Lesson Isosceles triangles

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Isosceles triangles

Definition. A triangle is called isosceles if it has two sides of equal length.

An example of an isosceles triangle is shown in Figure 1.

Theorem 1
If a triangle has two sides of equal length, then the angles opposite to these sides are congruent.

Proof
Let ABC be a triangle with sides AC and BC of equal length (Figure 1).
We need to prove that angles BAC and ABC are congruent.

Consider the triangle BAC.
The side AC of the triangle ABC corresponds to the side BC of the triangle BAC.
The side BC of the triangle ABC corresponds to the side AC of the triangle BAC.
The side AB is common to the triangles ABC and BAC.
Since all three of the corresponding sides of the triangles ABC and BAC are of equal length,
these two triangles are congruent, in accordance to the postulate 3 (SSS) of the triangle
congruency (see the lesson Congruence tests for triangles of this topic).
Hence, the corresponding angles BAC and ABC are congruent.

The proof is completed.

Figure 1. To the Theorem 1

Thus we proved that in isosceles triangle two angles opposite to the equal sides are congruent.

The statement opposite to the Theorem 1 is true also.

Theorem 2
If in a triangle two angles are congruent, then the sides opposite to these angles are of equal length.

Proof
Let ABC be a triangle with congruent angles BAC and ABC (Figure 2).
We need to prove that angles BAC and ABC are congruent.

Consider the triangle BAC.
The angle BAC of the triangle ABC corresponds to the angle ABC of the triangle BAC.
The angle ABC of the triangle ABC corresponds to the angle BAC of the triangle BAC.
The side AB is common to the triangles ABC and BAC.
Since two angles and the included sides of the triangles ABC and BAC are congruent,
these two triangles are congruent, in accordance to the postulate 2 (ASA) of the triangle
congruency (see the lesson Congruence tests for triangles of this topic).
Hence, the corresponding sides AC and BC are of equal length.

The proof is completed.

Figure 2. To the Theorem 2

Thus we proved that in isosceles triangle two sides opposite to the congruent angles are of equal length.

Summary
It follows from the two previous Theorems that
    - the triangle is isosceles if and only if it has two equal sides and/or two congruent angles;
    - in a triangle, the sides opposite to the congruent angles are of equal length;
    - in a triangle, the angles opposite to the equal sides are congruent.

When the isosceles triangle is considered, its side included between congruent angles is often called the base.

Problem 1
In an isosceles triangle the angle at the base is 40�.
Find the angle opposite to the base.

Solution
In an isosceles triangle two angles adjacent to the base are congruent, so both angles adjacent to the base are 40�.
Since the sum of interior angles of the triangle is equal to 180�, the angle opposite to the base is equal to
180� - (40�+40�) = 180� - 80� = 100�.

Answer. The angle opposite to the base is 100�.

Problem 2
In an isosceles triangle the angle opposite to the base is 40�.
Find the angle adjacent to the base.

Solution
Since the sum of interior angles of the triangle is equal to 180�, the sum of two angles adjacent to the base is equal to
180� - 40� = 140�.
In an isosceles triangle two angles adjacent to the base are congruent, so each of these angles is equal to
140�/2 = 70�.

Answer. The angle adjacent to the base is 70�.

For navigation over the lessons on Properties of Triangles use this file/link Properties of Trianles.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link GEOMETRY - YOUR ONLINE TEXTBOOK.

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