Worksheet: Tangents and Angles of Tangency

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In this worksheet, we will practice using the properties of tangents to find the measure of an angle of tangency or an inscribed angle with it in the same arc.

Q1:

⃖ ⃗ 𝐵 𝐶 is tangent to the circle 𝑀 at 𝐵 . Find 𝑚 ∠ 𝐴 𝐵 𝐶 .

Q2:

Given that 𝐵 𝐶 is a tangent to the circle, find 𝑚 ∠ 𝐴 𝐵 𝐶 .

Q3:

Given that 𝐵 𝐶 is a tangent to the circle below, find 𝑚 ∠ 𝐴 𝐵 𝐶 and 𝑦 ∘ .

  • A 2 2 6 ∘ , 226
  • B 1 1 3 ∘ , 113
  • C 2 2 6 ∘ , 113
  • D 1 1 3 ∘ , 226

Q4:

Given that 𝐴 𝐷 is a tangent to the circle, find 𝑚 ∠ 𝐵 𝐴 𝐷 .

Q5:

Given that 𝑚 ∠ 𝑀 𝐵 𝐶 = 2 8 ∘ , find 𝑚 ∠ 𝐴 .

Q6:

Given that 𝐴 𝐷 is a tangent to the circle and 𝑚 ∠ 𝐶 𝐴 𝐷 = 9 7 ∘ , find 𝑚 ∠ 𝐴 𝐵 𝐶 and 𝑦 .

  • A 𝑚 ∠ 𝐴 𝐵 𝐶 = 8 3 ∘ , 𝑦 = 8 3
  • B 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 6 6 ∘ , 𝑦 = 1 6 6
  • C 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 6 6 ∘ , 𝑦 = 8 3
  • D 𝑚 ∠ 𝐴 𝐵 𝐶 = 8 3 ∘ , 𝑦 = 1 6 6

Q7:

Find 𝑚 ∠ 𝐴 𝐶 𝐷 .

Q8:

Given that 𝐵 𝐶 is a tangent to the circle, and 𝑚 ∠ 𝐷 𝐴 𝐵 = 5 1 ∘ , determine 𝑚 ∠ 𝐴 𝐵 𝐶 and 𝑚 ∠ 𝐴 𝐸 𝐵 .

  • A 7 8 ∘ , 3 9 ∘
  • B 5 1 ∘ , 5 1 ∘
  • C 7 8 ∘ , 7 8 ∘
  • D 3 9 ∘ , 3 9 ∘
  • E 3 9 ∘ , 7 8 ∘

Q9:

If 𝑚 ∠ 𝐴 𝐵 𝐶 = 8 3 ∘ and 𝑚 ∠ 𝑋 𝐴 𝐵 = 3 8 ∘ , find 𝑚 ∠ 𝐶 𝐷 𝐵 .

Q10:

Given that ⃖ ⃗ 𝐵 𝐶 is a tangent to the circle, and 𝑚 ∠ 𝐵 𝐴 𝐷 = 6 3 ∘ , find 𝑚 ∠ 𝐴 𝐵 𝐶 and 𝑚 ∠ 𝐴 𝐸 𝐵 .

  • A 𝑚 ∠ 𝐴 𝐵 𝐶 = 2 7 ∘ , 𝑚 ∠ 𝐴 𝐸 𝐵 = 5 4 ∘
  • B 𝑚 ∠ 𝐴 𝐵 𝐶 = 5 4 ∘ , 𝑚 ∠ 𝐴 𝐸 𝐵 = 5 4 ∘
  • C 𝑚 ∠ 𝐴 𝐵 𝐶 = 5 4 ∘ , 𝑚 ∠ 𝐴 𝐸 𝐵 = 2 7 ∘
  • D 𝑚 ∠ 𝐴 𝐵 𝐶 = 2 7 ∘ , 𝑚 ∠ 𝐴 𝐸 𝐵 = 2 7 ∘

Q11:

Given that 𝑚 ∠ 𝐸 𝐶 𝐷 = 5 4 ∘ and 𝑚 ∠ 𝐹 𝐵 𝐷 = 7 8 ∘ , find 𝑥 and 𝑦 .

