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Isosceles and Equilateral Triangles

Sep 10, 2022
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Introduction

Isosceles Triangle: 

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

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. 

parallel
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:  

parallel

Proof: 

Let us consider an isosceles triangle ABC, 

Proof: 

Draw a bisector of ∠ACB., i.e., CD. 

Now in ∆ACD and ∆BCD, 

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, 

Let us consider an isosceles triangle ABC, 
Let us consider an isosceles triangle ABC, 

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

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. 

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. 

Example 3

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°. 

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.
Find the value of x in the given figure.
  1. Find the value of b in the given diagram.
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.
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.
Find the values of x and y in the diagram.
  1. Find the perimeter of the given triangle.
Find the perimeter of the given triangle.
  1. Find the values of x and y in the given figure.
Find the values of x and y in the given figure.
  1. Find the value of the variables in the diagram.
Find the value of the variables in the diagram.
  1. Find the value of x in the figure.
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.

Comments:

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