Worksheet: Tangents and Angles of Tangency

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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