A triangle is a polygon with 3 vertices and 3 sides which makes 3 angles .The total sum of the three angles of the triangle is 180 degrees. There are three types of triangle which are differentiated based on length of their vertex.

1.  Equilateral Triangle

2.  Isosceles triangle

3.  Scalene Triangle

In this article we will learn about Isosceles and the Equilateral triangle and their theorem and based on which we will solve some examples.

Isosceles Triangles

[Image will be Uploaded Soon]

An isosceles triangle is a triangle which has at least two congruent sides. These congruent sides are called the legs of the triangle. The point at which these legs join is called the vertex of the isosceles triangle, and the angle opposite to the hypotenuse is called the vertex angle and the other two angles are called base angles.

Properties of Isosceles Triangle:

1. Isosceles triangle has two equal sides.

2. It has  two equal base angles

3. An isosceles triangle which has 90 degrees is called a right isosceles triangle.

From the properties of Isosceles triangle, Isosceles triangle theorem is derived.

Isosceles Triangle Theorem:

• If two sides of a triangle are congruent, then the corresponding angles are congruent.

• (Converse) If two angles of a triangle are congruent, then the sides corresponding to those angles are congruent.

Proving of Theorem

Theorem 1: If two sides of a triangle are congruent, then the corresponding angles are congruent

Proof: Assume an isosceles triangle ABC where AC = BC. We need to prove that the angles corresponding to the sides AC and BC are equal, that is, ∠CAB = ∠CBA.

[Image will be Uploaded Soon]

First we draw a bisector of angle ∠ACB and name it as CD.

Now in ∆ACD and ∆BCD we have,

AC = BC                                                              (Given)

∠ACD = ∠BCD                                                   (By construction)

CD = CD                                                             (Common in both)

Thus, ∆ACD ≅∆BCD                                         (By congruence)

So, ∠CAB = ∠CBA                                             (By congruence)

Theorem 2: (Converse) If two angles of a triangle are congruent, then the sides corresponding those angles are congruent  

Proof: Assume an Isosceles triangle ABC. We have to prove that AC = BC and ∆ABC is isosceles.

Construct a bisector CD which meets the side AB at right angles.

Now in ∆ACD and ∆BCD we have,

∠ACD = ∠BCD                                                    (By construction)

CD = CD                                                               (Common in both)

∠ADC = ∠BDC = 90°                                          (By construction)

Thus, ∆ACD ≅ ∆BCD                                         (By ASA congruence)

So, AB = AC                                                         (By Congruence)

Or ∆ABC is isosceles.

Example

Question: Find angle X

Solution:   Let triangle be ABC

In ∆ABC

AB=BC     (Given)

So,

∠A=∠C      (angle corresponding to congruent sides are equal)

45 degree =∠C 

∠A+∠B+∠C=180 degree (Angle sum property)

45 + x +45 =180

X= 180-90

X= 90 degrees.

Equilateral Triangles

In equilateral triangle all three sides of the triangle are equal which makes all the three internal angles of the triangle to be equal. Equilateral triangle is also known as an equiangular triangle. Equilateral triangles have unique characteristics. The following characteristics of equilateral triangles are known as corollaries. 

[Image will be Uploaded Soon]

Properties of Equilateral Triangle.

1. The Equilateral Triangle has 3 equal sides.

2. The Equilateral Triangle has 3 equal angles.

3. The total sum of the interior angles of a triangle is 180 degrees, therefore, every angle of an equilateral triangle is 60 degrees.

4. It is a 3 sided regular polygon.

The following corollaries of equilateral triangles are derived from the properties of equilateral triangle and Isosceles triangle theorem. 

Isosceles Triangle Theorem:

•  A triangle is said to be equilateral if and only if it is equiangular.

•  Each angle of an equilateral triangle is the same and measures 60 degrees each.

Theorem1: Each angle of an equilateral triangle is the same and measures 60 degrees each.

Proof: Let an equilateral triangle be ABC

 AB=AC=>∠C=∠B. — (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   — (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have 

∠A=∠B=∠C — (3)

In △ABC,

∠A+∠B+∠C=180 degree (Angle sum property)

=>∠A+∠A+∠A=180 degree

=>∠A=180/3 =60 degree

Therefore, ∠A=∠B=∠C=60 degree

Therefore the angles of the equilateral triangle are 60 degrees each.

Hence Proved

Theorem 2: A triangle is said to be equilateral if and only if it is equiangular.

Proof: Let an equilateral triangle be ABC

 AB=AC=>∠C=∠B. — (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   — (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have 

Therefore, ∠A=∠B=∠C — (3)

Therefore, an equilateral triangle is an equiangular triangle 

Hence Proved

Solved Example-

Question: show that angles of equilateral triangle are 60 degree each

Solution: Let an equilateral triangle be ABC

 AB=AC=>∠C=∠B. — (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>∠B=∠A   — (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

From (1) and (2) we have 

∠A=∠B=∠C — (3)

In △ABC,

∠A+∠B+∠C=180 degree (Angle sum property)

=>∠A+∠A+∠A=180 degree

=>∠A=180/3 =60 degree

Therefore, ∠A=∠B=∠C=60 degree