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## 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.

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