16.3 Angles in triangles and quadrilaterals — Learn

Prerequisites: use straight-line and parallel-line facts, and solve one-step equations.

Objectives: use triangle and quadrilateral angle sums, including the exterior angle of a triangle. This covers G3 and G4.

Triangles

The interior angles of a triangle add to $180^\circ$. An exterior angle is formed by extending one side. It equals the sum of the two opposite interior angles.

Example 1. A triangle has angles $48^\circ$, $73^\circ$ and $x^\circ$.

Example 2. A triangle has two opposite interior angles of $42^\circ$ and $69^\circ$. Find the exterior angle.

The exterior angle is $111^\circ$, the sum of the opposite interior angles.

Quadrilaterals

A quadrilateral is a four-sided polygon. Its interior angles add to $360^\circ$. You can split it into two triangles, each with angle sum $180^\circ$.

Example 3. Three angles in a quadrilateral are $90^\circ$, $105^\circ$ and $83^\circ$. Find the fourth.

Watch out. A quadrilateral does not have angle sum $180^\circ$. That is the sum for one triangle; a diagonal makes two triangles, so $180+180=360^\circ$.

Progress check

The courtyard tile is triangular. Two angles are $68^\circ$ and $47^\circ$. Find the last angle and state the sum used.

You should now be able to

• find a missing interior angle in a triangle;
• use the exterior angle rule for a triangle;
• find a missing interior angle in a quadrilateral and explain the sum.