Back to Use facts about supplementary, complementary, vertical, and adjacent angles to solve simple equations

Angle Relationships and Equations

For each problem, identify the angle relationship, write an equation, solve, and verify your answer.

Grade 7·21 problems·~35 min·Common Core Math - Grade 7·standard·7-g-b-5

Work through problems with immediate feedback

Recall / Warm-Up

Two angles are supplementary. One angle measures 47°. What is the measure of the other angle?

43°

133°

53°

143°

Solve for x: 3x + 12 = 90. What is the value of x?

Two angles are complementary. One angle measures 62°. What is the measure of the other angle?

Fluency Practice

$A straight horizontal line divided by a ray into two angles: (x + 24)° on the left and 82° on the right.$

Two angles form a straight line. One angle measures $(x + 24)\degree$ and the other measures 82°. Find the value of $x$ .

Two angles are complementary. One angle measures $(2x - 10)\degree$ and the other measures 56°. Find the value of $x$ .

$Two intersecting lines with vertical angles labeled (5x + 3)° and (3x + 19)°.$

Two intersecting lines form vertical angles. One angle measures $(5x + 3)\degree$ and its vertical angle measures $(3x + 19)\degree$ . Find the measure of the angle.

Find the measure of the supplement of a 73° angle.

Two straight lines cross at a point. One of the four angles measures 68°. What are the measures of the other three angles, listed in clockwise order starting from the 68° angle?

112°, 68°, 112°

68°, 68°, 68°

112°, 112°, 112°

112°, 68°, 68°

Varied Practice

$Two intersecting lines at point P. ∠1 = 55°; ∠2, ∠3, ∠4 are unknown.$

Two lines intersect at point P. The four angles are labeled ∠1, ∠2, ∠3, and ∠4 going clockwise, starting from the top-left. If ∠1 measures 55°, which other angle also measures 55°?

∠2

∠3

∠4

Both ∠2 and ∠4

$A straight line divided by a ray into two supplementary angles labeled (2x + 15)° and (3x − 5)°.$

Two supplementary angles measure $(2x + 15)\degree$ and $(3x - 5)\degree$ . The value of $x$ is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . The larger angle measures ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ °.

larger angle:

Angle P and angle Q are complementary. Angle Q and angle R are supplementary. If angle P measures 32°, find the measure of angle R. Show each step and state which angle relationship you used.

Two adjacent angles on a straight line measure $(4x + 6)\degree$ and $(2x + 18)\degree$ . Find the value of $x$ .

Word Problems

$Two planks meeting at a point on a straight baseline, forming supplementary angles labeled (6x + 4)° and (3x + 14)°.$

Marta is building a wooden frame. Two planks meet at a point on a straight edge, forming two supplementary angles. One plank creates an angle of $(6x + 4)\degree$ with the edge. The other creates an angle of $(3x + 14)\degree$ .

How many degrees does the smaller angle measure?

Two streets intersect. One angle at the intersection measures $(8x + 4)\degree$ . The angle directly across from it (a vertical angle) measures $(10x - 12)\degree$ .

What is the degree measure of this vertical angle pair?

$A straight line with three rays from one point creating angles a = 65°, b = ?, and c = 40° above the line.$

A horizontal line has three rays extending upward from a single point on the line, creating three angles above it: angle a on the left (measuring 65°), angle b in the middle (unknown), and angle c on the right (measuring 40°).

What is the measure of angle b?

Are angles a and c supplementary? Explain how you know.

A parking lot has two rows of spaces that meet at an intersection. The angle between Row 1 and the main drive measures $(2x + 5)\degree$ . The angle between Row 2 and the main drive, directly across the intersection, measures $(3x - 20)\degree$ . The two row angles are vertical angles.

Find the value of x, then find the measure of the angle each row makes with the main drive. Show all work and verify your answer.

Error Analysis

$Contrast card: student incorrectly uses 180 for complementary angles; correct method uses 90.$

A student solved this problem: "Find the complement of 54°."

The student's work:

x + 54 = 180
x = 126°

What error did the student make?

The student confused complementary with supplementary. Complementary angles sum to 90°, so the equation should be x + 54 = 90, giving x = 36°.

The student set up the equation correctly but made an arithmetic error when solving.

The student should not use an equation; complements can only be found by measuring with a protractor.

The student's work is correct — the complement of 54° is 126°.

$Contrast card: Deon sums vertical angles to 180; correct method sets them equal to each other.$

Deon solved this problem: 'Two lines intersect. One angle measures $(3x + 7)\degree$ and the vertical angle measures $(5x - 9)\degree$ . Find x.'

Deon's work:

$(3x + 7) + (5x - 9) = 180$
$8x - 2 = 180$
$x = 22.75$

What error did Deon make?

Deon treated the vertical angles as supplementary instead of equal. The correct equation is 3x + 7 = 5x − 9.

Deon set up the right type of equation but made an algebra error combining like terms.

Deon should have set each angle equal to 90°, since vertical angles are complementary.

Deon's equation is correct — vertical angles add to 180°.

Challenge / Extension

$A straight line with three rays from one point creating three adjacent angles labeled (3x + 20)°, (x + 15)°, and (2x − 5)°.$

Three rays extend from a single point on a straight line, creating three angles above the line. The angles measure $(3x + 20)\degree$ , $(x + 15)\degree$ , and $(2x - 5)\degree$ from left to right. What is the measure of the largest angle?

Two lines intersect. One angle measures $(4x + 15)\degree$ and the adjacent angle measures $(x + 30)\degree$ . (a) Write an equation and solve for x. (b) Find all four angle measures at the intersection. (c) Verify your answer by confirming that each adjacent pair sums to 180° and each vertical pair is equal.

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