Last modified on September 6th, 2022

Isosceles Triangle Theorem

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 Base Angle Theorem
Proof of Isosceles Triangle Theorem

Isosceles Triangle Theorem Proof

To prove: ∠ABC = ∠ACB

Isosceles Triangle Theorem Proof

Proof:

A bisector of ∠BAC, AE is drawn

Given: AB = AC

S.NoStatementReason
1.∠BAE = ∠CAEBy Drawing
2.AE  =AECommon Side (Reflective Property)
3.∆BAE ≅ ∆CAEBy SAS Congruence
4.∠ABC = ∠ACBCorresponding 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 of the Converse of the Isosceles Triangle Theorem

Converse of the Isosceles Triangle Theorem Proof

To prove: AB = AC

Converse of the Isosceles Triangle Theorem Proof

Proof:

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

Given: ∠ABC = ∠ACB

S.NoStatementReason
1.∠BAE = ∠CAEBy Drawing
2.AE  =AECommon Side (Reflective Property)
3.∠BEA = ∠CEA = 90°By Drawing
4.∆BAE ≅ ∆CAEBy ASA Congruence
5.AB = ACCorresponding 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,
AB2 = AE2 + BE2, here AE = 3, BE = 4
x2 = (3)2 + (4)2
x2 = 9 + 16 = √25
x = 5 units
Again, since ∆ABC is an isosceles triangle, with sides AB = AC
Hence, AC = x = 5 units

Last modified on September 6th, 2022

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