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Worksheet: Finding the Measures of Angles Formed by Lines Intersecting outside a Circle

Q1:

Given that π‘₯ ∘ is the measure of the major arc 𝐡 𝐢 , find the value of π‘₯ .

Q2:

Find π‘₯ .

Q3:

Find π‘₯ .

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

Q4:

In the figure, find the value of π‘₯ .

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

Q5:

Find π‘š ∠ π‘₯ .

Q6:

Given that π‘š ∠ 𝐴 𝐸 𝐢 = 2 6 ∘ and π‘š 𝐴 𝐢 = 8 5 ∘ , find π‘š 𝐢 𝐷 .

Q7:

Find the value of 𝑦 .

Q8:

Find π‘₯ .

Q9:

Given that οƒͺ 𝐡 𝐢 is a tangent to the circle below, find π‘š ∠ 𝐴 𝐡 𝐢 .

Q10:

In the given figure, find the values of π‘₯ and 𝑦 .

  • A π‘₯ = 1 2 0 , 𝑦 = 2 4 0
  • B π‘₯ = 2 4 0 , 𝑦 = 6 0
  • C π‘₯ = 2 4 0 , 𝑦 = 1 2 0
  • D π‘₯ = 6 0 , 𝑦 = 2 4 0
  • E π‘₯ = 1 4 0 , 𝑦 = 2 2 0

Q11:

Given that and are two tangents to the circle , and , determine .

  • A
  • B
  • C
  • D

Q12:

Given that , find .

  • A
  • B
  • C
  • D

Q13:

Lines and are tangent to circle . If , what is ?

  • A
  • B
  • C
  • D

Q14:

Given that 𝐴 𝐡 is a diameter in the circle 𝑀 ,  𝐴 𝐢 is a tangent to the circle at 𝐴 , and π‘š ∠ 𝐷 𝑀 𝐡 = 1 3 1 ∘ , find π‘š ∠ 𝐴 𝐢 𝐡 .

  • A 2 4 . 5 ∘
  • B 1 3 1 ∘
  • C 4 9 ∘
  • D 6 5 . 5 ∘

Q15:

Given that , where and are tangent to the circle at and , find .

  • A
  • B
  • C
  • D

Q16:

In the figure, and are tangents to . What is ?

  • A
  • B
  • C
  • D

Q17:

𝐴 𝐡 𝐢 𝐷 𝐸 is the regular pentagon drawn inside the circle 𝑀 , βƒ–      βƒ— 𝐴 𝑋 is a tangent to the circle at 𝐴 , and βƒ–     βƒ— 𝐸 𝑋 is a tangent to the circle at 𝐸 . Find π‘š ∠ 𝐴 𝑋 𝐸 .

Q18:

Circle 𝑀 has radius 65. Suppose 𝐴 is on a line 𝐿 and 𝑀 𝐴 is perpendicular to 𝐿 . If 2 𝑀 𝐴 βˆ’ 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

Q19:

In the figure, is a diameter of the circle . Given that , find .

  • A
  • B
  • C
  • D

Q20:

If , find .

  • A
  • B
  • C
  • D

Q21:

If 𝐸 𝐢 = 1 0 c m , 𝐸 𝐷 = 6 c m , 𝐸 𝐡 = 5 c m , find the length of 𝐸 𝐴 .

Q22:

If 𝐸 𝐴 𝐸 𝐡 = 5 3 , 𝐸 𝐢 = 1 2 c m , and 𝐸 𝐷 = 5 c m , find the lengths of 𝐸 𝐡 and 𝐡 𝐴 .

  • A 𝐸 𝐡 = 4 c m , 𝐡 𝐴 = 6 c m
  • B 𝐸 𝐡 = 6 c m , 𝐡 𝐴 = 1 0 c m
  • C 𝐸 𝐡 = 1 0 c m , 𝐡 𝐴 = 6 c m
  • D 𝐸 𝐡 = 6 c m , 𝐡 𝐴 = 4 c m
  • E 𝐸 𝐡 = 2 4 c m , 𝐡 𝐴 = 2 0 c m

Q23:

Given 𝐸 𝐷 = 2 7 c m , and 𝐢 𝐷 = 1 7 c m , find the length of 𝐸 𝐴 rounded to the nearest hundredth.

Q24:

Find length 𝐡 𝐢 to the nearest integer.

Q25:

In the figure shown, the circle has a radius of 12 cm, 𝐴 𝐡 = 1 2 c m , and 𝐴 𝐢 = 3 5 c m . Determine the distance from 𝐡 𝐢 to the center of the circle, 𝑀 , and the length of 𝐴 𝐷 , rounding your answers to the nearest tenth.

  • A 19.6 cm, 20.2 cm
  • B 11.5 cm, 20.2 cm
  • C 3.4 cm, 26.6 cm
  • D 3.4 cm, 20.5 cm