An isosceles triangle is a triangle having two equal sides, no matter in what direction the apex or peak of the triangle points. The key properties of isosceles triangle are:

Contains two equal sides with the base being the unequal, third side

The angles opposite the two equal sides are equal

When the third angle is 90°, it is called a right isosceles triangle

Using the properties of isosceles triangle, the two theorems along with their proofs are given below.

What is the Isosceles Triangle Theorem

Also known as the base angle theorem.

Isosceles Triangle Theorem Proof

To prove: ∠ABC = ∠ACB

Proof:

A bisector of ∠BAC, AE is drawn

Given: AB = AC

S.No

Statement

Reason

1.

∠BAE = ∠CAE

By Drawing

2.

AE =AE

Common Side (Reflective Property)

3.

∆BAE ≅ ∆CAE

By SAS Congruence

4.

∠ABC = ∠ACB

Corresponding parts of Congruent triangles (CPCTC) are congruent

Hence proved that, the angles opposite to the equal sides of an isosceles triangle are also equal.

What is the Converse of the Isosceles Triangle Theorem

Converse of the Isosceles Triangle Theorem Proof

To prove: AB = AC

Proof:

A bisector of ∠BAC, AE is drawn that meets BC at right angles

Given: ∠ABC = ∠ACB

S.No

Statement

Reason

1.

∠BAE = ∠CAE

By Drawing

2.

AE =AE

Common Side (Reflective Property)

3.

∠BEA = ∠CEA = 90°

By Drawing

4.

∆BAE ≅ ∆CAE

By ASA Congruence

5.

AB = AC

Corresponding parts of Congruent triangles (CPCTC) are congruent

Hence proved that, the sides opposite to the two equal angles of a triangle are equal.

Isosceles Triangle Theorem Examples

Let us solve some problems to understand the theorems better.

Find the value of x in the given isosceles triangle.

Solution:

As we know, according to the isosceles triangle theorem, ∠ACB = ∠ABC = x Now, by the triangle sum theorem, ∠ABC + ∠ACB + ∠BAC = 180° x + x + 80° = 180° 2x = 180° – 80° = 100° x = 50°

In ∆XYZ, XY = XZ, and ∠XYZ = 55°. Find ∠YXZ.

Solution:

As we know, according to the isosceles triangle theorem, ∠XYZ = ∠XZY = 55° Now, by the triangle sum theorem, ∠XYZ + ∠XZY + ∠YXZ = 180° 55° + 55° + ∠YXZ = 180° ∠YXZ = 180° – (55° + 55°) ∠YXZ = 180° – 110° ∠YXZ = 70°

Solve the value of x in the given isosceles triangle.

Solution:

Here we will use the Pythagorean Theorem to find the side lengths of the given isosceles triangle. Since AE divides BC into BE and EC Thus, BE = EC = 4cm Applying Pythagorean Theorem in ∆ABE, AB^{2} = AE^{2} + BE^{2}, here AE = 3, BE = 4 x^{2} = (3)^{2} + (4)^{2} x^{2} = 9 + 16 = √25 x = 5 units Again, since ∆ABC is an isosceles triangle, with sides AB = AC Hence, AC = x = 5 units