Worksheet: The Sum of Angles in Quadrilaterals

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In this worksheet, we will practice finding a missing angle's measure in a quadrilateral knowing that the sum of angles in a quadrilateral is 360 degrees.

Q1:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝑚∠𝐴=165∘, 𝑚∠𝐴𝐵𝐶=4𝑥∘, 𝑚∠𝐵𝐶𝐷=4𝑥∘, and 𝑚∠𝐶𝐷𝐴=5𝑥∘. What is 𝑥?

Q2:

In the given figure, 𝑚∠𝐴=74∘ and 𝑚∠𝐶=92∘. Find 𝑚∠𝐷.

Q3:

Determine the value of 𝑥 in the given quadrilateral.

Q4:

Find 𝑚∠𝐴.

Q5:

In a quadrilateral𝐴𝐵𝐶𝐷, if 𝑚∠𝐴=3𝑚∠𝐵=3𝑚∠𝐶=72∘, find 𝑚∠𝐷.

Q6:

The sum of the measures of the angles of a trapezoid is 360∘. Write an equation to find the missing measure 𝑥, and then solve it.

  • A 𝑥 + 1 0 2 = 1 8 0 , 𝑥 = 7 8
  • B 𝑥 + 1 0 2 = 1 8 0 , 𝑥 = 1 0 2
  • C 𝑥 + 1 0 2 + 1 1 6 + 7 8 = 3 6 0 , 𝑥 = 6 4
  • D 𝑥 + 1 0 2 + 1 1 6 = 3 6 0 , 𝑥 = 1 0 2
  • E 𝑥 + 1 0 2 + 1 1 6 + 7 8 = 3 6 0 , 𝑥 = 7 8

Q7:

From the figure, in which 𝑚∠𝐶𝐷𝐴=5𝑥∘, 𝑚∠𝐵𝐶𝐷=7𝑦∘, and 𝑚∠𝐴𝐵𝐶=8𝑦∘, find the values of 𝑥 and 𝑦.

  • A 𝑥 = 9 5 , 𝑦 = 8 4
  • B 𝑥 = 9 5 , 𝑦 = 1 2
  • C 𝑥 = 1 9 , 𝑦 = 1 2
  • D 𝑥 = 1 9 , 𝑦 = 9 5

Q8:

Given that 𝑚∠𝐷=23𝑚∠𝐵, find 𝑚∠𝐵 and 𝑚∠𝐷.

  • A 𝑚 ∠ 𝐵 = 1 0 8 ∘ , 𝑚 ∠ 𝐷 = 7 2 ∘
  • B 𝑚 ∠ 𝐵 = 5 3 ∘ , 𝑚 ∠ 𝐷 = 1 2 7 ∘
  • C 𝑚 ∠ 𝐵 = 7 2 ∘ , 𝑚 ∠ 𝐷 = 1 0 8 ∘
  • D 𝑚 ∠ 𝐵 = 1 2 7 ∘ , 𝑚 ∠ 𝐷 = 5 3 ∘

Q9:

Determine 𝑚∠𝐶𝐴𝐵.

Q10:

From the figure, determine 𝑚∠𝐵𝐸𝐷.

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

Q11:

In the figure, 𝑚∠𝐴=34∘, 𝑚∠𝐹=48∘, and 𝑚∠𝐵𝐸𝐹=34∘. Determine the measure of all angles in 𝐷𝐸𝐵𝐶.

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

Q12:

Determine 𝑚∠𝐹𝐸𝐶.

Q13:

In the figure, if 𝑚∠𝐴=26∘, 𝑚∠𝐴𝐶𝐵=54∘, 𝑚∠𝐹=146∘, determine 𝑚∠𝐸.

Q14:

The internal angles of a quadrilateral are in the ratio 3∶4∶5∶6. What is the largest angle?

  • A 1 2 0 ∘
  • B 8 0 ∘
  • C 6 0 ∘
  • D 2 0 ∘

Q15:

In the given figure, find 𝑥 and 𝑦.

  • A 𝑥 = 1 6 , 𝑦 = 3 1
  • B 𝑥 = 1 4 , 𝑦 = 3 1
  • C 𝑥 = 1 9 , 𝑦 = 4 8
  • D 𝑥 = 1 6 , 𝑦 = 5 9
  • E 𝑥 = 9 6 , 𝑦 = 1 1 8

Q16:

Find the measures of the angles of a quadrilateral given that the ratio between them is 691213:::.

  • A 8 7 ∘ , 8 1 ∘ , 9 3 ∘ , 9 9 ∘
  • B 6 3 ∘ , 1 1 7 ∘ , 5 4 ∘ , 1 2 6 ∘
  • C 2 2 ∘ , 2 8 ∘ , 3 7 ∘ , 4 9 ∘
  • D 1 5 ∘ , 1 8 ∘ , 2 1 ∘ , 2 2 ∘
  • E 5 4 ∘ , 8 1 ∘ , 1 0 8 ∘ , 1 1 7 ∘

Q17:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝑚∠𝐴=162∘, 𝑚∠𝐴𝐵𝐶=4𝑥∘, 𝑚∠𝐵𝐶𝐷=2𝑥∘, and 𝑚∠𝐶𝐷𝐴=3𝑥∘. What is 𝑥?

Q18:

Find 𝑚∠𝐴.

Q19:

Find 𝑚∠𝐴.

Q20:

Determine the value of 𝑥 in the given quadrilateral.

Q21:

Determine the value of 𝑥 in the given quadrilateral.

Q22:

In a quadrilateral𝐴𝐵𝐶𝐷, if 𝑚∠𝐴=9𝑚∠𝐵=5𝑚∠𝐶=45∘, find 𝑚∠𝐷.

Q23:

In the given figure, 𝑚∠𝐴=60∘ and 𝑚∠𝐶=118∘. Find 𝑚∠𝐷.

Q24:

In the given figure, 𝑚∠𝐴=75∘ and 𝑚∠𝐶=96∘. Find 𝑚∠𝐷.

Q25:

Three angles in a quadrilateral are 156 rad, 215 rad and 31∘. Find the measure of the fourth angle in radians giving the answer to three decimal places.

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