## 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)

2*x* = 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)

3*x* = 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

- Find the value of
*x*in the given figure.

- Find the value of
*b*in the given diagram.

- Find the value of
*y*in the given figure.

- Below figure shows that ∆ABC is an equilateral triangle and ∠ABE ∠CAD ∠BCF. Prove that ∆DEF is also an equilateral triangle.

- Use the below diagram to prove that ∠ABE ∠DCE, and also identify the isosceles triangles.

- Find the values of x and y in the diagram.

- Find the perimeter of the given triangle.

- Find the values of x and y in the given figure.

- Find the value of the variables in the diagram.

- 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

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

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