Worksheet: Angles Formed by Lines Intersecting outside a Circle

In this worksheet, we will practice finding the measure of an angle that is formed by lines intersecting outside a circle.

Q1:

Given that 𝑥 is the measure of the major arc 𝐵 𝐶 , find the value of 𝑥 .

Q2:

Find 𝑥 .

Q3:

In the given figure, find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 1 2 0 , 𝑦 = 2 4 0
  • B 𝑥 = 2 4 0 , 𝑦 = 6 0
  • C 𝑥 = 2 4 0 , 𝑦 = 1 2 0
  • D 𝑥 = 6 0 , 𝑦 = 2 4 0
  • E 𝑥 = 1 4 0 , 𝑦 = 2 2 0

Q4:

Find length 𝐵 𝐶 to the nearest integer.

Q5:

Given that 𝐴 𝐵 is tangent to the circle and that 𝐷 and 𝐶 lie on the circle, determine the value of 𝑥 to the nearest tenth.

Q6:

Given that 𝐸 𝐵 = 9 c m , 𝐴 𝐵 = 2 1 c m , 𝐸 𝐶 = 2 9 c m , and 𝐸 𝐺 = 1 0 c m , find the lengths of 𝐸 𝐷 and 𝐹 𝐺 .

  • A 𝐸 𝐷 = 9 . 3 1 c m , 𝐹 𝐺 = 2 7 c m
  • B 𝐸 𝐷 = 1 7 c m , 𝐹 𝐺 = 9 . 3 1 c m
  • C 𝐸 𝐷 = 2 7 c m , 𝐹 𝐺 = 9 . 3 1 c m
  • D 𝐸 𝐷 = 9 . 3 1 c m , 𝐹 𝐺 = 1 7 c m
  • E 𝐸 𝐷 = 9 6 . 6 7 c m , 𝐹 𝐺 = 2 7 0 . 3 4 c m

Q7:

Find 𝑥 .

Q8:

Find 𝑥 .

  • A 9 6
  • B 3 6
  • C 6 6
  • D 4 8

Q9:

In the figure, find the value of 𝑥 .

  • A 2 1 0
  • B 1 0 5
  • C 1 1 0
  • D 5 5

Q10:

Find 𝑥 .

Q11:

Given that 𝑚 𝐴 𝐸 𝐶 = 2 6 and 𝑚 𝐴 𝐶 = 8 5 , find 𝑚 𝐶 𝐷 .

Q12:

Find the value of 𝑦 .

Q13:

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

Q14:

Given that 𝐴 𝐵 and 𝐴 𝐶 are two tangents to the circle 𝑀 , and 𝑚 𝑀 𝐴 𝐶 = 2 1 , determine 𝑚 𝐶 𝐴 𝐵 .

Q15:

Given that 𝑚 𝑋 𝑌 𝑀 = 3 7 , find 𝑚 𝐶 𝐴 𝐵 .

Q16:

Lines 𝐴 𝐵 and 𝐴 𝐶 are tangent to circle 𝑀 . If 𝑚 𝑀 𝐶 𝐵 = 3 2 , what is 𝑚 𝐵 𝐴 𝐶 ?

Q17:

Given that 𝐴 𝐵 is a diameter in the circle 𝑀 , 𝐴 𝐶 is a tangent to the circle at 𝐴 , and 𝑚 𝐷 𝑀 𝐵 = 1 3 1 , find 𝑚 𝐴 𝐶 𝐵 .

Q18:

Given that 𝑚 𝑀 𝐶 𝐵 = 4 9 , where 𝐴 𝐵 and 𝐴 𝐶 are tangent to the circle at 𝐵 and 𝐶 , find 𝑚 𝐵 𝐴 𝑀 .

Q19:

In the figure, 𝐴 𝐵 and 𝐴 𝐶 are tangents to 𝑀 . What is 𝑚 𝐶 𝐴 𝐵 ?

Q20:

𝐴 𝐵 𝐶 𝐷 𝐸 is the regular pentagon drawn inside the circle 𝑀 , 𝐴 𝑋 is a tangent to the circle at 𝐴 , and 𝐸 𝑋 is a tangent to the circle at 𝐸 . Find 𝑚 𝐴 𝑋 𝐸 .

Q21:

Circle 𝑀 has radius 65. Suppose 𝐴 is on a line 𝐿 and 𝑀 𝐴 is perpendicular to 𝐿 . If 2 𝑀 𝐴 5 6 = 1 8 what can be said of how 𝐿 lies with respect to the circle?

  • A 𝐿 is outside of circle 𝑀
  • B 𝐿 is a tangent to circle 𝑀
  • C 𝐿 is a secant to circle 𝑀

Q22:

In the figure, 𝐴 𝐵 is a diameter of the circle 𝑀 . Given that 𝑚 𝐴 𝐸 𝐷 = 5 4 , find 𝑚 𝐸 𝐶 𝐷 .

Q23:

If 𝑚 𝐵 𝑀 𝐶 = 9 9 . 5 , find 𝑚 𝐶 𝑌 𝑀 .

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