Worksheet: 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 subtended by the same arc.

Q1:

⃖⃗𝐵𝐶 is tangent to the circle of center 𝑀 at 𝐵. Find 𝑚∠𝐴𝐵𝐶.

Q2:

Given that 𝐵𝐶 is a tangent to the circle, find 𝑚∠𝐴𝐵𝐶.

Q3:

Given that 𝐵𝐶 is a tangent to the circle below, find 𝑚∠𝐴𝐵𝐶 and 𝑦∘.

  • A226∘, 226
  • B113∘, 113
  • C113∘, 226
  • D226∘, 113

Q4:

Given that 𝐴𝐷 is a tangent to the circle, find 𝑚∠𝐵𝐴𝐷.

Q5:

Given that 𝑚∠𝑀𝐵𝐶=28∘, find 𝑚∠𝐴.

Q6:

Given that 𝐴𝐷 is a tangent to the circle and 𝑚∠𝐶𝐴𝐷=97∘, find 𝑚∠𝐴𝐵𝐶 and 𝑦.

  • A𝑚∠𝐴𝐵𝐶=83∘, 𝑦=83
  • B𝑚∠𝐴𝐵𝐶=166∘, 𝑦=83
  • C𝑚∠𝐴𝐵𝐶=166∘, 𝑦=166
  • D𝑚∠𝐴𝐵𝐶=83∘, 𝑦=166

Q7:

Find 𝑚∠𝐴𝐶𝐷.

Q8:

Given that 𝐵𝐶 is a tangent to the circle, and 𝑚∠𝐷𝐴𝐵=51∘, determine 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝐴𝐸𝐵.

  • A39∘, 78∘
  • B78∘, 39∘
  • C51∘, 51∘
  • D39∘, 39∘
  • E78∘, 78∘

Q9:

If 𝑚∠𝐴𝐵𝐶=83∘ and 𝑚∠𝑋𝐴𝐵=38∘, find 𝑚∠𝐶𝐷𝐵.

Q10:

Given that ⃖⃗𝐵𝐶 is a tangent to the circle, and 𝑚∠𝐵𝐴𝐷=63∘, find 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝐴𝐸𝐵.

  • A𝑚∠𝐴𝐵𝐶=54∘, 𝑚∠𝐴𝐸𝐵=27∘
  • B𝑚∠𝐴𝐵𝐶=27∘, 𝑚∠𝐴𝐸𝐵=27∘
  • C𝑚∠𝐴𝐵𝐶=27∘, 𝑚∠𝐴𝐸𝐵=54∘
  • D𝑚∠𝐴𝐵𝐶=54∘, 𝑚∠𝐴𝐸𝐵=54∘

Q11:

Given that 𝑚∠𝐸𝐶𝐷=54∘ and 𝑚∠𝐹𝐵𝐷=78∘, find 𝑥 and 𝑦.

  • A54, 84
  • B48, 84
  • C78, 84
  • D78, 102
  • E54, 126

Q12:

Given that 𝑚∠𝐵𝐸𝐶=31∘, find 𝑚∠𝐶 and 𝑚∠𝐵𝐷𝐴.

  • A𝑚∠𝐶=28∘, 𝑚∠𝐵𝐷𝐴=62∘
  • B𝑚∠𝐶=28∘, 𝑚∠𝐵𝐷𝐴=59∘
  • C𝑚∠𝐶=28∘, 𝑚∠𝐵𝐷𝐴=28∘
  • D𝑚∠𝐶=28∘, 𝑚∠𝐵𝐷𝐴=118∘
  • E𝑚∠𝐶=59∘, 𝑚∠𝐵𝐷𝐴=59∘

Q13:

Given that 𝑚∠𝐴𝐶𝐵=80∘, find 𝑚∠𝐴𝐷𝑀.

Q14:

In the figure, 𝐴𝐵 is a diameter of circle 𝑀. Given that 𝑚∠𝐶𝐵𝐷=43∘, find 𝑚∠𝐴𝑀𝐷.

Q15:

Find 𝑚∠𝐵𝐴𝐶.

Q16:

Find 𝑚∠𝐶𝐴𝐵.

Q17:

If 𝑚∠𝑀𝐴𝐵=47∘, find 𝑚∠𝐴𝐶𝐷.

Q18:

If 𝐴𝐷 and 𝐴𝐸 are tangents to a circle with center 𝑀 and 𝑚∠𝐴=62∘, find 𝑚∠𝐸𝑀𝐷.

Q19:

Find 𝑚∠𝐵𝑀𝐶.

Q20:

Given the figure below, find 𝑚∠𝑀𝐶𝐵 and 𝑚∠𝐶𝐵𝐴.

  • A68∘, 112∘
  • B34∘, 56∘
  • C56∘, 34∘
  • D34∘, 34∘

Q21:

Given that 𝑚∠𝐴=84∘, where 𝐴𝐵 and 𝐴𝐶 are tangent to the circle at 𝐵 and 𝐶, find 𝑚∠𝐸.

Q22:

Given that 𝑚∠𝑀𝐴𝐶=36∘, determine 𝑚∠𝐵𝐴𝑀 and 𝑚∠𝐴𝑀𝐶.

  • A36∘, 54∘
  • B54∘, 54∘
  • C54∘, 36∘
  • D36∘, 36∘

Q23:

Given that 𝑚∠𝐵𝐴𝑀=35∘, find 𝑚∠𝐵𝑀𝐶.

Q24:

A circle with center 𝑀 has diameter 𝐴𝐵. If ⃖⃗𝐴𝐶 and ⃖⃗𝐵𝐷 are two tangents to the circle, what can you say about them?

  • AThey are perpendicular.
  • BThey are parallel.
  • CThey are intersecting.

Q25:

Triangle 𝐴𝐵𝐶 is right angled at 𝐵. Given that 𝑚∠𝐴=30∘, 𝐴𝐵=37.6cm, and 𝑀 is the circle that lies on 𝐴 and 𝐵 and is tangent to 𝐴𝐶 at 𝐴, determine the area of △𝐴𝐵𝑀 to the nearest tenth of a square centimeter.

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