Isosceles Triangle Theorems

You have already learnt about the properties and types of triangles. One of the special types of triangle is the isosceles triangle. Isosceles triangle is a triangle which has two equal sides, no matter in what direction the apex (or peak) of the triangle points. Some pointers about isosceles triangles are:

• It has two equal sides.
• It has two equal angles, that is, the base angles.
• When the third angle is 90 degree, it is called a right isosceles triangle.

In this article, we will state two theorems regarding the properties of isosceles triangles and discuss their proofs.

Isosceles Triangle Theorems and Proofs

Theorem 1: Angles opposite to the equal sides of an isosceles triangle are also equal.

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

We first draw a bisector of ∠ACB and name it as CD.

Now in ∆ACD and ∆BCD we have,

AC = BC                                                                (Given)

∠ACD = ∠BCD                                                    (By construction)

CD = CD                                                               (Common to both)

Thus,  ∆ACD ≅∆BCD                                        (By SAS congruency)

So, ∠CAB = ∠CBA                                              (By CPCTC)

Theorem 2: Sides opposite to the equal angles of a triangle are equal.

Proof: Consider an isosceles triangl ABC. We need 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 to both)

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

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

So, AB = AC                                                         (By CPCTC)

Or ∆ABC is isosceles.