ BC = 15 m

Hence, the ladder is 15 m away from the wall.

## Properties of Triangle Test

Properties of Triangle Test - 1

Properties of Triangle Test - 2

Properties of Triangle Test - 3

## Class-7 Properties of Triangle Worksheet

Properties of Triangle Worksheet - 1

Properties of Triangle Worksheet - 2

Properties of Triangle Worksheet - 3

## Answer Sheet

**Properties-Of-Triangle-Answer**Download the pdf

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