Practice the problems of Math in Focus Grade 7 Workbook Answer Key Chapter 6 Lesson 6.1 Complementary, Supplementary, and Adjacent Angles to score better marks in the exam.

## Math in Focus Grade 7 Course 2 B Chapter 6 Lesson 6.1 Answer Key Complementary, Supplementary, and Adjacent Angles

### Math in Focus Grade 7 Chapter 6 Lesson 6.1 Guided Practice Answer Key

**Technology Activity**

**Materials:**

- geometry software

Explore The Relationship Of Complementary Angles Using Geometry Software

Work in pairs.

Step 1.

Construct \(\overline{\mathrm{AB}}\). Then construct a second line segment, \(\overline{\mathrm{BC}}\), that is perpendicular to \(\overline{\mathrm{AB}}\). Finally, construct line segment \(\overline{\mathrm{BD}}\).

Step 2.

Select ∠ABC and find

Step 3.

Select ∠ABD and find its measure. Then select ∠DBC and find its measure.

Step 4.

Use the calculate function of the program to find the sum of the measures of ∠ABD and ∠DBC. What do you notice about the sum of their measures?

Step 5.

Select the point D and drag it so that you change the measures of ∠ABD and ∠DBC. Record your results in a table as shown below.

Step 6.

As the angle measures change, how does the sum of the angle measures change?

**Math Journal**

Describe what you notice about the sum of the measures of complementary angles.

**Solve.**

Question 1.

Name three pairs of complementary angles.

Answer:

m∠ABC = 61° We have:

m∠PQR = 29°

m∠ABC + m∠PQR = 61° + 29° = 90° we add the measures of the two angles:

Since the sum of their measures is 90° ∠ABC and ∠PQR are complementary angles

m∠DEF = 38° We have:

m∠MNO = 52°

m∠DEF + m∠MNO = 38° + 52° = 90° we add the measures of the two angles:

Since the sum of their measures is 90°, ∠DEF and ∠MNO are complementary angles

m∠GIH = 21° We have:

m∠JKL = 69°

m∠GIH + m∠JKL = 21° + 69° = 90° We add the measures of the two angles:

Since the sum of their measures is 90°, ∠GIH and ∠JKL are complementary angles

∠ABC and ∠PQR

∠DEF and ∠MNO

∠GIH and ∠JKL

**Copy and complete the table.**

Question 2.

Angles A and 8 are complementary. Find m∠B for each measure of ∠A.

Answer:

m∠A + m∠B = 90° We are given:

We determine each measure of ∠A:

62° ;17° ;54° ; 75°

**Technology Activity**

**Materials:**

- geometry software

Explore The Relationship Of Supplementary Angles Using Geometry Software

Work in pairs.

Step 1.

Construct \(\overline{P R}\). Then construct a second line segment, \(\overline{S Q}\), as shown below.

Step 2.

Select ∠PQR and find its measure.

Step 3.

Select ∠SQP and find its measure. Then select ∠SQR and find its measure.

Step 4.

Use the calculate function of the program to find the sum of the measures of ∠SQP and ∠SQR. What do you notice about the sum of their measures?

Step 5.

Select the point S and drag it so that you change the measures of ∠SQP and ∠SQR. Record your results in a table as shown below.

Step 6.

As the angle measures change, how does the sum of the angle measures change?

**MathJournal**

Describe what you notice about the sum of the measures of supplementary angles.

**Solve.**

Question 3.

Tell whether each pair of angles is supplementary.

a) m∠X = 32° and m∠Y = 108°

Answer:

m∠X = 32° a) We are given:

m∠Y = 108°

m∠X + m∠Y = 32° + 108° = 140° Add the measures of the two angles:

As m∠X + m∠Y ≠ 180°. ∠X and ∠Y are not supplementary

b) m∠A = 45° and m∠B = 45°

Answer:

m∠A = 45° b) We are given:

m∠B = 45°

m∠A + m∠B = 45° + 45° = 90° Add the measures of the two angles:

As m∠A + m∠B ≠ 180°, ∠A and ∠B are not supplementary.

c) m∠D = 12° and m∠E = 168°

Answer:

m∠D = 12° c) We are given:

m∠E = 168°

m∠D + m∠E = 12° + 168° = 180° Add the measures of the two angles:

As m∠D + m∠E = 180°, ∠D and ∠E are supplementary.

d) m∠V = 85° and m∠W = 95°

Answer:

m∠V = 85° d) We are given:

m∠W = 95°

m∠V + m∠W = 85° + 95° = 180° Add the measures of the two angles:

As m∠V + m∠W = 180°. ∠V and ∠W are supplementary.

