Worksheet: Angle Relationships

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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 ∠𝑍𝐹𝑌?

Q2:

Find 𝑚∠𝐶.

Q3:

Find 𝑚∠𝐴𝑂𝐵, 𝑚∠𝐴𝑂𝐸, and 𝑚∠𝐷𝑂𝐸.

  • A 𝑚 ∠ 𝐴 𝑂 𝐵 = 3 8 ∘ , 𝑚 ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , 𝑚 ∠ 𝐷 𝑂 𝐸 = 1 4 2 ∘
  • B 𝑚 ∠ 𝐴 𝑂 𝐵 = 3 8 ∘ , 𝑚 ∠ 𝐴 𝑂 𝐸 = 1 4 2 ∘ , 𝑚 ∠ 𝐷 𝑂 𝐸 = 3 8 ∘
  • C 𝑚 ∠ 𝐴 𝑂 𝐵 = 1 4 2 ∘ , 𝑚 ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , 𝑚 ∠ 𝐷 𝑂 𝐸 = 3 8 ∘
  • D 𝑚 ∠ 𝐴 𝑂 𝐵 = 1 4 2 ∘ , 𝑚 ∠ 𝐴 𝑂 𝐸 = 3 8 ∘ , 𝑚 ∠ 𝐷 𝑂 𝐸 = 1 4 2 ∘

Q4:

Determine 𝑚∠6 and 𝑚∠7 in the given figure.

  • A 𝑚 ∠ 6 = 2 3 ∘ , 𝑚 ∠ 7 = 2 0 3 ∘
  • B 𝑚 ∠ 6 = 2 3 ∘ , 𝑚 ∠ 7 = 6 7 ∘
  • C 𝑚 ∠ 6 = 1 1 3 ∘ , 𝑚 ∠ 7 = 6 7 ∘
  • D 𝑚 ∠ 6 = 2 0 3 ∘ , 𝑚 ∠ 7 = 2 3 ∘
  • E 𝑚 ∠ 6 = 6 7 ∘ , 𝑚 ∠ 7 = 1 1 3 ∘

Q5:

Find 𝑚∠𝑋𝐴𝐶.

Q6:

Determine 𝑚∠𝐷𝑋𝐶.

Q7:

If 𝑚∠𝑥=37∘ and 𝑚∠𝑦=54∘, find 𝑚∠𝑧.

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 𝑚∠𝐵𝑂𝐴.

Q10:

⃖ ⃗ 𝑃 𝐿 and ⃖⃗𝐽𝑀 are two intersecting straight lines. Given that 𝑚∠𝐿𝑁𝑀=(2𝑥+2)∘ and 𝑚∠𝐽𝑁𝐿=(4𝑥−11)∘, find 𝑚∠𝐽𝑁𝑃.

Q11:

If ∠𝑎 and ∠𝑏 form a linear pair, where 𝑚∠𝑎=(3𝑥−6)∘ and 𝑚∠𝑏=(7𝑥+6)∘, determine 𝑚∠𝑎 and 𝑚∠𝑏.

  • A 𝑚 ∠ 𝑎 = 2 1 ∘ , 𝑚 ∠ 𝑏 = 6 9 ∘
  • B 𝑚 ∠ 𝑎 = 1 3 2 ∘ , 𝑚 ∠ 𝑏 = 4 8 ∘
  • C 𝑚 ∠ 𝑎 = 5 4 ∘ , 𝑚 ∠ 𝑏 = 1 2 6 ∘
  • D 𝑚 ∠ 𝑎 = 6 9 ∘ , 𝑚 ∠ 𝑏 = 2 1 ∘
  • E 𝑚 ∠ 𝑎 = 4 8 ∘ , 𝑚 ∠ 𝑏 = 1 3 2 ∘

Q12:

These two angles are supplementary. Write and solve an equation to find 𝑥.

