Interior Angles of a Triangle

Interior Angles of a Triangle

How to solve the interior angles of a triangle problems: formula, proof, example, and its solution.

Formula

The sum of the measures of the interior angles of a triangle is 180.

The sum of the measures of the interior angles of a triangle is 180.

Proof

The Proof of the formula by drawing an auxiliary line that is parallel to the base of the triangle.

Draw an auxiliary line
that passes through the upper vertex
and that is parallel to the base.

Draw a green angle
on the left side of the blue angle.

The green angles are congruent
because they are
alternate interior angles in parallel lines.

Draw a red angle
on the right side of the blue angle.

By the same reason,
the red angles are also congruent.

The angles at the upper vertex
(green, blue, and red)
are on the auxiliary line.

So m∠(blue) + m∠(green) + m∠(red) = 180.

Example

Find the value of x. The measures of the interior angles: 60, 3x + 30, 7x + 10.

The colored angles are the interior angles of a triangle.

So (60) + (3x + 30) + (7x + 10) = 180.