Worksheet: Angle Relationships

In this worksheet, we will practice solving problems involving algebraic expressions and equations using the relationships between complementary, supplementary, adjacent, and vertically opposite angles.

Q1:

What is the measure of βˆ π‘πΉπ‘Œ?

Q2:

Find π‘šβˆ πΆ.

Q3:

Find π‘šβˆ π΄π‘‚π΅, π‘šβˆ π΄π‘‚πΈ, and π‘šβˆ π·π‘‚πΈ.

  • A π‘š ∠ 𝐴 𝑂 𝐡 = 3 8 ∘ , π‘š ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , π‘š ∠ 𝐷 𝑂 𝐸 = 1 4 2 ∘
  • B π‘š ∠ 𝐴 𝑂 𝐡 = 3 8 ∘ , π‘š ∠ 𝐴 𝑂 𝐸 = 1 4 2 ∘ , π‘š ∠ 𝐷 𝑂 𝐸 = 3 8 ∘
  • C π‘š ∠ 𝐴 𝑂 𝐡 = 1 4 2 ∘ , π‘š ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , π‘š ∠ 𝐷 𝑂 𝐸 = 3 8 ∘
  • D π‘š ∠ 𝐴 𝑂 𝐡 = 1 4 2 ∘ , π‘š ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , π‘š ∠ 𝐷 𝑂 𝐸 = 1 4 2 ∘

Q4:

Determine π‘šβˆ 6 and π‘šβˆ 7 in the given figure.

  • A π‘š ∠ 6 = 2 3 ∘ , π‘š ∠ 7 = 2 0 3 ∘
  • B π‘š ∠ 6 = 2 3 ∘ , π‘š ∠ 7 = 6 7 ∘
  • C π‘š ∠ 6 = 1 1 3 ∘ , π‘š ∠ 7 = 6 7 ∘
  • D π‘š ∠ 6 = 2 0 3 ∘ , π‘š ∠ 7 = 2 3 ∘
  • E π‘š ∠ 6 = 6 7 ∘ , π‘š ∠ 7 = 1 1 3 ∘

Q5:

Find π‘šβˆ π‘‹π΄πΆ.

Q6:

Determine π‘šβˆ π·π‘‹πΆ.

Q7:

If π‘šβˆ π‘₯=37∘ and π‘šβˆ π‘¦=54∘, find π‘šβˆ π‘§.

Q8:

If the measure of an angle’s supplement is 6 more than three times the measure of its complement, what is the measure of that angle?

Q9:

Given the following figure, find π‘šβˆ π΅π‘‚π΄.

Q10:

βƒ–    βƒ— 𝑃 𝐿 and ⃖⃗𝐽𝑀 are two intersecting straight lines. Given that π‘šβˆ πΏπ‘π‘€=(2π‘₯+2)∘ and π‘šβˆ π½π‘πΏ=(4π‘₯βˆ’11)∘, find π‘šβˆ π½π‘π‘ƒ.

Q11:

If βˆ π‘Ž and βˆ π‘ form a linear pair, where π‘šβˆ π‘Ž=(3π‘₯βˆ’6)∘ and π‘šβˆ π‘=(7π‘₯+6)∘, determine π‘šβˆ π‘Ž and π‘šβˆ π‘.

  • A π‘š ∠ π‘Ž = 2 1 ∘ , π‘š ∠ 𝑏 = 6 9 ∘
  • B π‘š ∠ π‘Ž = 1 3 2 ∘ , π‘š ∠ 𝑏 = 4 8 ∘
  • C π‘š ∠ π‘Ž = 5 4 ∘ , π‘š ∠ 𝑏 = 1 2 6 ∘
  • D π‘š ∠ π‘Ž = 6 9 ∘ , π‘š ∠ 𝑏 = 2 1 ∘
  • E π‘š ∠ π‘Ž = 4 8 ∘ , π‘š ∠ 𝑏 = 1 3 2 ∘

Q12:

These two angles are supplementary. Write and solve an equation to find π‘₯.

Recall that two angles are supplementary if the sum of their measures is 180∘.