**Copy and complete the table.**

Question 4.

Angles A and B are supplementary. Find m∠B for each measure of ∠A.

Answer:

m∠A + m∠B = 180° We are given:

We determine each measure of ∠A:

98°; 154°; 44°; 75°

**Complete.**

Question 5.

In the diagram, m∠ABC = 90°. Find the value of x.

Answer:

We are given

m∠ABD + m∠DBC = 90° Complementary angles:

x° + 23° = 90° Substitute:

x + 23° – 23° = 90° — 23° Subtract 23° from bothsides:

x = 67 Simplify:

x = 67

Given that \(\overleftrightarrow{P Q}\) is a straight line, find the value of y.

Question 6.

Answer:

we are given

m∠PQR + m∠ROQ = 180° Adjacent angles on a straight line:

37° + y° = 180° Substitute:

37° + y° – 37° = 180° – 37° Subtract 37° from both sides:

y = 143 Simplify:

y = 143

Question 7.

Answer:

we are given

m∠POS + m∠SQR + m∠ROQ = 180° Adjacent angles on a straight line:

2y° + 70° + 3y° = 180° substitute

5y° + 70° = 180° Simplify:

5y° + 70° – 70° = 180° — 70° Subtract 70° from both sides:

5g = 110 Simplify:

\(\frac{5 y}{5}\) = \(\frac{110}{5}\) Divide both sides by 5:

y = 22 Simplify:

y = 22

**Complete.**

Question 8.

In the diagram, m∠PQR = 90° and the ratio x : y = 1: 4. Find the values of x and y.

Method I

Use bar models.

Answer:

m∠PQR = 90°

x : y = 1 : 4

we are given

x° + y° = 90° Method 1; use bar models

Complementary angles:

We use bar models:

5 units → 90 We have:

1 unit → \(\frac{90}{5}\) = 18

x = 18

y = 4 ∙ 18 = 72

Method 2; use a variable to represent the measure of the angle

y = 4 • x = 4x The ratio x : y = 1 : 4 So we have:

x° + y° = 90° Complementary angles

x + 4x = 90 Substitute:

5x = 90 Simplify:

\(\frac{5 x}{5}\) = \(\frac{90}{5}\) Divide both sides by 5:

x = 18 Simplify:

y = 4 • x Substitute:

= 4 • 18

= 72 Simptify:

x = 18; y = 72

### Math in Focus Course 2B Practice 6.1 Answer Key

**Tell whether each pair of angles is complementary.**

Question 1.

m∠A = 25° and m∠B = 65°

Answer:

m∠A = 25° We are given:

m∠B = 65°

m∠A + m∠B = 25° + 65° = 90° We compute the sum:

Because the sum of the measures of the two angles is exactly 90°, the given angles are complementary.

Yes

Question 2.

m∠C = 105° and m∠D = 7°

Answer:

m∠C = 105° We are given:

m∠D = 7°

m∠C + m∠D = 105° + 7° = 112° We compute the sum:

Because the sum of the measures of the two angles is not exactly 90°, the given angles are not complementary.

No

Question 3.

m∠E = 112° and m∠F = 68°

Answer:

m∠E = 112° We are given:

m∠F = 68°

m∠E + m∠F = 112° + 68° = 180° We compute the sum:

Because the sum of the measures of the two angles is not exactly 90° the given angles are not complementary.

No

Question 4.

m∠G = 45° and m∠H = 45°

Answer:

m∠G = 45° We are given:

m∠H = 45°

m∠G + m∠H = 45° + 45° = 90° We compute the sum:

Because the sum of the measures of the two angles is exactly 90°, the given angles are complementary.

