Grade 8 Workbook — Chapter 9: Angle Pair Relationships

SOL 8.MG.1 · Companion to Textbook Chapter 9

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 116.

PAGE 1 — Chapter opener

Chapter 9 · Angle Pair Relationships

Standard 8.MG.1

In this chapter you will:

Name an angle, its vertex, and its sides, and classify it by measure
Identify adjacent angles and state the two conditions they must meet
Identify complementary angles (sum $90^\circ$ ) and supplementary angles (sum $180^\circ$ )
Identify vertical angles at an intersection and explain why they are equal
Find an unknown angle measure from a diagram
Write and solve an equation for an unknown angle, including in real situations

Words to know: angle · vertex · side · degree · measure · acute · right · obtuse · straight · adjacent angles · linear pair · complementary · complement · supplementary · supplement · vertical angles · perpendicular

Not in this chapter: parallel lines cut by a third line. Every relationship on these pages happens at a single point.

PAGE 2 — The parts of an angle

9.1 Angles, Adjacent Angles, and Adding Measures

FIGURE: fig1-angle-parts.png (full width)

Fill in the blanks.

The shared endpoint of the two rays is the ____________.

The two rays are the ____________ of the angle.

Three ways to name this angle: ____________, ____________, ____________.

The measure is written $m\angle ABC =$ ____________.

Classify by measure.

Measure	Name
greater than $0^\circ$ , less than $90^\circ$
exactly $90^\circ$
greater than $90^\circ$ , less than $180^\circ$
exactly $180^\circ$

Name the angle in the figure three different ways, and give its measure.

_______________ _______________ _______________ Measure: ______
Name the vertex and both sides.

Vertex: ______ Sides: _______________ _______________

PAGE 3 — Adjacent angles

Adjacent Angles

FIGURE: fig2-adjacent-angles.png (full width)

Two conditions for adjacent angles.

First condition: they share a ____________ and share a ____________.
Second condition: their ____________ do not overlap.

Name the two adjacent angles and the side they share.

_______________ and _______________ Shared side: ______
$m\angle COA =$ ______ $+$ ______ $=$ ______

FIGURE: fig3-adjacent-or-not.png (full width)

Why are $\angle 3$ and $\angle 4$ not adjacent?
Classify each measure.

$55^\circ$	$90^\circ$	$118^\circ$	$180^\circ$

PAGE 4 — Adding and subtracting adjacent measures

Putting Angles Together

Classify each angle.

a) $12^\circ$	b) $90^\circ$	c) $145^\circ$	d) $89^\circ$	e) $180^\circ$

$m\angle RST = 74^\circ$ , and $S$ is the vertex of no other angle. Two other names:
$\angle FGH$ and $\angle HGJ$ are adjacent, $m\angle FGH = 52^\circ$ , $m\angle HGJ = 64^\circ$ .

$m\angle FGJ =$ ______
$\overrightarrow{BD}$ is in the interior of $\angle ABC$ , $m\angle ABD = 25^\circ$ , $m\angle DBC = 61^\circ$ .

$m\angle ABC =$ ______
$\overrightarrow{YW}$ is in the interior of $\angle XYZ$ , $m\angle XYZ = 96^\circ$ , $m\angle XYW = 43^\circ$ .

$m\angle WYZ =$ ______
Why can $\angle ABC$ and $\angle ABD$ never be adjacent when $D$ is inside $\angle ABC$ ?
Two angles share a vertex but no side. Adjacent? ______ Why?

PAGE 5 — Apply it and check your thinking

9.1 Apply and Explain

Apply it. A laptop lid makes a $105^\circ$ angle with the base. A stylus lies on the base inside that angle, making $30^\circ$ with the base.

WORK SPACE: 1.5 in tall, full width

Angle between stylus and lid: ______
Explain. Can two adjacent angles both be obtuse? Give an example or say why not.
Find the error. A student says $\angle 1$ and $\angle 2$ in the left panel are not adjacent because their measures are different.

What went wrong? _______________________________________________

Exit ticket 9.1

Vertex and sides of $\angle PQR$ : ______ _______________ _______________
$\angle FGH$ and $\angle HGJ$ adjacent, $52^\circ$ and $64^\circ$ . $m\angle FGJ =$ ______
The two conditions for adjacency:
1. _______________________________ 2. _______________________________
An angle of $179^\circ$ is ____________, because _______________________________

PAGE 6 — Two sums to know

9.2 Complementary and Supplementary Angles

Fill in the definitions.

Two angles are complementary when their measures add to ______.

Two angles are supplementary when their measures add to ______.

To find a complement, subtract from ______. To find a supplement, subtract from ______.

