Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 2: Angles and Triangles

Lesson 2: Angles of Triangles

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students explore the interior and exterior angle relationships of triangles, learning that the sum of a triangle's interior angles is always 180° and that an exterior angle equals the sum of the two nonadjacent interior angles. Students apply these properties to write and solve algebraic equations to find missing angle measures. The lesson connects geometric reasoning with algebra through hands-on activities and real-world problems like the Bermuda Triangle example.

Section 1

Sum of Angles in a Triangle

Property

The sum of the angles of a triangle is $180^\circ$ .
If you cut out the angles and put all the vertices together, so that the angles are adjacent and not overlapping, you will get a straight angle.

Examples

A triangle has angles measuring $50^\circ$ and $70^\circ$ . The third angle is $180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ$ .
A right triangle has one angle of $90^\circ$ . If another angle is $35^\circ$ , the third angle must be $180^\circ - 90^\circ - 35^\circ = 55^\circ$ .
An isosceles triangle has two equal angles. If the unique angle is $40^\circ$ , the other two angles together are $180^\circ - 40^\circ = 140^\circ$ . Each equal angle is $140^\circ / 2 = 70^\circ$ .

Explanation

No matter what a triangle looks like—tall, short, wide, or skinny—if you add its three corner angles together, you will always get exactly $180^\circ$ . It is a fundamental rule for all triangles!

Section 2

Solving for Angles Given as a Ratio

Property

If the interior angles of a triangle are in the ratio $a:b:c$ , their measures can be represented as $ax$ , $bx$ , and $cx$ . The sum of these angles is $180^\circ$ .

ax + bx + cx = 180

Examples

Section 3

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

Sum of Angles in a Triangle

Property

The sum of the angles of a triangle is $180^\circ$ .
If you cut out the angles and put all the vertices together, so that the angles are adjacent and not overlapping, you will get a straight angle.

Examples

A triangle has angles measuring $50^\circ$ and $70^\circ$ . The third angle is $180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ$ .
A right triangle has one angle of $90^\circ$ . If another angle is $35^\circ$ , the third angle must be $180^\circ - 90^\circ - 35^\circ = 55^\circ$ .
An isosceles triangle has two equal angles. If the unique angle is $40^\circ$ , the other two angles together are $180^\circ - 40^\circ = 140^\circ$ . Each equal angle is $140^\circ / 2 = 70^\circ$ .

Explanation

Section 2

Solving for Angles Given as a Ratio

Property

If the interior angles of a triangle are in the ratio $a:b:c$ , their measures can be represented as $ax$ , $bx$ , and $cx$ . The sum of these angles is $180^\circ$ .

ax + bx + cx = 180

Examples

Section 3

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

Sum of Angles in a Triangle

Property

The sum of the angles of a triangle is $180^\circ$ .
If you cut out the angles and put all the vertices together, so that the angles are adjacent and not overlapping, you will get a straight angle.

Examples

A triangle has angles measuring $50^\circ$ and $70^\circ$ . The third angle is $180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ$ .
A right triangle has one angle of $90^\circ$ . If another angle is $35^\circ$ , the third angle must be $180^\circ - 90^\circ - 35^\circ = 55^\circ$ .
An isosceles triangle has two equal angles. If the unique angle is $40^\circ$ , the other two angles together are $180^\circ - 40^\circ = 140^\circ$ . Each equal angle is $140^\circ / 2 = 70^\circ$ .

Explanation

Section 2

Solving for Angles Given as a Ratio

Property

If the interior angles of a triangle are in the ratio $a:b:c$ , their measures can be represented as $ax$ , $bx$ , and $cx$ . The sum of these angles is $180^\circ$ .

ax + bx + cx = 180