Properties of Triangles

Properties of an Isosceles Triangle

An isosceles triangle is a triangle with two sides of equal length. In an isosceles triangle, the angle formed by the two equal sides is called the vertex angle, the side opposite the vertex angle is called the base, and the angles adjacent to the base are called the base angles.
Key Properties:
  1. The two base angles of an isosceles triangle are equal.
  2. The angle bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base.

[Explanation]

In the isosceles triangle , where , let the angle bisector of intersect the base at point .
In and :
Thus, from (1), (2), and (3), by the SAS congruence theorem. Therefore,
This proves that the two base angles of an isosceles triangle are equal.
Since ,
Additionally, , and since ,
Thus, from (4) and (5), the segment is the perpendicular bisector of . This proves that the angle bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

Condition for a Triangle to be Isosceles

A triangle with two equal interior angles is an isosceles triangle.
In , if , let the angle bisector of intersect at point . In and :
Additionally, since the sum of the interior angles of a triangle is always constant and , it follows that :
Thus, from (1), (2), and (3), by the ASA congruence theorem. Therefore,
This proves that a triangle with two equal interior angles is an isosceles triangle.