What the sums force. A complementary pair is made of two ____________ angles, so an obtuse angle has ____________ complement.

FIGURE: fig4-complementary-adjacent.png (half width, left)

FIGURE: fig5-supplementary-linear-pair.png (half width, right)

Complement of $35^\circ$ : ______
Supplement of $130^\circ$ : ______
Complement of $12^\circ$ : ______
Supplement of $90^\circ$ : ______
In the right-angle figure, the complementary pair is _______________ and _______________, and together they form the right angle _______________.
In the straight-line figure, the supplementary pair is _______________ and _______________.

Why must they be supplementary? _______________________________________________

PAGE 7 — Touching is optional

Complementary Without Touching

FIGURE: fig6-nonadjacent-complementary.png (full width)

Are $25^\circ$ and $65^\circ$ complementary? ______ Are they adjacent? ______

Explain both answers: _______________________________________________
Can an obtuse angle have a complement? ______ Why?
Find the complement of each.

a) $5^\circ$	b) $41^\circ$	c) $68.5^\circ$	d) $89^\circ$

Find the supplement of each.

a) $15^\circ$	b) $90^\circ$	c) $117^\circ$	d) $179^\circ$

PAGE 8 — Describing one angle with the other

9.2 Practice

Two complementary angles, one three times the other.

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
Two supplementary angles, one five times the other.

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
$\angle A$ and $\angle B$ complementary, $m\angle A = 54^\circ$ . $m\angle B =$ ______
$\angle C$ and $\angle D$ supplementary, $m\angle C = 88^\circ$ . $m\angle D =$ ______
Complementary, supplementary, or neither?

a) $47^\circ$ and $43^\circ$	b) $62^\circ$ and $118^\circ$	c) $30^\circ$ and $60^\circ$	d) $105^\circ$ and $85^\circ$

Two complementary angles with equal measures. Each measures ______

PAGE 9 — Apply it and check your thinking

9.2 Apply and Explain

Explain. Why are two right angles always supplementary but never complementary?
Apply it. A kite string makes a $68^\circ$ angle with level ground. A flagpole stands straight up beside it.

Relationship used: ____________________ Angle with the flagpole: ______
Apply it. A staircase handrail meets a vertical post at $55^\circ$ .

Angle with the floor: ______
Find the error. A student says the supplement of $40^\circ$ is $50^\circ$ .

What went wrong? _______________________________________________

Correct supplement: ______

Exit ticket 9.2

Complement of $73^\circ$ : ______
Supplement of $73^\circ$ : ______
Two supplementary angles with equal measures: ______ each, and each is a ____________ angle.
Why does every obtuse angle have a supplement but no complement?

PAGE 10 — Four angles from two lines

9.3 Vertical Angles

FIGURE: fig7-vertical-angles.png (full width)

Fill in the blanks.

Vertical angles are the two angles that sit ____________ each other at the intersection. They share only the ____________.

Vertical angles are never ____________, because they share no side.

Each pair of neighbors at an intersection is a ____________ pair, so each neighboring pair adds to ______.

The two pairs of vertical angles: _______________ and _______________
All four linear pairs:

FIGURE: fig8-why-vertical-equal.png (full width)

$m\angle 1 = 55^\circ$ . Then $m\angle 2 =$ ______, $m\angle 3 =$ ______, $m\angle 4 =$ ______
If $m\angle 1 = 90^\circ$ : the other three are ______, ______, ______, and the lines are ____________.

PAGE 11 — Why they are equal

One Measure Determines All Four

Explain, using linear pairs, why $m\angle 1 = m\angle 3$ .

$m\angle 1 = 180^\circ -$ ______ and $m\angle 3 = 180^\circ -$ ______, so ____________________
Are vertical angles ever adjacent? ______ Why?
One angle at an intersection is $42^\circ$ . The other three: ______, ______, ______
One angle at an intersection is $118^\circ$ . The other three: ______, ______, ______
Vertical angles $(3x)^\circ$ and $51^\circ$ . Equation: ____________ $x =$ ______
Vertical angles $(7x - 4)^\circ$ and $(5x + 12)^\circ$ .

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Each angle: ______
All four angles equal. Each measures ______, and the lines are ____________.
$\angle 1$ and $\angle 3$ are vertical and $m\angle 1 + m\angle 3 = 130^\circ$ . Each: ______

PAGE 12 — Apply it and check your thinking

9.3 Apply and Explain

Explain. Why can a pair of vertical angles be supplementary only when both are $90^\circ$ ?
Apply it.