  • A78, 84
  • B54, 84
  • C54, 126
  • D48, 84
  • E78, 102

Q12:

Given that 𝑚 ∠ 𝐵 𝐸 𝐶 = 3 1 ∘ , find 𝑚 ∠ 𝐶 and 𝑚 ∠ 𝐵 𝐷 𝐴 .

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

Q13:

Given that 𝑚 ∠ 𝐴 𝐶 𝐵 = 8 0 ∘ , find 𝑚 ∠ 𝐴 𝐷 𝑀 .

  • A 8 0 ∘
  • B 1 0 0 ∘
  • C 4 0 ∘
  • D 5 0 ∘

Q14:

In the figure, 𝐴 𝐵 is a diameter of circle 𝑀 . Given that 𝑚 ∠ 𝐶 𝐵 𝐷 = 4 3 ∘ , find 𝑚 ∠ 𝐴 𝑀 𝐷 .

Q15:

Find 𝑚 ∠ 𝐵 𝐴 𝐶 .

Q16:

Find 𝑚 ∠ 𝐶 𝐴 𝐵 .

Q17:

If 𝑚 ∠ 𝑀 𝐴 𝐵 = 4 7 ∘ , find 𝑚 ∠ 𝐴 𝐶 𝐷 .

Q18:

If 𝐴 𝐷 and 𝐴 𝐸 are tangents to the circle 𝑀 and 𝑚 ∠ 𝐴 = 6 2 ∘ , find 𝑚 ∠ 𝐸 𝑀 𝐷 .

Q19:

Find 𝑚 ∠ 𝐵 𝑀 𝐶 .

Q20:

Given the figure below, find 𝑚 ∠ 𝑀 𝐶 𝐵 and 𝑚 ∠ 𝐶 𝐵 𝐴 .

  • A 5 6 ∘ , 3 4 ∘
  • B 3 4 ∘ , 3 4 ∘
  • C 6 8 ∘ , 1 1 2 ∘
  • D 3 4 ∘ , 5 6 ∘

Q21:

Given that 𝑚 ∠ 𝐴 = 8 4 ∘ , where 𝐴 𝐵 and 𝐴 𝐶 are tangent to the circle at 𝐵 and 𝐶 , find 𝑚 ∠ 𝐸 .

Q22:

Given that 𝑚 ∠ 𝑀 𝐴 𝐶 = 3 6 ∘ , determine 𝑚 ∠ 𝐵 𝐴 𝑀 and 𝑚 ∠ 𝐴 𝑀 𝐶 .

  • A 3 6 ∘ , 3 6 ∘
  • B 5 4 ∘ , 3 6 ∘
  • C 5 4 ∘ , 5 4 ∘
  • D 3 6 ∘ , 5 4 ∘

Q23:

Given that 𝑚 ∠ 𝐵 𝐴 𝑀 = 3 5 ∘ , find 𝑚 ∠ 𝐵 𝑀 𝐶 .

Q24:

A circle with centre 𝑀 has diameter 𝐴 𝐵 . If ⃖ ⃗ 𝐴 𝐶 and ⃖ ⃗ 𝐵 𝐷 are two tangents to the circle, what can you say about them?

  • Athey are intersecting
  • Bthey are perpendicular
  • Cthey are parallel

Q25:

Triangle 𝐴 𝐵 𝐶 is right angled at 𝐵 . Given that 𝑚 ∠ 𝐴 = 3 0 ∘ , 𝐴 𝐵 = 3 7 . 6 c m , and 𝑀 is the circle that lies on 𝐴 and 𝐵 and is tangent to 𝐴 𝐶 at 𝐴 , determine the area of △ 𝐴 𝐵 𝑀 to the nearest tenth of a square centimeter.

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