Worksheet: Properties of Cyclic Quadrilaterals

Practice

In this worksheet, we will practice identifying whether a quadrilateral is cyclic and using cyclic quadrilateral properties to find a missing angle.

Q1:

Is 𝐴 𝐵 𝐶 𝐷 a cyclic quadrilateral?

  • Ano
  • Byes

Q2:

In cyclic quadrilateral 𝐴 𝐵 𝐶 𝐷 , if 𝑚 ∠ 𝐶 = 1 2 1 ∘ , what is 𝑚 ∠ 𝐴 ?

Q3:

Is 𝐴 𝐵 𝐶 𝐷 a cyclic quadrilateral?

  • Ayes
  • Bno

Q4:

Given that 𝐴 𝐵 𝐶 𝐷 is a cyclic quadrilateral, find 𝑚 ∠ 𝐵 𝐴 𝐶 .

Q5:

Find 𝑚 ∠ 𝐵 𝐶 𝐷 .

Q6:

Find 𝑚 ∠ 𝐵 .

Q7:

Find 𝑚 ∠ 𝐸 𝐶 𝐹 and 𝑚 ∠ 𝐴 𝐵 𝐹 .

  • A 𝑚 ∠ 𝐸 𝐶 𝐹 = 7 6 ∘ , 𝑚 ∠ 𝐴 𝐵 𝐹 = 1 0 0 ∘
  • B 𝑚 ∠ 𝐸 𝐶 𝐹 = 1 0 0 ∘ , 𝑚 ∠ 𝐴 𝐵 𝐹 = 1 0 4 ∘
  • C 𝑚 ∠ 𝐸 𝐶 𝐹 = 1 0 4 ∘ , 𝑚 ∠ 𝐴 𝐵 𝐹 = 8 0 ∘
  • D 𝑚 ∠ 𝐸 𝐶 𝐹 = 8 0 ∘ , 𝑚 ∠ 𝐴 𝐵 𝐹 = 7 6 ∘
  • E 𝑚 ∠ 𝐸 𝐶 𝐹 = 7 6 ∘ , 𝑚 ∠ 𝐴 𝐵 𝐹 = 8 0 ∘

Q8:

𝐴 𝐵 𝐶 𝐷 is a cyclic quadrilateral with 𝑚 ∠ 𝐵 𝐴 𝐶 = 5 8 ∘ and 𝑚 ∠ 𝐴 𝐷 𝐸 = 8 7 ∘ . What is 𝑚 ∠ 𝐵 𝐶 𝐴 ?

Q9:

Is the quadrilateral 𝐴 𝐵 𝐶 𝐷 cyclic?

  • Ayes
  • Bno

Q10:

Find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 1 7 , 𝑦 = 6 8
  • B 𝑥 = 2 8 , 𝑦 = 6 8
  • C 𝑥 = 1 7 , 𝑦 = 5 6
  • D 𝑥 = 2 8 , 𝑦 = 5 6

Q11:

Given that 𝑚 ∠ 𝐷 𝐶 𝐻 = 9 2 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐷 = 4 𝑥 ∘ , and 𝑚 ∠ 𝐶 𝐷 𝐴 = 2 𝑦 ∘ , find 𝑥 + 𝑦 .

Q12:

In a cyclic quadrilateral 𝐴 𝐵 𝐶 𝐷 , if 𝑚 ∠ 𝐴 = 3 𝑚 ∠ 𝐵 = 2 𝑚 ∠ 𝐶 , find 𝑚 ∠ 𝐷 .

Q13:

Given that 𝑚 ∠ 𝐴 = 𝑦 ∘ , 𝑚 ∠ 𝐵 = ( 4 𝑥 − 3 ) ∘ , and 𝑚 ∠ 𝐶 = 5 𝑥 ∘ , find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 3 0 , 𝑦 = 1 5 0
  • B 𝑥 = 2 0 , 𝑦 = 8 0
  • C 𝑥 = 2 3 , 𝑦 = 8 9
  • D 𝑥 = 1 7 , 𝑦 = 9 5
  • E 𝑥 = 1 3 , 𝑦 = 1 2 5

Q14:

Calculate the values of 𝑥 and 𝑦 .

