Sum of angles of a triangle is \(180\) degrees (Proof)

A triangle is a polygon with three sides and three interior angles.

One of the most important properties in geometry is that the sum of the interior angles of any triangle is \(180^\circ\).

This rule helps us find missing angles and solve many geometry problems.

In this lesson, we will prove why the sum of the interior angles of a triangle equals \(180^\circ\).

Key Ideas Used in the Proof

To prove this property, we use two basic geometry facts:

A straight angle measures \(180^\circ\).
Alternate interior angles are equal when two parallel lines are cut by a transversal.

Consider a triangle with interior angles:

\(a, b,\) and \(c\).

Extend one side of the triangle to form a straight line.

Then draw a line through the opposite vertex that is parallel to the base of the triangle.

Using the properties of alternate interior angles, we find that the angles formed on the straight line correspond to the triangle’s angles.

Since a straight line measures \(180^\circ\), the three interior angles of the triangle must satisfy:

\(a + b + c = 180^\circ\)

Therefore, the sum of the interior angles of a triangle is \(180^\circ\).

Two angles of a triangle are \(50^\circ\) and \(60^\circ\).

Find the third angle.

\(50^\circ + 60^\circ + x = 180^\circ\)

\(110^\circ + x = 180^\circ\)

\(x = 70^\circ\)

So the third angle is \(70^\circ\).