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.

2. Find the value of b in the given diagram.

3. Find the value of y in the given figure.

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

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

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

7. Find the perimeter of the given triangle.

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

9. Find the value of the variables in the diagram.

10. 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.