  • A 𝑥 = 9 0 , 𝑦 = 9 0
  • B 𝑥 = 1 8 0 , 𝑦 = 2 0
  • C 𝑥 = 1 0 , 𝑦 = 9 0
  • D 𝑥 = 9 0 , 𝑦 = 1 0

Q15:

Is 𝐴 𝐵 𝐶 𝐷 a cyclic quadrilateral?

  • Ayes
  • Bno

Q16:

In the figure, 𝐴 𝐵 𝐶 𝐷 is a quadrilateral with 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 1 7 ∘ and 𝑚 ∠ 𝐴 𝐶 𝐵 = 3 2 . 5 ∘ . What is 𝑚 ∠ 𝐵 𝐷 𝐶 ?

Q17:

Given that 𝐴 𝐵 𝐶 𝐷 is a cyclic quadrilateral, 𝑚 ∠ 𝐴 𝐶 𝐵 = 7 0 ∘ , and 𝐴 𝐵 = 𝐴 𝐶 , find 𝑚 ∠ 𝐵 𝐷 𝐶 .

Q18:

Find the remaining internal angles of quadrilateral 𝐴 𝐵 𝐶 𝐷 .

  • A 𝑚 ∠ 𝐴 𝐵 𝐶 = 7 7 ∘ , 𝑚 ∠ 𝐶 = 1 5 0 ∘ , 𝑚 ∠ 𝐶 𝐷 𝐴 = 4 7 ∘
  • B 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 0 3 ∘ , 𝑚 ∠ 𝐶 = 7 7 ∘ , 𝑚 ∠ 𝐶 𝐷 𝐴 = 7 3 ∘
  • C 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 5 0 ∘ , 𝑚 ∠ 𝐶 = 1 0 3 ∘ , 𝑚 ∠ 𝐶 𝐷 𝐴 = 7 7 ∘
  • D 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 0 7 ∘ , 𝑚 ∠ 𝐶 = 7 7 ∘ , 𝑚 ∠ 𝐶 𝐷 𝐴 = 7 3 ∘
  • E 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 3 3 ∘ , 𝑚 ∠ 𝐶 = 7 7 ∘ , 𝑚 ∠ 𝐶 𝐷 𝐴 = 4 7 ∘

Q19:

Find 𝑚 ∠ 𝐸 𝑀 𝑁 , given that 𝐿 𝑀 𝑁 𝐸 is a cyclic quadrilateral with 𝑚 ∠ 𝑀 𝐿 𝐸 = 6 4 ∘ and 𝑚 ∠ 𝑀 𝐸 𝑁 = 3 8 ∘ .

Q20:

If 𝐴 𝐵 𝐶 𝐷 is a cyclic quadrilateral, where 𝑚 ∠ 𝐵 = 5 4 𝑚 ∠ 𝐷 , find 𝑚 ∠ 𝐵 .

Q21:

Is 𝐴 𝐵 𝐶 𝐷 a cyclic quadrilateral?

  • Ayes
  • Bno

Q22:

is a cyclic quadrilateral.

Work out the size of .

  • A
  • B
  • C
  • D
  • E

Work out the size of .

  • A
  • B
  • C
  • D
  • E

Q23:

Is there a circle passing through the vertices of the quadrilateral 𝐴 𝐵 𝐶 𝐷 ?

  • Ayes
  • Bno

Q24:

In the figure below, 𝑚 ∠ 𝐴 𝐷 𝐸 = 2 6 . 9 ∘ , 𝑚 ∠ 𝐴 𝐸 𝐵 = 8 2 . 9 ∘ , and 𝑚 ∠ 𝐴 𝐵 𝐸 = 5 9 . 1 ∘ . Is 𝐴 𝐵 𝐶 𝐷 a cyclic quadrilateral?

  • Ano
  • Byes

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