Learn on PengiBig Ideas Math, Course 3Chapter 3: Angles and Triangles

Lesson 2: Angles of Triangles

In this Grade 8 lesson from Big Ideas Math Course 3, students explore the interior and exterior angle relationships of triangles, discovering that the sum of interior angle measures always equals 180°. Students also learn the Exterior Angle Theorem, which states that an exterior angle of a triangle equals the sum of the two nonadjacent interior angles. They apply these concepts algebraically to find missing angle measures in a variety of triangle configurations.

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

Exterior Angle Theorem

Property

An exterior angle is formed when one side of a triangle is extended outward. It forms a linear pair (summing to 180°) with its adjacent interior angle.

The Exterior Angle Theorem states that the measure of an exterior angle equals the sum of the two non-adjacent (remote) interior angles:

m\angle\text{ext} = m\angle A + m\angle B

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

Exterior Angle Theorem

Property

An exterior angle is formed when one side of a triangle is extended outward. It forms a linear pair (summing to 180°) with its adjacent interior angle.

The Exterior Angle Theorem states that the measure of an exterior angle equals the sum of the two non-adjacent (remote) interior angles:

m\angle\text{ext} = m\angle A + m\angle B

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

Exterior Angle Theorem

Property

An exterior angle is formed when one side of a triangle is extended outward. It forms a linear pair (summing to 180°) with its adjacent interior angle.

The Exterior Angle Theorem states that the measure of an exterior angle equals the sum of the two non-adjacent (remote) interior angles:

m\angle\text{ext} = m\angle A + m\angle B