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

  • A 𝑥 − 4 0 = 9 0 , 𝑥 = 1 3 0
  • B 𝑥 + 4 0 − 4 5 = 1 8 0 , 𝑥 = 1 8 5
  • C 𝑥 + 4 0 = 4 5 , 𝑥 = 8 5
  • D 𝑥 + 4 0 = 1 8 0 , 𝑥 = 1 4 0
  • E 𝑥 + 4 0 = 9 0 , 𝑥 = 5 0

Q13:

The rays 𝐴𝐵 and 𝐵𝐶 are perpendicular. A point 𝐷 lies in the interior of ∠𝐴𝐵𝐶. Given that 𝑚∠𝐴𝐵𝐷=(5𝑟+20)∘ and 𝑚∠𝐷𝐵𝐶=(8𝑟−8)∘, find 𝑚∠𝐴𝐵𝐷 and 𝑚∠𝐷𝐵𝐶.

  • A 𝑚 ∠ 𝐴 𝐵 𝐷 = 5 9 ∘ , 𝑚 ∠ 𝐷 𝐵 𝐶 = 5 4 ∘
  • B 𝑚 ∠ 𝐴 𝐵 𝐷 = 2 4 ∘ , 𝑚 ∠ 𝐷 𝐵 𝐶 = 6 6 ∘
  • C 𝑚 ∠ 𝐴 𝐵 𝐷 = 9 3 ∘ , 𝑚 ∠ 𝐷 𝐵 𝐶 = 1 1 0 ∘
  • D 𝑚 ∠ 𝐴 𝐵 𝐷 = 5 0 ∘ , 𝑚 ∠ 𝐷 𝐵 𝐶 = 4 0 ∘
  • E 𝑚 ∠ 𝐴 𝐵 𝐷 = 8 4 ∘ , 𝑚 ∠ 𝐷 𝐵 𝐶 = 9 5 ∘

Q14:

In the given diagram, 𝐴𝐷 is a straight line, 𝑚∠𝐴𝐵𝐶=(4𝑥+12)∘, and 𝑚∠𝐷𝐵𝐶=128∘.

Form an equation that will allow you to calculate 𝑥.

  • A 4 𝑥 + 1 2 = 1 8 0
  • B 4 𝑥 + 1 4 0 = 1 8 0
  • C 4 𝑥 + 1 2 = 1 2 8
  • D 4 𝑥 + 1 1 6 = 1 8 0
  • E 4 𝑥 + 1 1 6 = 1 2 8

Solve for 𝑥.

  • A 𝑥 = 3
  • B 𝑥 = 1 0
  • C 𝑥 = 4 2
  • D 𝑥 = 1 6
  • E 𝑥 = 2 9

Q15:

Given that 𝑚∠𝐽=(3𝑥−10)∘ and 𝑚∠𝐾=(4𝑥+6)∘, find 𝑚∠𝐾.

Q16:

In the given diagram, 𝐴𝐵 and 𝐶𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate 𝑥.

  • A 9 𝑥 = 9 0
  • B 9 𝑥 = 2 7 0
  • C 9 𝑥 = 1 3 9
  • D 9 𝑥 = 4 1
  • E 9 𝑥 = 1 8 0

Find the value of 𝑥.

  • A 𝑥 = 1 0
  • B 𝑥 = 2 0
  • C 𝑥 = 1 5
  • D 𝑥 = 5
  • E 𝑥 = 3 0

Q17:

Using the fact that the lines in the figure intersect, find the values of 𝑥 and 𝑦.

  • A 𝑥 = 1 8 , 𝑦 = 2 2
  • B 𝑥 = 1 8 , 𝑦 = 2 4
  • C 𝑥 = 5 0 , 𝑦 = 1 3 0
  • D 𝑥 = 1 7 , 𝑦 = 2 3
  • E 𝑥 = 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 𝑚 ∠ 𝐸 = 1 5 6 ∘ , 𝑚 ∠ 𝐹 = 2 4 ∘
  • B 𝑚 ∠ 𝐸 = 7 8 ∘ , 𝑚 ∠ 𝐹 = 1 0 2 ∘
  • C 𝑚 ∠ 𝐸 = 2 4 ∘ , 𝑚 ∠ 𝐹 = 1 5 6 ∘
  • D 𝑚 ∠ 𝐸 = 1 0 2 ∘ , 𝑚 ∠ 𝐹 = 7 8 ∘
  • E 𝑚 ∠ 𝐸 = 5 7 ∘ , 𝑚 ∠ 𝐹 = 3 3 ∘

Q19:

Find 𝑤 and 𝑧.

