#### Need Help?

Get in touch with us

# Isosceles and Equilateral Triangles

## Introduction

### Isosceles Triangle:

The two opposite sides of a triangle are equal is called an isosceles triangle.

In the above isosceles triangle diagram,

• The opposite congruent sides are called legs
• The angle formed using two legs is called the vertex angle
• The third side opposite to legs of an isosceles triangle is known as its base
• The angles adjacent to its base are known as base angles

#### Perimeter

The perimeter of an isosceles triangle is the sum of the lengths of its sides.

P = 2a + b

### Equilateral Triangle

A triangle in which all the three sides are equal is called as an equilateral triangle.

#### Perimeter

The sum of the lengths of an equilateral triangle around the boundary.

P = 3a.

### Apply the base angles theorem

#### Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite to them are congruent.

Given: To prove: Proof:

Let us consider an isosceles triangle ABC,

Draw a bisector of ∠ACB., i.e., CD.

Now in ∆ACD and ∆BCD,

#### Converse of Base Angles Theorem:

If two angles of a triangle are congruent, then the sides opposite to them are congruent.

Given: To prove: Proof:

Let us consider an isosceles triangle ABC,

Example 1: In the given diagram, find the values of X and Y.

Solution:

Step 1: Given ∆KLN is an equiangular, so  ¯KN≅¯KL.

∴ Y = 4.

Step 2: Now, find the value of X. If ∠LNM = ∠LMN and then ∆LMN is an isosceles triangle.

LN = LM (Definition of congruence segments)

4 = X + 1 (Since ∆KLN is an equilateral, then LN = 4)

Example 2: In the given picture, prove that ∆QPS ≅ ∆PQR.

Solution:

From the figure, PS ≅ QR and ∠QPS ≅ ∠PQR.

PQ ≅ QP and PS ≅QR (Corresponding parts of congruent triangles)

∠QPS ≅ ∠PQR (Corresponding parts of congruent triangles)

∆QPS ≅ ∆PQR (By SAS congruence postulate)

Hence proved.

Example 3: In the given figure, find the value of X.

Solution:

From the diagram, the triangle is an isosceles triangle. So, the base angles are congruent.

Let us consider the opposite angle also X,

x + x + 100° = 180° (Triangle sum property)

2x = 180° – 100° = 80°

Example 4: Find the values of x and y from the given diagram.

Solution:

In the given figure, x represents an angle of an equilateral triangle.

x + x + 100° = 180° (Triangle sum property)

3x  = 180°

3x=180°

Also, from the given figure, the vertex angle forms a linear pair with x which is 60° and its measure is 120°.

120° + 35° + y° = 180°

155 + 2y = 25°

y = 25°

## Exercise

1. Find the value of x in the given figure.
1. Find the value of b in the given diagram.
1. Find the value of y in the given figure.
1. Below figure shows that ∆ABC is an equilateral triangle and ∠ABE  ∠CAD  ∠BCF. Prove that ∆DEF is also an equilateral triangle.
1. Use the below diagram to prove that ∠ABE  ∠DCE, and also identify the isosceles triangles.
1. Find the values of x and y in the diagram.
1. Find the perimeter of the given triangle.
1. Find the values of x and y in the given figure.
1. Find the value of the variables in the diagram.
1. Find the value of x in the figure.

### What have we learned

• Understand the definition of an isosceles and an equilateral triangles.
• Prove the base angles theorem.
• Prove the converse of the base angles theorem.
• Apply the base angles theorem.
• Find the perimeter of an isosceles and an equilateral triangles.
• Solve the different problems involving base angles triangles.

### Summary

1. Base: The third side opposite to legs of an isosceles triangle is known as its base.
2. Base angles: The angles adjacent to its base are known as base angles.
3. Vertex angle: The angle formed using two legs is called the vertex angle.
4. Leg: The opposite congruent sides are called legs.
5. Base angles theorem: If two sides of a triangle are congruent, then the angles opposite to them are congruent.
6. Converse of the base angles theorem: If two angles of a triangle are congruent, then the sides opposite to them are congruent.

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   