# Properties of an Isosceles Triangle

A 2D figure having three sides in the cartesian plane is known as a trigon or popularly known as a triangle. Based on the sides, the triangle is divided into three types: Equilateral triangle, Isosceles triangle, and scalene triangle.

• Equilateral triangle: As the name suggests, such triangles have all three lengths identical to each other.
• Scalene triangle: These triangles have all sides of distinct lengths.
• Isosceles triangle: This is a special type of triangle that contains two equal sides facing opposite to each other, and the third side can be of any length.

The main purpose of this article is to define various properties of the isosceles triangle. Let us learn about the triangle in detail as we unravel its properties.

## Isosceles Triangle Theorem

Suppose we have a triangle PQR which is an isosceles triangle; the isosceles triangle theorem states that the angles opposite to two equal sides are equal and congruent if those lengths (sides) of a triangle are also congruent.

The Converse of the theorem is also true, which states that two sides of an isosceles triangle are congruent if the corresponding angles opposite to those lengths are equal or congruent. Therefore, we can conclude that ∠PQR = ∠PRQ and PQ = PR in the figure given above. We can prove this by drawing a perpendicular from vertex P. In an isosceles triangle, any perpendicular drawn from the vertex opposite to the base bisects the base. Therefore, let us mark the point where the perpendicular bisects the base QR as O. We can see that angle POQ and POR are 90 degrees in value (due to the perpendicular PO). Since the base is split into two halves, QO = OR. Angle PQO is also equal to angle PRO (angles of an isosceles triangle). Hence, due to the Angle Side Angle congruence rule, the triangles POQ and POR are congruent to each other. Therefore, side QP is congruent to side PR, which proves the theorem.

You may also want to learn about the Area of Triangle Formula.

## Properties of an Isosceles Triangle.

Below are listed all the important properties related to an isosceles triangle. They are:

• Two sides of an isosceles triangle (sides opposite to each other) are congruent to each other.
• The base of an isosceles triangle is the third side that is unequal to the other two sides.
• Because the base contains two congruent base angles, the angles opposite to the equal sides are congruent, respectively. This is also termed the isosceles triangle base angle theorem
• The left-out angle, which is opposite to the base, is known as the apex angle.
• The perpendicular drawn from the apex angle of an isosceles triangle bisects the base into two halves. This altitude splits the apex angle into identical angles too. This altitude from the apex angle is also responsible for dividing the isosceles triangle into two right-angled triangles that are congruent to each other.
• There are several ways to determine the area of an isosceles triangle, but the simplest way is ½ x base x height.
• The summation of the three sides helps us figure out the perimeter of the triangle.
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