  • A π‘₯ βˆ’ 4 0 = 9 0 , π‘₯ = 1 3 0
  • B π‘₯ + 4 0 βˆ’ 4 5 = 1 8 0 , π‘₯ = 1 8 5
  • C π‘₯ + 4 0 = 4 5 , π‘₯ = 8 5
  • D π‘₯ + 4 0 = 1 8 0 , π‘₯ = 1 4 0
  • E π‘₯ + 4 0 = 9 0 , π‘₯ = 5 0

Q13:

The rays 𝐴𝐡 and οƒͺ𝐡𝐢 are perpendicular. A point 𝐷 lies in the interior of ∠𝐴𝐡𝐢. Given that π‘šβˆ π΄π΅π·=(5π‘Ÿ+20)∘ and π‘šβˆ π·π΅πΆ=(8π‘Ÿβˆ’8)∘, find π‘šβˆ π΄π΅π· and π‘šβˆ π·π΅πΆ.

  • A π‘š ∠ 𝐴 𝐡 𝐷 = 5 9 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 5 4 ∘
  • B π‘š ∠ 𝐴 𝐡 𝐷 = 2 4 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 6 6 ∘
  • C π‘š ∠ 𝐴 𝐡 𝐷 = 9 3 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 1 1 0 ∘
  • D π‘š ∠ 𝐴 𝐡 𝐷 = 5 0 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 4 0 ∘
  • E π‘š ∠ 𝐴 𝐡 𝐷 = 8 4 ∘ , π‘š ∠ 𝐷 𝐡 𝐢 = 9 5 ∘

Q14:

In the given diagram, 𝐴𝐷 is a straight line, π‘šβˆ π΄π΅πΆ=(4π‘₯+12)∘, and π‘šβˆ π·π΅πΆ=128∘.

Form an equation that will allow you to calculate π‘₯.

  • A 4 π‘₯ + 1 2 = 1 8 0
  • B 4 π‘₯ + 1 4 0 = 1 8 0
  • C 4 π‘₯ + 1 2 = 1 2 8
  • D 4 π‘₯ + 1 1 6 = 1 8 0
  • E 4 π‘₯ + 1 1 6 = 1 2 8

Solve for π‘₯.

  • A π‘₯ = 3
  • B π‘₯ = 1 0
  • C π‘₯ = 4 2
  • D π‘₯ = 1 6
  • E π‘₯ = 2 9

Q15:

Given that π‘šβˆ π½=(3π‘₯βˆ’10)∘ and π‘šβˆ πΎ=(4π‘₯+6)∘, find π‘šβˆ πΎ.

Q16:

In the given diagram, 𝐴𝐡 and 𝐢𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate π‘₯.

  • A 9 π‘₯ = 9 0
  • B 9 π‘₯ = 2 7 0
  • C 9 π‘₯ = 1 3 9
  • D 9 π‘₯ = 4 1
  • E 9 π‘₯ = 1 8 0

Find the value of π‘₯.

  • A π‘₯ = 1 0
  • B π‘₯ = 2 0
  • C π‘₯ = 1 5
  • D π‘₯ = 5
  • E π‘₯ = 3 0

Q17:

Using the fact that the lines in the figure intersect, find the values of π‘₯ and 𝑦.

  • A π‘₯ = 1 8 , 𝑦 = 2 2
  • B π‘₯ = 1 8 , 𝑦 = 2 4
  • C π‘₯ = 5 0 , 𝑦 = 1 3 0
  • D π‘₯ = 1 7 , 𝑦 = 2 3
  • E π‘₯ = 1 0 , 𝑦 = 3 2

Q18:

If ∠𝐸 and ∠𝐹 are supplementary, and the measure of ∠𝐸 is 24∘ more than the measure of ∠𝐹, find the measure of each angle.

  • A π‘š ∠ 𝐸 = 1 5 6 ∘ , π‘š ∠ 𝐹 = 2 4 ∘
  • B π‘š ∠ 𝐸 = 7 8 ∘ , π‘š ∠ 𝐹 = 1 0 2 ∘
  • C π‘š ∠ 𝐸 = 2 4 ∘ , π‘š ∠ 𝐹 = 1 5 6 ∘
  • D π‘š ∠ 𝐸 = 1 0 2 ∘ , π‘š ∠ 𝐹 = 7 8 ∘
  • E π‘š ∠ 𝐸 = 5 7 ∘ , π‘š ∠ 𝐹 = 3 3 ∘

Q19:

Find 𝑀 and 𝑧.