Yes

**Tell whether each pair of angles is supplementary.**

Question 5.

m∠A = 130° and m∠8 = 50°

Answer:

m∠A = 130° We are given:

m∠B = 50°

m∠A + m∠B = 130° + 50° = 180° We compute the sum:

Because the sum of the measures of the two angles is exactly 180°, the given angles are supplementary.

Yes

Question 6.

m∠C = 90° and m∠D = 80°

Answer:

m∠C = 90° We are given:

m∠D = 80°

m∠C + m∠D = 90° + 80° = 170° We compute the sum:

Because the sum of the measures of the two angles is exactly 180°, the given angles are not supplementary.

No

Question 7.

m∠E = 120° and m∠F = 60°

Answer:

m∠E = 120° We are given:

m∠F = 60°

m∠E + m∠F = 120° + 60° = 180° We compute the sum:

Because the sum of the measures of the two angles is exactly 180°, the given angles are supplementary.

yes

Question 8.

m∠G = 60° and m∠H = 30°

Answer:

m∠G = 60° We are given:

m∠H = 30°

m∠G + m∠H = 60° + 30° = 90° We compute the sum:

Because the sum of the measures of the two angles is exactly 180°, the given angles are not supplementary.

yes

**Find the measure of the complement of the angle with the given measure.**

Question 9.

19°

Answer:

m∠θ = 19° We are given:

90° – m∠θ = 90° – 19 = 71° We determine the measure of the complement of the angle with the given

measure:

71°

Question 10.

64°

Answer:

m∠θ = 64° We are given:

90° – m∠θ = 90° – 64° = 26° We determine the measure of the complement of the angle with the given

measure:

26°

Question 11.

7°

Answer:

m∠θ = 7° We are given:

90° – m∠θ = 90° – 7° = 83° We determine the measure of the complement of the angle with the given

measure:

83°

Question 12.

35°

Answer:

m∠θ = 35° We are given the angle:

90° – m∠θ = 90° – 35° = 55° We determine the measure of the complement of the angle with the given

measure:

55°

**Find the measure of the supplement of the angle with the given measure.**

Question 13.

78°

Answer:

m∠θ = 78° We are given the angle:

180° – m∠θ = 180° – 78° = 102° We determine the measure of the complement of the angle with the given measure:

102°

Question 14.

4°

Answer:

m∠θ = 4° We are given the angle:

180° – m∠θ = 180° – 78° = 102° We determine the measure of the complement of the angle with the given measure:

176°

Question 15.

153°

Answer:

m∠θ = 153° We are given the angle:

180° – m∠θ = 180° – 153° = 27° We determine the measure of the complement of the angle with the given measure:

27°

Question 16.

101°

Answer:

m∠θ = 101° We are given the angle:

180° – m∠θ = 180° – 101° = 79° We determine the measure of the complement of the angle with the given measure:

79°

**∠ABD and ∠DBC are complementary angles. Find the value of x.**

Question 17.

Answer:

m∠ABD + m∠DBC = 90° ∠ABD and ∠DBC complementary:

65 + x = 90 Substitute:

65 + x – 65 = 90 – 65 Subtract 65 from both sides:

x = 25 Simplify:

x = 25

Question 18.

Answer:

m∠ABD + m∠DBC = 90° ∠ABD and ∠DBC complementary:

8 + x = 90 Substitute:

8 + x – 8 = 90 – 8 Subtract 8 from both sides:

x = 82 Simplify:

x = 82

**∠PQS and ∠SQR are complementary angles. Find the value of m.**

Question 19.

Answer:

m∠PQS + m∠SQR = 90° ∠PQS and ∠SQR supplementary:

m + 29 = 180 Substitute:

m + 29 – 29 = 180 – 29 Subtract 29 from both sides:

x = 151 Simplify:

x = 151

Question 20.

Answer:

m∠PQS + m∠SQR = 180° ∠PQS and ∠SQR supplementary:

152 + m = 180 Substitute:

152 + m – 152 = 180 – 152 Subtract 152 from both sides:

m = 28 Simplify:

m = 28

**Answer each of the following.**

Question 21.

The measure of an angle is 7°. Find the measure of its complement.

