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.

  • Aπ‘šβˆ π΄π‘‚π΅=38∘, π‘šβˆ π΄π‘‚πΈ=38∘, π‘šβˆ π·π‘‚πΈ=142∘
  • Bπ‘šβˆ π΄π‘‚π΅=38∘, π‘šβˆ π΄π‘‚πΈ=142∘, π‘šβˆ π·π‘‚πΈ=38∘
  • Cπ‘šβˆ π΄π‘‚π΅=142∘, π‘šβˆ π΄π‘‚πΈ=38∘, π‘šβˆ π·π‘‚πΈ=38∘
  • Dπ‘šβˆ π΄π‘‚π΅=142∘, π‘šβˆ π΄π‘‚πΈ=38∘, π‘šβˆ π·π‘‚πΈ=142∘

Q4:

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

  • Aπ‘šβˆ 6=23∘, π‘šβˆ 7=203∘
  • Bπ‘šβˆ 6=23∘, π‘šβˆ 7=67∘
  • Cπ‘šβˆ 6=113∘, π‘šβˆ 7=67∘
  • Dπ‘šβˆ 6=203∘, π‘šβˆ 7=23∘
  • Eπ‘šβˆ 6=67∘, π‘šβˆ 7=113∘

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:

⃖⃗𝑃𝐿 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‘.

  • Aπ‘šβˆ π‘Ž=21∘, π‘šβˆ π‘=69∘
  • Bπ‘šβˆ π‘Ž=132∘, π‘šβˆ π‘=48∘
  • Cπ‘šβˆ π‘Ž=54∘, π‘šβˆ π‘=126∘
  • Dπ‘šβˆ π‘Ž=69∘, π‘šβˆ π‘=21∘
  • Eπ‘šβˆ π‘Ž=48∘, π‘šβˆ π‘=132∘

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∘.

  • Aπ‘₯βˆ’40=90, π‘₯=130
  • Bπ‘₯+40βˆ’45=180, π‘₯=185
  • Cπ‘₯+40=45, π‘₯=85
  • Dπ‘₯+40=180, π‘₯=140
  • Eπ‘₯+40=90, π‘₯=50

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.

  • Aπ‘šβˆ π΄π΅π·=59∘, π‘šβˆ π·π΅πΆ=54∘
  • Bπ‘šβˆ π΄π΅π·=24∘, π‘šβˆ π·π΅πΆ=66∘
  • Cπ‘šβˆ π΄π΅π·=93∘, π‘šβˆ π·π΅πΆ=110∘
  • Dπ‘šβˆ π΄π΅π·=50∘, π‘šβˆ π·π΅πΆ=40∘
  • Eπ‘šβˆ π΄π΅π·=84∘, π‘šβˆ π·π΅πΆ=95∘

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‘₯.

  • A4π‘₯+12=180
  • B4π‘₯+140=180
  • C4π‘₯+12=128
  • D4π‘₯+116=180
  • E4π‘₯+116=128

Solve for π‘₯.

  • Aπ‘₯=3
  • Bπ‘₯=10
  • Cπ‘₯=42
  • Dπ‘₯=16
  • Eπ‘₯=29

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‘₯.

  • A9π‘₯=90
  • B9π‘₯=270
  • C9π‘₯=139
  • D9π‘₯=41
  • E9π‘₯=180

Find the value of π‘₯.

  • Aπ‘₯=10
  • Bπ‘₯=20
  • Cπ‘₯=15
  • Dπ‘₯=5
  • Eπ‘₯=30

Q17:

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

  • Aπ‘₯=18, 𝑦=22
  • Bπ‘₯=18, 𝑦=24
  • Cπ‘₯=50, 𝑦=130
  • Dπ‘₯=17, 𝑦=23
  • Eπ‘₯=10, 𝑦=32

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.

  • Aπ‘šβˆ πΈ=156∘, π‘šβˆ πΉ=24∘
  • Bπ‘šβˆ πΈ=78∘, π‘šβˆ πΉ=102∘
  • Cπ‘šβˆ πΈ=24∘, π‘šβˆ πΉ=156∘
  • Dπ‘šβˆ πΈ=102∘, π‘šβˆ πΉ=78∘
  • Eπ‘šβˆ πΈ=57∘, π‘šβˆ πΉ=33∘

Q19:

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.

Q20:

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

  • Aπ‘₯=5, 𝑦=29
  • Bπ‘₯=25, 𝑦=65
  • Cπ‘₯=5, 𝑦=65
  • Dπ‘₯=30, 𝑦=29
  • Eπ‘₯=8, 𝑦=29

Q21:

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‘₯.

  • A10π‘₯+20=90
  • B10π‘₯+20=180
  • C2π‘₯+20=180
  • D2π‘₯+20=90
  • E4π‘₯+12=90

Solve for π‘₯.

  • Aπ‘₯=35
  • Bπ‘₯=7
  • Cπ‘₯=16
  • Dπ‘₯=19.5
  • Eπ‘₯=8

Q22:

Answer the following questions using the given diagram.

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

  • A3π‘₯+48=90
  • B3π‘₯+44=90
  • C3π‘₯+48=46
  • D3π‘₯+46=90
  • E3π‘₯+2=46

Find the value of π‘₯.

  • Aπ‘₯=30
  • Bπ‘₯=14
  • Cπ‘₯=16
  • Dπ‘₯=2
  • Eπ‘₯=42

Q23:

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‘₯.

  • A4π‘₯=90∘
  • B2π‘₯=90∘
  • C2π‘₯=180∘
  • D3π‘₯=90∘
  • E3π‘₯=180∘

Solve for π‘₯.

  • Aπ‘₯=30∘
  • Bπ‘₯=22.5∘
  • Cπ‘₯=45∘
  • Dπ‘₯=90∘
  • Eπ‘₯=60∘

Q24:

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

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

  • A6π‘₯=90
  • B11π‘₯+2=90
  • C11π‘₯+2=180
  • D5π‘₯+2=90
  • E6π‘₯=180

Find the value of π‘₯.

  • Aπ‘₯=8
  • Bπ‘₯=18
  • Cπ‘₯=30
  • Dπ‘₯=15
  • Eπ‘₯=16

Find the value of 𝑦.

  • A𝑦=30
  • B𝑦=42
  • C𝑦=132
  • D𝑦=138
  • E𝑦=48

Find the value of 𝑧.

  • A𝑧=30
  • B𝑧=48
  • C𝑧=138
  • D𝑧=150
  • E𝑧=132

Q25:

Quadrilateral 𝐽𝐾𝐿𝑀 is a rectangle. If π‘šβˆ πΎπ½πΏ=(4π‘₯+7)∘ and π‘šβˆ π½πΏπΎ=(9π‘₯+5)∘, find π‘₯.

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