Isosceles Triangle Theorem

Learn All About Isosceles Triangle in 10 Minutes

In this blog, we will get to know about the ABCs of isosceles triangle. After you go through this article, you will be able to solve any numerical related to isosceles triangle. Furthermore, you will get to know about the proof as well.

Definition of Isosceles Triangle

An isosceles triangle is a triangle where two sides have equal lengths.

The 'legs' of an isosceles triangle are the two equal sides.

The third and uneven side of an isosceles triangle serves as the 'base.'

An isosceles triangle's 'vertex angle' is the angle created by two equal sides.

The angles that include the base of an isosceles triangle are known as 'base angles.' 

Isosceles Triangle Theorem

If two triangle sides are congruent, the angles opposite these sides are also congruent. The pons asinorum is a statement in geometry that the angles opposing the equal sides of an isosceles triangle are also equal. This assertion or concept is known as the isosceles triangle theorem and is found in Proposition 5 of Book 1 of Euclid's Elements. 

Isosceles Triangle Theorem Converse

As per the Isosceles Triangle Theorem, when two triangle angles are congruent, the sides opposite these angles are also congruent.

Proof

First you need to draw SR¯¯¯¯¯, the bisector of the vertex angle ∠PRQ .

Since SR¯¯¯¯¯ is the angle bisector , ∠PRS≅∠QRS .

By the Reflexive Property ,

RS¯¯¯¯¯≅RS¯¯¯¯¯

It is given that ∠P≅∠Q .

Therefore, by AAS congruent, ΔPRS≅ΔQRS .

Since corresponding parts of congruent triangles are congruent,

PR¯¯¯¯¯≅QR¯¯¯¯¯

A conditional statement is reversed by replacing the hypothesis (if...) with the conclusion (then...). You might need to fiddle with it to make sure it works. 

Isosceles Triangle Theorem Proof

In this section, we will take a look at the proof of Isosceles Triangle Theorem. 

∠P≅∠Q

Proof:

Let S be the midpoint of PQ¯¯¯¯¯ .

Join R and S.

Since S is the midpoint of  PQ¯¯¯¯¯ , PS¯¯¯¯¯≅QS¯¯¯¯¯ .

By Reflexive Property ,

RS¯¯¯¯¯≅RS¯¯¯¯¯

It is given that PR¯¯¯¯¯≅RQ¯¯¯¯¯

Therefore, by SSS ,

ΔPRS≅ΔQRS

You have to understand that the corresponding parts of congruent triangles are congruent,

∠P≅∠Q

The converse of the Isosceles Triangle Theorem is also true.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If ∠A≅∠B , then AC¯¯¯¯¯≅BC¯¯¯¯¯ . 

Isosceles Triangle Theorem Example

In this section, we will take a look at examples related to isosceles triangle theorem.

Q.1. Find the value of x using the Isosceles Triangle Theorem in the figure below.

As per the given figure, we can see that,

In ∆XYZ, we see that XY = XZ = 12 cm

If two sides of a triangle are found to be congruent, the angles opposite the congruent sides are identical, according to the isosceles triangle theorem.

Thus, ∠Y = ∠Z [Since XY = XZ]

∠Y = 35º, ∠Z = x

Thus, ∠Y = ∠Z = 35º.

Hence the value of x is 35º.

Q.2. Find the value of PR, if QR = 7.5 cm and ∠P and ∠Q of ∆PQR are equal to 70º

In order to solve the question, we will draw a figure first. 

Given that, in ∆PQR, ∠P = ∠Q = 70º.

If two angles of a triangle are congruent, the sides opposing the congruent angles are equal, as far as the converse of isosceles triangle theorem is concerned.

Thus, PR = QR [Since, ∠P = ∠Q]

But, QR = 7.5 cm

Therefore, the value of PR = 7.5 cm.

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