Answer:

m∠θ = 7° We are given the angle:

90° – m∠θ = 90° – 7° = 83° We determine the measure of the complement of the angle with the given measure:

83°

Question 22.

The measure of an angle is 84°. Find the measure of its supplement.

Answer:

m∠θ = 84° We are given the angle:

180° – m∠θ = 180° – 84° = 96° We determine the measure of the supplement of the angle with the given measure:

83°

Question 23.

**Math Journal**

a) Find the measures of the complement and the supplement of each of the following angles, where possible.

m∠W = 2° m∠X = 40° m∠Y = 32° m∠Z = 115°

Answer:

m∠W = 2° a) we are given the angle:

90° – m∠W = 90° – 2° = 88° We determine the complement of ∠W:

180° – m∠W = 180° – 2° = 178° We determine the supplement of ∠W:

m∠X = 40° We are given the angle:

90° — m∠X = 90° – 40° = 50° We determine the complement of ∠X:

180° — m∠X = 180° – 40° = 140° We determine the supplement of ∠X:

m∠Y = 32° We are given the angle:

90° — m∠Y = 90° — 32° = 58° We determine the complement of ∠Y:

180° — m∠Y = 180° — 32° = 148° We determine the supplement of ∠Y:

m∠Z = 115° We are given the angle:

As m∠Z > 90°: ∠Z has no complement

180° — m∠Z = 180 — 115° = 65° We determinethe supplement of ∠Z:

b) Which angle in a) does not have both a complement and a supplement?

Answer:

∠Z does not have both complement and supplement

c) In general, what must be true about the measure of an angle that has both a complement and a supplement?

Answer:

In order that an angle has both complement and supplement, its measure must be greater or equal than 0° and less or equal than 90°.

Question 24.

**Math Journal** Identify all the angles in each diagram. Tell which angles are adjacent. Explain your reasoning.

The measure of ∠ABC = 90°. Find the value of x.

Answer:

∠GOH We identify all the angles in the first diagram:

∠HOK

∠GOK

∠GOH and ∠HOK The adjacent angles in the diagram is the pair of angles which have a common side and a common vertex and don’t overlap:

∠WOX We identify all the angles in the second diagram:

∠XOY

∠YOZ

∠WOY

∠XOZ

∠WOZ

∠WOX and ∠XOY (common side OX, common vertex O) We identify the adjacent angles in the second diagram:

∠XOY and ∠YOZ (common side OY, common vertex O)

∠WOY and ∠YOZ (common side OY, common vertex O)

∠WOX and ∠XOZ (common side OX, common vertex O)

Question 25.

Answer:

m∠ABC = 90° We are given:

x° + 42° + 30° = 90° we have

x + 72° = 90°

x + 72 – 72 = 90 – 72 Subtract 72 from both sides

x = 18 simplify

x = 18

Question 26.

Answer:

m∠ABC = 90° We are given:

2x° + 22° = 90° we have

2x + 22 – 22 = 90 – 22 Subtract 22 from both sides

2x = 68 simplify

\(\frac{2 x}{2}\) = \(\frac{68}{2}\) Divide by 2:

x = 34 simplify:

x = 34

Question 27.

Answer:

m∠ABC = 90° We are given:

2x° + 45° + 21° = 90° we have

2x° + 66 = 90°

2x + 66 — 66 = 90 — 66 Subtract 66 from both sides:

2x = 24 Simplify:

\(\frac{2 x}{2}\) = \(\frac{24}{2}\) Divide by 2:

x = 12 Simplify:

x = 12

Question 28.

Answer:

m∠ABC = 90° We are given:

18° + 3x° + 21° = 90° we have

39° + 3x° = 90°

39 + 3x — 39 = 90 — 39 Subtract 39 from both sides:

3x = 51 Simplify:

\(\frac{3 x}{3}\) = \(\frac{51}{3}\) Divide by 3:

x = 17 Simplify:

x = 17

\(\overleftrightarrow{P R}\) is a straight line. Find the value of m.

Question 29.

Answer:

m∠PQR = 90° Because \(\overline{P R}\) is a straight line, we have

m∠PQT + m∠TQS + m∠SQR = m∠PQR we have

49° + m° + 57° = 180° substitute

106° + m° = 180°

106 + m – 106 = 180 – 106 subtract 106 from both sides

m = 74

Question 30.

