Lesson 11-4: Understand Angle Relationships and Triangles

Property

To prove the Triangle Angle-Sum Theorem, we construct an auxiliary line through one vertex of the triangle parallel to the opposite side.

This creates alternate interior angles that allow us to represent the triangle's three angles as a single straight line:

Examples

• Draw $\Delta ABC$ and construct an auxiliary line $\overleftrightarrow{DE}$ through vertex $B$ so that $\overleftrightarrow{DE} \parallel \overline{AC}$.
• Because $\overline{AB}$ acts as a transversal cutting the parallel lines, $\angle DAB \cong \angle A$ (alternate interior angles).
• Similarly, with transversal $\overline{BC}$, $\angle EBC \cong \angle C$.
• The three angles meeting at vertex $B$ form a straight angle, proving that $m\angle A + m\angle B + m\angle C = 180^\circ$.

Property

An exterior angle of a triangle is formed when one side of the triangle is extended beyond a vertex. The exterior angle and its adjacent interior angle are supplementary, meaning they sum to $180°$.

Examples