Introduction
Isosceles Triangle:
The two opposite sides of a triangle are equal is called an isosceles triangle.
![Isosceles Triangle:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1671.png)
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.
![Equilateral Triangle:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1655.png)
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,
![Proof:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1654.png)
Draw a bisector of ∠ACB., i.e., CD.
Now in ∆ACD and ∆BCD,
![Now in ∆ACD and ∆BCD](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2361.png)
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,](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1657.png)
![Let us consider an isosceles triangle ABC,](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2362.png)
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.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1656.png)
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)
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1658.png)
Example 2: In the given picture, prove that ∆QPS ≅ ∆PQR.
![Example 2: In the given picture, prove that ∆QPS ≅ ∆PQR.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1659.png)
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](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1660.png)
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°
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1662.png)
Example 4: Find the values of x and y from the given diagram.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1661.png)
Solution:
In the given figure, x represents an angle of an equilateral triangle.
x + x + 100° = 180° (Triangle sum property)
3x = 180°
3x=180°
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1662.png)
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°.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1663.png)
120° + 35° + y° = 180°
155 + 2y = 25°
y = 25°
Exercise
- Find the value of x in the given figure.
![Find the value of x in the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1685.png)
- Find the value of b in the given diagram.
![Find the value of b in the given diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1686.png)
- Find the value of y in the given figure.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1687.png)
- Below figure shows that ∆ABC is an equilateral triangle and ∠ABE ∠CAD ∠BCF. Prove that ∆DEF is also an equilateral triangle.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1688.png)
- 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.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1689.png)
- Find the values of x and y in the diagram.
![Find the values of x and y in the diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1690.png)
- Find the perimeter of the given triangle.
![Find the perimeter of the given triangle.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1691.png)
- Find the values of x and y in the given figure.
![Find the values of x and y in the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1692.png)
- Find the value of the variables in the diagram.
![Find the value of the variables in the diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1693.png)
- Find the value of x in the figure.
![Find the value of x in the figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1694.png)
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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