  • A 𝑀 = 2 0 , 𝑧 = 1 2 . 5
  • B 𝑀 = 1 7 . 5 , 𝑧 = 2 1 . 2 5
  • C 𝑀 = 2 2 . 5 , 𝑧 = 3 . 7 5
  • D 𝑀 = 1 5 , 𝑧 = 3 0
  • E 𝑀 = 1 7 . 5 , 𝑧 = 2 1 . 5

Q20:

If 𝐸𝐢 is an altitude of △𝐴𝐸𝐷, π‘šβˆ πœƒ=(5π‘₯+2)∘, and π‘šβˆ πœ™=(3π‘₯+8)∘, find π‘šβˆ πœƒ.

Q21:

Find the values of π‘₯ and 𝑦 so that ⃖⃗𝑃𝑅 and ⃖⃗𝑆𝑄 are perpendicular.

  • A π‘₯ = 5 , 𝑦 = 2 9
  • B π‘₯ = 2 5 , 𝑦 = 6 5
  • C π‘₯ = 5 , 𝑦 = 6 5
  • D π‘₯ = 3 0 , 𝑦 = 2 9
  • E π‘₯ = 8 , 𝑦 = 2 9

Q22:

In the given diagram, 𝐴𝐷 is a straight line, π‘šβˆ π΄π΅πΆ=(4π‘₯+12)∘, and π‘šβˆ π·π΅πΆ=(6π‘₯+8)∘.

Form an equation that will allow you to calculate π‘₯.

  • A 1 0 π‘₯ + 2 0 = 9 0
  • B 1 0 π‘₯ + 2 0 = 1 8 0
  • C 2 π‘₯ + 2 0 = 1 8 0
  • D 2 π‘₯ + 2 0 = 9 0
  • E 4 π‘₯ + 1 2 = 9 0

Solve for π‘₯.

  • A π‘₯ = 3 5
  • B π‘₯ = 7
  • C π‘₯ = 1 6
  • D π‘₯ = 1 9 . 5
  • E π‘₯ = 8

Q23:

Answer the following questions using the given diagram.

Form an equation that will allow you to calculate π‘₯.

  • A 3 π‘₯ + 4 8 = 9 0
  • B 3 π‘₯ + 4 4 = 9 0
  • C 3 π‘₯ + 4 8 = 4 6
  • D 3 π‘₯ + 4 6 = 9 0
  • E 3 π‘₯ + 2 = 4 6

Find the value of π‘₯.

  • A π‘₯ = 3 0
  • B π‘₯ = 1 4
  • C π‘₯ = 1 6
  • D π‘₯ = 2
  • E π‘₯ = 4 2

Q24:

In the given diagram, ∠𝐴𝐡𝐢 is a right angle and π‘šβˆ π΄π΅π· is twice π‘šβˆ π·π΅πΆ. Let π‘šβˆ π·π΅πΆ=π‘₯.

Form an equation that will allow you to calculate π‘₯.

  • A 4 π‘₯ = 9 0 ∘
  • B 2 π‘₯ = 9 0 ∘
  • C 2 π‘₯ = 1 8 0 ∘
  • D 3 π‘₯ = 9 0 ∘
  • E 3 π‘₯ = 1 8 0 ∘

Solve for π‘₯.

  • A π‘₯ = 3 0 ∘
  • B π‘₯ = 2 2 . 5 ∘
  • C π‘₯ = 4 5 ∘
  • D π‘₯ = 9 0 ∘
  • E π‘₯ = 6 0 ∘

Q25:

In the given diagram, 𝐴𝐡 and 𝐢𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate π‘₯.

  • A 6 π‘₯ = 9 0
  • B 1 1 π‘₯ + 2 = 9 0
  • C 1 1 π‘₯ + 2 = 1 8 0
  • D 5 π‘₯ + 2 = 9 0
  • E 6 π‘₯ = 1 8 0

Find the value of π‘₯.

  • A π‘₯ = 8
  • B π‘₯ = 1 8
  • C π‘₯ = 3 0
  • D π‘₯ = 1 5
  • E π‘₯ = 1 6

Find the value of 𝑦.

  • A 𝑦 = 3 0
  • B 𝑦 = 4 2
  • C 𝑦 = 1 3 2
  • D 𝑦 = 1 3 8
  • E 𝑦 = 4 8

Find the value of 𝑧.

  • A 𝑧 = 3 0
  • B 𝑧 = 4 8
  • C 𝑧 = 1 3 8
  • D 𝑧 = 1 5 0
  • E 𝑧 = 1 3 2

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