Learn on PengienVision, Mathematics, Grade 8Chapter 6: Congruence and Similarity

Lesson 8: Angles, Lines, and Transversals

In this Grade 8 lesson from enVision Mathematics Chapter 6, students learn to identify and find angle measures formed when a transversal intersects parallel lines, including corresponding angles, alternate interior angles, and same-side interior angles. Students apply these relationships to solve for unknown angle measures using algebraic equations and determine conditions that prove two lines are parallel.

Section 1

Corresponding Angles are Congruent

Property

When a transversal intersects two parallel lines, corresponding angles are congruent. Corresponding angles occupy the same relative position at each intersection point.

Examples

Section 2

Using Algebra to Solve for Unknown Angles

Property

The sum of the measures of interior angles in a triangle is $180^\circ$ .
Vertical angles have equal measure.
Complementary angles add up to $90^\circ$ .
Supplementary angles add up to $180^\circ$ .

The use of algebraic language can help us to write relationships and solve problems.

Examples

Two angles lie on a straight line. One is $115^\circ$ and the other is $x$ . Since they are supplementary, their sum is $180^\circ$ . The equation is $x + 115 = 180$ , so $x = 65^\circ$ .
The angles in a triangle are $x$ , $2x$ , and $60^\circ$ . They must sum to $180^\circ$ , so $x + 2x + 60 = 180$ . This gives $3x = 120$ , so $x = 40^\circ$ . The angles are $40^\circ$ , $80^\circ$ , and $60^\circ$ .
An angle of $50^\circ$ and an angle of $y$ are complementary. This means their sum is $90^\circ$ . The equation is $50 + y = 90$ , so the unknown angle $y$ is $40^\circ$ .

Explanation

Geometric rules about angles are like secret codes for making equations. If you know angles add up to $180^\circ$ or are equal, you can set up a problem to find the missing piece, $x$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Corresponding Angles are Congruent

Property

When a transversal intersects two parallel lines, corresponding angles are congruent. Corresponding angles occupy the same relative position at each intersection point.

Examples

Section 2

Using Algebra to Solve for Unknown Angles

Property

The sum of the measures of interior angles in a triangle is $180^\circ$ .
Vertical angles have equal measure.
Complementary angles add up to $90^\circ$ .
Supplementary angles add up to $180^\circ$ .

The use of algebraic language can help us to write relationships and solve problems.

Examples

Two angles lie on a straight line. One is $115^\circ$ and the other is $x$ . Since they are supplementary, their sum is $180^\circ$ . The equation is $x + 115 = 180$ , so $x = 65^\circ$ .
The angles in a triangle are $x$ , $2x$ , and $60^\circ$ . They must sum to $180^\circ$ , so $x + 2x + 60 = 180$ . This gives $3x = 120$ , so $x = 40^\circ$ . The angles are $40^\circ$ , $80^\circ$ , and $60^\circ$ .
An angle of $50^\circ$ and an angle of $y$ are complementary. This means their sum is $90^\circ$ . The equation is $50 + y = 90$ , so the unknown angle $y$ is $40^\circ$ .

Explanation

Geometric rules about angles are like secret codes for making equations. If you know angles add up to $180^\circ$ or are equal, you can set up a problem to find the missing piece, $x$ .

Overview

Lesson overview

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

Overview

Section 1

Corresponding Angles are Congruent

Property

When a transversal intersects two parallel lines, corresponding angles are congruent. Corresponding angles occupy the same relative position at each intersection point.

Examples

Section 2

Using Algebra to Solve for Unknown Angles

Property

The sum of the measures of interior angles in a triangle is $180^\circ$ .
Vertical angles have equal measure.
Complementary angles add up to $90^\circ$ .
Supplementary angles add up to $180^\circ$ .

The use of algebraic language can help us to write relationships and solve problems.

Examples

Two angles lie on a straight line. One is $115^\circ$ and the other is $x$ . Since they are supplementary, their sum is $180^\circ$ . The equation is $x + 115 = 180$ , so $x = 65^\circ$ .
The angles in a triangle are $x$ , $2x$ , and $60^\circ$ . They must sum to $180^\circ$ , so $x + 2x + 60 = 180$ . This gives $3x = 120$ , so $x = 40^\circ$ . The angles are $40^\circ$ , $80^\circ$ , and $60^\circ$ .
An angle of $50^\circ$ and an angle of $y$ are complementary. This means their sum is $90^\circ$ . The equation is $50 + y = 90$ , so the unknown angle $y$ is $40^\circ$ .

Explanation

Geometric rules about angles are like secret codes for making equations. If you know angles add up to $180^\circ$ or are equal, you can set up a problem to find the missing piece, $x$ .