Learn on PengiReveal Math, AcceleratedUnit 11: Angles

Lesson 11-4: Understand Angle Relationships and Triangles

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 11), students learn that the interior angles of a triangle always sum to 180 degrees and that an exterior angle of a triangle equals the sum of its two remote interior angles. The lesson uses parallel lines and alternate interior angles to prove these relationships, then extends them to show that the exterior angles of a triangle sum to 360 degrees. Students apply these concepts to real-world problems involving parallel streets and intersecting transversals.

Section 1

Proving the Triangle Angle-Sum Theorem

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:

m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ

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 $m\angle A + m\angle B + m\angle C = 180^\circ$ .

Section 2

Defining Exterior Angles of a Triangle

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

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Proving the Triangle Angle-Sum Theorem

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:

m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ

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 $m\angle A + m\angle B + m\angle C = 180^\circ$ .

Section 2

Defining Exterior Angles of a Triangle

Property

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Proving the Triangle Angle-Sum Theorem

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:

m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ

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 $m\angle A + m\angle B + m\angle C = 180^\circ$ .

Section 2

Defining Exterior Angles of a Triangle