ANGLE RELATIONSHIPS IN CIRCLES WORKSHEET

Problem 1 :

Line m is tangent to the circle. Find the measure of the red angle.

Problem 2 :

Line m is tangent to the circle. Find the measure of the red arc.

Problem 3 :

In the diagram below, BC is tangent to the circle.  Find m∠CBD.

Problem 4 :

Find the value of x in the diagram shown below. 

Problem 5 :

Find the value of x in the diagram shown below. 

Problem 6 :

Find the value of x in the diagram shown below. 

Answers

Problem 1 :

Line m is tangent to the circle. Find the measure of the red angle.

Answer :

m∠1  =  1/2 ⋅ 150°

m∠1  =  75°

Problem 2 : 

Line m is tangent to the circle. Find the measure of the red arc.

Answer :

m∠arc RSP  =  2 ⋅ 130°

m∠arc RSP  =  260°

Problem 3 :

In the diagram below, BC is tangent to the circle.  Find m∠CBD.

Answer :

m∠ CBD  =  1/2 ⋅ m∠arc DAB

5x  =  1/2 ⋅ (9x + 20)

Multiply each side by 2.

10x  =  9x + 20

Subtract 9x from each side.

x  =  20

So, the angle measure CBD is

m∠ CBD  =  5(20°)

m∠ CBD  =  100°

Problem 4 : 

Find the value of x in the diagram shown below. 

Answer :

x°  =  1/2 ⋅ (m∠arc PS + m∠arc RQ)

x°  =  1/2 ⋅ (106° + 174°)

x°  =  1/2 ⋅ (280°)

x°  =  140°

So, the value of x is 140. 

Problem 5 : 

Find the value of x in the diagram shown below. 

Answer :

Using Theorem 2, we have 

m∠ GHF  =  1/2 ⋅ (m∠EDG - m∠GF)

72°  =  1/2 ⋅ (200° - x°)

Multiply each side by 2. 

144  =  200 - x

Solve for x.

x  =  56

Problem 6 :

Find the value of x in the diagram shown below. 

Answer :

Arcs MN and MLN make a whole circle. So, we have

m∠arc MLN + m∠MN  =  360°

Plug m∠MN  =  92°.

m∠arc MLN + 92°  =  360°

Subtract 92° from each side. 

m∠arc MLN  =  360° - 92°

m∠arc MLN  =  268°

Using Theorem 2, we have

x°  =  1/2 ⋅ (m∠MLN - m∠MN)

x°  =  1/2 ⋅ (268° - 92°)

x  =  1/2 ⋅ (176)

x  =  88

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