Answer:

m∠PQR = 180° Because \(\overline{P R}\) is a straight line, we have

m∠PQT + m∠TQS + m∠SQR = m∠PQR we have

20° + m° + 87° = 180° substitute

150° + m° = 180°

150 + m – 150 = 180 – 150 subtract 150 from both sides

m = 30 Simplify:

m = 30

In the diagram, the ratio a: b = 2: 3. Find the values of a and b.

Question 31.

Answer:

a : b = 2 : 3 we are given

\(\frac{a}{b}\) = \(\frac{2}{3}\) we rewrite the ratio

3a = 2b cross-multiply

\(\frac{3 a}{3}\) = \(\frac{2 b}{3}\) Divide both sides by 3

a = \(\frac{2 b}{3}\)

m∠RQS + m∠SQP = 90° ∠RQS and ∠SQP are complementary

b + a = 90

b + \(\frac{2 b}{3}\) = 90 substitute

\(\frac{3 b+2 b}{3}\) = 90 Determine b

\(\frac{5 b}{3}\) = 90

5b = 3 ∙ 90

5b = 270

\(\frac{5 b}{5}\) = \(\frac{270}{5}\)

b = 54

a = \(\frac{2 \cdot 54}{3}\) = 36 Determine a

a = 36

b = 54

Question 32.

\(\overleftrightarrow{P R}\) is a straight line.

Answer:

a : b = 2 : 3 we are given

\(\frac{a}{b}\) = \(\frac{2}{3}\) we rewrite the ratio

3a = 2b cross-multiply

\(\frac{3 a}{3}\) = \(\frac{2 b}{3}\) Divide bothsides by 3

a = \(\frac{2 b}{3}\)

m∠RQS + m∠SQP = 180° ∠RQS and ∠SQP are supplementary

b + a = 180 substitute

b + \(\frac{2 b}{3}\) = 180

\(\frac{3 b+2 b}{3}\) = 180

\(\frac{5 b}{3}\) = 180

5b = 3 ∙ 180

5b = 540

\(\frac{5 b}{5}\) = \(\frac{540}{5}\)

b = 108

a = \(\frac{2 \cdot 108}{3}\) = 72

a = 72

b = 108

Solve.

Question 33.

The diagram shows the pattern on a stained glass window. \(\overleftrightarrow{A C}\) is a straight line. ∠EBD and ∠DBA are complementary angles and m∠DBA = 30°. Find the measures of ∠EBD and ∠CBD.

Answer:

we are given

m∠EBD + m∠DBA = 90° ∠EBD and ∠DBA complementary:

m∠EBD + 30° = 90° Substitute:

m∠EBD + 30° — 30° = 90° – 30° Subtract 30° from both sides:

m∠EBD = 60° Simplify:

m∠CBD + m∠DBA = 180° \(\overline{A C}\) straight line:

m∠CBD + 30° = 180° Substitute

m∠CBD + 30° – 30° = 180° – 30° Subtract 30° from both sides

m∠CBD = 150° Simplify

m∠EBD = 60°

m∠CBD = 150°

Question 34.

The diagram shows a kite. The two diagonals \(\overline{M P}\) and \(\overline{Q T}\) are perpendicular to each other. Identify all pairs of complementary angles and all pairs of supplementary angles that are not pairs of right angles.

Answer:

\(\overline{M P}\) ⊥ \(\overline{Q T}\) We are given:

∠MNR and ∠RNQ We identify the pairs of complementary angles:

∠QNS and ∠SNP

∠NMQ and ∠MQN

∠NQP and ∠QPN

∠TMN and ∠MTN

∠NTP and ∠NPT

∠MNR and ∠RNQ We identify the pairs of supplementary angLes that are not pairs of right angles:

∠MNS and ∠SNP

Complementary angles: ∠MNR and ∠RNQ

∠QNS and ∠SNP

∠NMQ and ∠MQN

∠NQP and ∠QPN

∠TMN and ∠MTN

∠NTP and ∠NPT

Supplementary angles: ∠MNR and ∠RNP

∠MNS and ∠SNP