FIGURE: fig14-context-scissors.png (full width)

 $x =$  ______  Relationship used: ____________________

 $y =$  ______  Relationship used: ____________________

Explain. Two intersecting lines form one obtuse angle. How many of the four are obtuse? ______

Justify: _______________________________________________
Find the error. A student writes " $\angle 1$ and $\angle 2$ are vertical angles, so $m\angle 1 + m\angle 2 = 180^\circ$ ."

What went wrong? _______________________________________________

Counterexample: _______________________________________________

Exit ticket 9.3

One angle at an intersection is $73^\circ$ . The other three: ______, ______, ______
Vertical angles $(5x)^\circ$ and $85^\circ$ . $x =$ ______
Difference between vertical and adjacent angles:
Why can a pair of vertical angles never be adjacent?

PAGE 13 — Three steps, every time

9.4 Solving for Unknown Angle Measures

The routine.

Step one: find the ____________ — a straight line means ____________, a right angle means ____________, two lines crossing means ____________.
Step two: write the ____________.
Step three: solve, then ____________ back and check.

FIGURE: fig9-supplementary-equation.png (full width)

Equation: ____________________________

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Measures: ______ and ______ Check: ______ $+$ ______ $= 180$

FIGURE: fig10-complementary-equation.png (full width)

Equation: ____________________________

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Measures: ______ and ______ Check: ______ $+$ ______ $= 90$

PAGE 14 — Vertical and multi-angle diagrams

Equations from a Diagram

FIGURE: fig11-vertical-equation.png (full width)

Equation: ____________________________

$x =$ ______ Each marked angle: ______

FIGURE: fig12-three-angles-line.png (full width)

Equation: ____________________________

$x =$ ______ Middle angle: ______

FIGURE: fig13-context-ramp.png (full width)

Relationship: ____________________ Marked angle: ______

PAGE 15 — Guided practice

9.4 Guided Practice

Supplementary: $(2x)^\circ$ and $(4x + 12)^\circ$ .

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Measures: ______ and ______
Complementary: $(x + 15)^\circ$ and $(2x)^\circ$ .

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
Vertical: $(6x + 5)^\circ$ and $(8x - 21)^\circ$ .

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Each measure: ______

PAGE 16 — Independent practice

9.4 Independent Practice

Supplementary: $(3x - 6)^\circ$ and $(x + 10)^\circ$ . $x =$ ______ Measures: ______ and ______
Complementary: $(5x)^\circ$ and $(x + 12)^\circ$ . $x =$ ______ Measures: ______ and ______
Vertical: $(9x - 14)^\circ$ and $(7x + 2)^\circ$ . $x =$ ______ Each: ______
Along a line: $55^\circ$ , $(3x)^\circ$ , $65^\circ$ . $x =$ ______ Middle angle: ______
Two complementary angles, one $18^\circ$ more than the other.

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
Two supplementary angles, one $24^\circ$ less than three times the other.

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______

PAGE 17 — Apply it and check your thinking

9.4 Apply and Explain

Explain. The supplement of an angle is four times its complement.

Equation: ____________________________ Angle: ______

Check: supplement ______, complement ______, and $4 \times$ ______ $=$ ______
Apply it. A skateboard ramp rises from level ground at $22^\circ$ . A vertical post runs to the underside of the ramp.

Relationship: ____________________ Angle between ramp and post: ______
Apply it. Two straight streets cross. The acute angle is $(4x)^\circ$ and the obtuse angle beside it is $(5x + 45)^\circ$ .

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Four angles: ______, ______, ______, ______
Apply it. Three adjacent angles along one straight tile edge: $30^\circ$ , $(2x)^\circ$ , $(x + 15)^\circ$ .

$x =$ ______ Measures: ______, ______, ______
Find the error. Told $(3x)^\circ$ and $(x + 20)^\circ$ are complementary, a student writes $4x + 20 = 180$ and gets $x = 40$ .

What went wrong? _______________________________________________

Correct: $x =$ ______ Measures: ______ and ______
Explain. Why does a straight line in a diagram let you write an equation even when no measure is marked on it?

Exit ticket 9.4

Supplementary: $(4x + 10)^\circ$ and $(2x + 20)^\circ$ . $x =$ ______ Measures: ______ and ______
Complementary: $(7x)^\circ$ and $(2x)^\circ$ . $x =$ ______ Measures: ______ and ______
Vertical: $(5x - 12)^\circ$ and $(3x + 8)^\circ$ . $x =$ ______ Each: ______
Set the equation equal to what?

complementary	supplementary	vertical

PAGE 18 — Chapter 9 review, part 1

Chapter 9 Review

Part A · Identifying and describing the relationships

Describe each relationship in one sentence.