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

Q20:

If 𝐸𝐶 is an altitude of △𝐴𝐸𝐷, 𝑚∠𝜃=(5𝑥+2)∘, and 𝑚∠𝜙=(3𝑥+8)∘, find 𝑚∠𝜃.

Q21:

Find the values of 𝑥 and 𝑦 so that ⃖⃗𝑃𝑅 and ⃖⃗𝑆𝑄 are perpendicular.

  • A 𝑥 = 5 , 𝑦 = 2 9
  • B 𝑥 = 2 5 , 𝑦 = 6 5
  • C 𝑥 = 5 , 𝑦 = 6 5
  • D 𝑥 = 3 0 , 𝑦 = 2 9
  • E 𝑥 = 8 , 𝑦 = 2 9

Q22:

In the given diagram, 𝐴𝐷 is a straight line, 𝑚∠𝐴𝐵𝐶=(4𝑥+12)∘, and 𝑚∠𝐷𝐵𝐶=(6𝑥+8)∘.

Form an equation that will allow you to calculate 𝑥.

  • A 1 0 𝑥 + 2 0 = 9 0
  • B 1 0 𝑥 + 2 0 = 1 8 0
  • C 2 𝑥 + 2 0 = 1 8 0
  • D 2 𝑥 + 2 0 = 9 0
  • E 4 𝑥 + 1 2 = 9 0

Solve for 𝑥.

  • A 𝑥 = 3 5
  • B 𝑥 = 7
  • C 𝑥 = 1 6
  • D 𝑥 = 1 9 . 5
  • E 𝑥 = 8

Q23:

Answer the following questions using the given diagram.

Form an equation that will allow you to calculate 𝑥.

  • A 3 𝑥 + 4 8 = 9 0
  • B 3 𝑥 + 4 4 = 9 0
  • C 3 𝑥 + 4 8 = 4 6
  • D 3 𝑥 + 4 6 = 9 0
  • E 3 𝑥 + 2 = 4 6

Find the value of 𝑥.

  • A 𝑥 = 3 0
  • B 𝑥 = 1 4
  • C 𝑥 = 1 6
  • D 𝑥 = 2
  • E 𝑥 = 4 2

Q24:

In the given diagram, ∠𝐴𝐵𝐶 is a right angle and 𝑚∠𝐴𝐵𝐷 is twice 𝑚∠𝐷𝐵𝐶. Let 𝑚∠𝐷𝐵𝐶=𝑥.

Form an equation that will allow you to calculate 𝑥.

  • A 4 𝑥 = 9 0 ∘
  • B 2 𝑥 = 9 0 ∘
  • C 2 𝑥 = 1 8 0 ∘
  • D 3 𝑥 = 9 0 ∘
  • E 3 𝑥 = 1 8 0 ∘

Solve for 𝑥.

  • A 𝑥 = 3 0 ∘
  • B 𝑥 = 2 2 . 5 ∘
  • C 𝑥 = 4 5 ∘
  • D 𝑥 = 9 0 ∘
  • E 𝑥 = 6 0 ∘

Q25:

In the given diagram, 𝐴𝐵 and 𝐶𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate 𝑥.

  • A 6 𝑥 = 9 0
  • B 1 1 𝑥 + 2 = 9 0
  • C 1 1 𝑥 + 2 = 1 8 0
  • D 5 𝑥 + 2 = 9 0
  • E 6 𝑥 = 1 8 0

Find the value of 𝑥.

  • A 𝑥 = 8
  • B 𝑥 = 1 8
  • C 𝑥 = 3 0
  • D 𝑥 = 1 5
  • E 𝑥 = 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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