Isosceles Triangle - Properties And 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

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.

Isosceles Triangle

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 triangle ABC. We need to prove that AC = BC and ∆ABC is isosceles.

Isosceles Triangle Theorem 2

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.

To learn more about isosceles triangles, their properties and examples based on the theorems discussed above, download Byju’s The Learning App.


Practise This Question

Integers are not closed under which operation?