Vertical: _______________________________________________

Adjacent: _______________________________________________

Supplementary: _______________________________________________

Complementary: _______________________________________________

FIGURE: fig15-review-angles-on-line.png (full width)

An adjacent pair: _______________ and _______________

A complementary pair: _______________ and _______________

A supplementary pair: _______________ and _______________
$a =$ ______ (relationship: ____________) $b =$ ______ (relationship: ____________)

PAGE 19 — Chapter 9 review, part 2

Chapter 9 Review (continued)

FIGURE: fig16-review-intersection.png (full width)

The two pairs of vertical angles: _______________ and _______________
$m\angle 2 =$ ______ $m\angle 3 =$ ______ $m\angle 4 =$ ______
Which numbered angles are adjacent to $\angle 1$ in the Lesson 9.3 four-angle figure? ______
Always, sometimes, or never also adjacent?

vertical	supplementary	complementary

True or false: every supplementary pair is adjacent. ______ Reason:
How can two angles be adjacent but neither complementary nor supplementary? Example:
Which panel of the Lesson 9.1 two-panel figure shows adjacent angles, and why not the other?

PAGE 20 — Chapter 9 review, part 3

Chapter 9 Review (continued)

Part B · Solving for unknown angle measures

Complement of $26^\circ$ : ______ Supplement of $26^\circ$ : ______
Supplement of $149^\circ$ : ______
Supplementary: $(2x + 5)^\circ$ and $(3x - 20)^\circ$ . $x =$ ______ Measures: ______ and ______
Complementary: $(4x + 2)^\circ$ and $(x - 7)^\circ$ . $x =$ ______ Measures: ______ and ______
Vertical: $(10x - 3)^\circ$ and $(7x + 18)^\circ$ . $x =$ ______ Each: ______
Along a line: $x^\circ$ , $62^\circ$ , $(2x + 4)^\circ$ . $x =$ ______ Measures: ______, ______, ______
One angle at an intersection is $33^\circ$ . The other three: ______, ______, ______

PAGE 21 — Chapter 9 review, part 4

Chapter 9 Review (continued)

Two complementary angles in the ratio $2 : 3$ .

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
Two supplementary angles that differ by $46^\circ$ .

WORK SPACE: 1.5 in tall, full width

Measures: ______ and ______
An angle is three times its own supplement.

WORK SPACE: 1.5 in tall, full width

Angle: ______ Supplement: ______
Along a line: $25^\circ$ , $35^\circ$ , $(3x)^\circ$ , $30^\circ$ . $x =$ ______ Unknown angle: ______
Vertical: $(x + 40)^\circ$ and $(3x)^\circ$ . $x =$ ______ Each: ______

PAGE 22 — Chapter 9 review, part 5

Chapter 9 Review (continued)

Part C · Application and reasoning

Apply it. The ramp is rebuilt to rise at $34^\circ$ instead of $28^\circ$ .

Marked angle: ______ Relationship: ____________________
Apply it. The scissors open so the marked angle is $48^\circ$ . $x =$ ______ $y =$ ______
Apply it. Two roads cross. One angle is $(6x - 9)^\circ$ and the angle vertical to it is $(4x + 15)^\circ$ .

WORK SPACE: 1.5 in tall, full width

$x =$ ______ Four angles: ______, ______, ______, ______
Explain. The supplement of an angle is three times its complement.

Equation: ____________________________ Angle: ______

Check: supplement ______, complement ______
Find the error. " $\angle 1$ and $\angle 2$ are vertical angles, so $m\angle 1 + m\angle 2 = 180^\circ$ ."

What went wrong? _______________________________________________

The student was thinking of a ____________ pair. Correct statement:
Explain. How do you decide which equation to write? Name what you look for in each case.

Straight line means _______________ Right angle means _______________ Crossing lines mean _______________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Figure widths: every angle diagram is full width except pages 6, where fig4-complementary-adjacent.png and fig5-supplementary-linear-pair.png sit side by side at half width. The two-panel figures (fig3-adjacent-or-not.png, fig6-nonadjacent-complementary.png) are wide and short; give them their own block with no text beside them.
Do not redraw the diagrams. Every figure is generated from its stated measures by make_grade8_ch09.py, so a hand-redrawn version risks showing an angle that contradicts its label. Place the PNG as-is and never rescale one axis only.
Degree signs: set every measure with a true degree symbol, not a superscript letter o.
Work space: plain white boxes with a hairline border. Items where a student writes and solves an equation need the full 1.5 in; single-blank items do not.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.
Optional protractor page: if the workbook is printed as a packet, a blank protractor practice page can be added after page 2, but no chapter item depends on physical measurement.