Isosceles Triangle Theorems

An isosceles triangle is one of the many varieties of triangle differentiated by the length of their sides. Today we will learn more about the isosceles triangle and its theorem. An isosceles triangle is known for its two equal sides. The peak or the apex of the triangle can point in any direction. Check this example:

Here we have an isosceles triangle ABC, where side AB is congruent to side BC, which is the main reason the triangle is isosceles. These are called legs; if they are equal, it is an isosceles triangle. 

Hence, length of AB = length of BC

Some of the Basics About the Isosceles Triangle are as Follows:

  1. The triangle has two equal sides, with the base as the third, unequal side.

  2. The angles opposite the two equal sides will always match

  3. An isosceles triangle is called a right isosceles triangle when its third angle is 90 degrees. 

Properties of an Isosceles Triangle

Here we have an isosceles triangle ABC to explore the parts. Let us discuss some of the properties of an isosceles triangle.

  1. Like any triangle, ABC has three interior angles those are ∠A, ∠B, and ∠C

  2. All three interior angles are acute angles.

  3. It also has three sides, and those are AB, BC, and AC

  4. Side AB is congruent to side BC; therefore, we refer to those twins as legs.

  5. The third side AC is known as the base, even if the triangle is not sitting on that side.

  6. The angles formed between the base and leg ∠A and ∠C are called base angles

Isosceles Triangle Theorem and Its Proof

Theorem 1 - “Angle opposite to the two equal sides of an isosceles triangle are also equal.”

Proof: consider an isosceles triangle ABC, where AC=BC. we will have to prove that angles opposite to the sides AC and BC are equal, i.e., ∠CAB = ∠CBA

To test this mathematically, we will have to introduce a median line. Here, a line must be constructed from an interior angle to the midpoint of the opposite side, which is the base side AB.

We will find Point M on the base side AB, where we can construct the line segment CM.

After constructing, we get two triangles; those are CAM and CBM.

Now, 

CA = CB (Given)

AM= BM (median)

CM = CM (reflective property)

Thus, triangle CAM = triangle CBM.

SO, ∠CAM = ∠CBM (CPCTC)

Putting this in words we have shown that the three sides of triangle CAM are congruent to triangle CBM, which means here you have the SIDE SIDE SIDE Postulate, which gives the congruence. So, if two triangles are congruent, then the parts corresponding to the congruent triangle are congruent (CPCTC). This gives you A is congruent to ∠B.

Theorem 2

It is the converse of the isosceles triangle theorem

“Sides opposite to the two equal angles of a triangle are equal.”

Proof: Consider an isosceles triangle DEF, here we must prove that side DE = side DF and DEF is isosceles.

The converse of the isosceles triangle theorem says that if two angles of the triangle are equal, then the opposite side is the same.

Now you are wondering whether this statement is true or not. Not every converse statement of an original statement is true. The converse could be either true or false if the original statement is false.

To prove the converse statement lets construct a bisector DG which meets the side EF at right angles.

∠DEF = ∠DFE (given)

DG = DG (reflective property)

∠DGE = ∠DGF = 90 degrees (By Construction)

Thus, triangle DEG = triangle DFG (By ASA congruency)

So, DE = DF (By CPCTC)

In simpler words, we have two small right-angle triangles, DEG and DFG, instead of one big isosceles triangle. Since the line DG is an angle bisector, it makes <DGE and ∠DGF congruent to each other. Since the line segment, DG, is used in both right-angle triangles, it is congruent to itself. This would be angle, side, angle (ASA) theorem. With the triangles being congruent, their corresponding part also becomes congruent, which makes DE=DF. It is worth noting that isosceles theorem converse is true.

Solved Isosceles Triangle Questions

  1. In triangle PQR, we have PQ=PR and ∠PQR =47degrees. Find ∠QPR

Solution: By the isosceles triangle theorem, we have 47 degrees =PQR =PRQ. As the angles in a triangle sum up to 180 degrees, we have

∠QPR = 180 degree – ( ∠PQR + ∠PRQ)

     =180 – 2 x 47 = 86 degree

Fun facts About Isosceles Triangle

  1. The word ‘isosceles’ is derived from the Latin word “īsoscelēs’ and the ancient Greek word ‘ἰσοσκελής (isoskelḗs)’ which means ‘equal-legged.’ 

  2. The Babylonian and Egyptian mathematics knew how to calculate ‘area’ much before the ancient Greek mathematicians studied the isosceles triangle.

  3. If a building is shaped like isosceles, it not only makes them attractive but also earthquake resistant.

  4. The shapes of this triangle are often used in construction due to their high strength.

  5. Always remember that the sum of the three angles of the isosceles triangle is always 180 degrees. So, if you know the value of two angles, finding the value of the third angle is easy.

FAQ (Frequently Asked Questions)

1. What are the Angles in an Isosceles Triangle?

An isosceles triangle has two base angles and one other angle. Both the base angles are equal to each other. Here, you have an isosceles triangle where two sides of the triangle are similar to each other. All three angles will always add up to 360 degrees. An isosceles triangle has two congruent sides. It also has congruent base angles. It has an altitude that (1) meets the base at a right angle (2) bisects the apex angle (3) divides the original isosceles triangle into two congruent halves.

2. Is Every Isosceles Triangle Equilateral?

A triangle with three equal sides is known as the equilateral triangle, and an isosceles triangle has two equal sides. So, every equilateral triangle is isosceles, but not all isosceles triangles are equilateral. Suppose if we consider a triangle where base length doesn’t match the two legs of the triangle. This will not be termed as an equilateral triangle, but still be called an isosceles triangle. If we consider a triangle where base length matches the two legs of the triangle. This will be termed as an equilateral triangle as well as an isosceles triangle.