Circle Theorems worksheet with 12 problems to find missing angles in circles.

To solve the missing angles in these circle problems, we will use various circle theorems. Here are the key theorems we might use:

1. Angle at the Center Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the circumference.
2. Angles in the Same Segment: Angles subtended by the same arc in the same segment are equal.
3. Cyclic Quadrilateral: Opposite angles in a cyclic quadrilateral sum to 180°.
4. Tangent-Secant Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
5. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is 180°.

Let's solve each problem step by step.

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Problem 1

- Given: Central angle = 118°
- To Find: Angle \( a \)

Using the Angle at the Center Theorem:
\[ a = \frac{1}{2} \times 118^\circ = 59^\circ \]

Answer: \( a = 59^\circ \)

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Problem 2

- Given: Angles \( 62^\circ \) and \( 47^\circ \)
- To Find: Angles \( b \) and \( c \)

Using the Sum of Angles in a Triangle:
\[ b + 62^\circ + 47^\circ = 180^\circ \]
\[ b = 180^\circ - 62^\circ - 47^\circ = 71^\circ \]

Since \( b \) and \( c \) are opposite angles in a cyclic quadrilateral:
\[ c = 180^\circ - b = 180^\circ - 71^\circ = 109^\circ \]

Answers: \( b = 71^\circ \), \( c = 109^\circ \)

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Problem 3

- Given: Angles \( 12^\circ \), \( 24^\circ \)
- To Find: Angles \( d \), \( e \), \( f \)

Using the Angles in the Same Segment:
\[ d = 12^\circ \]
\[ e = 24^\circ \]

Using the Sum of Angles in a Triangle for the triangle containing \( f \):
\[ f + 12^\circ + 24^\circ = 180^\circ \]
\[ f = 180^\circ - 12^\circ - 24^\circ = 144^\circ \]

Answers: \( d = 12^\circ \), \( e = 24^\circ \), \( f = 144^\circ \)

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Problem 4

- Given: Angle \( 49^\circ \)
- To Find: Angle \( g \)

Using the Angle at the Center Theorem:
\[ g = \frac{1}{2} \times 49^\circ = 24.5^\circ \]

Answer: \( g = 24.5^\circ \)

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Problem 5

- Given: Angles \( 43^\circ \)
- To Find: Angles \( h \) and \( i \)

Using the Tangent-Secant Theorem:
\[ h = 43^\circ \]

Using the Sum of Angles in a Triangle:
\[ i + 43^\circ + 43^\circ = 180^\circ \]
\[ i = 180^\circ - 43^\circ - 43^\circ = 94^\circ \]

Answers: \( h = 43^\circ \), \( i = 94^\circ \)

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Problem 6

- Given: Angle \( 116^\circ \)
- To Find: Angle \( j \)

Using the Cyclic Quadrilateral property:
\[ j = 180^\circ - 116^\circ = 64^\circ \]

Answer: \( j = 64^\circ \)

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Problem 7

- Given: Angles \( 67^\circ \) and \( 81^\circ \)
- To Find: Angles \( k \), \( l \), \( m \)

Using the Angles in the Same Segment:
\[ k = 67^\circ \]
\[ l = 81^\circ \]

Using the Sum of Angles in a Triangle for the triangle containing \( m \):
\[ m + 67^\circ + 81^\circ = 180^\circ \]
\[ m = 180^\circ - 67^\circ - 81^\circ = 32^\circ \]

Answers: \( k = 67^\circ \), \( l = 81^\circ \), \( m = 32^\circ \)

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Problem 8

- Given: Angles \( 50^\circ \) and \( 142^\circ \)
- To Find: Angle \( n \)

Using the Cyclic Quadrilateral property:
\[ n = 180^\circ - 50^\circ = 130^\circ \]

Answer: \( n = 130^\circ \)

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Problem 9

- Given: Central angle \( 232^\circ \)
- To Find: Angle \( p \)

Using the Angle at the Center Theorem:
\[ p = \frac{1}{2} \times (360^\circ - 232^\circ) = \frac{1}{2} \times 128^\circ = 64^\circ \]

Answer: \( p = 64^\circ \)

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Problem 10

- Given: Angle \( 76^\circ \)
- To Find: Angle \( q \)

Using the Angle at the Center Theorem:
\[ q = \frac{1}{2} \times 76^\circ = 38^\circ \]

Answer: \( q = 38^\circ \)

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Problem 11

- Given: Angle \( 122^\circ \)
- To Find: Angle \( r \)

Using the Cyclic Quadrilateral property:
\[ r = 180^\circ - 122^\circ = 58^\circ \]

Answer: \( r = 58^\circ \)

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Problem 12

- Given: Angles \( 44^\circ \), \( 25^\circ \)
- To Find: Angles \( s \) and \( t \)

Using the Sum of Angles in a Triangle for the triangle containing \( t \):
\[ t + 44^\circ + 25^\circ = 180^\circ \]
\[ t = 180^\circ - 44^\circ - 25^\circ = 111^\circ \]

Using the Cyclic Quadrilateral property:
\[ s = 180^\circ - 111^\circ = 69^\circ \]

Answers: \( s = 69^\circ \), \( t = 111^\circ \)

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Final Answers

\[
\boxed{
\begin{array}{ll}
a = 59^\circ & b = 71^\circ, c = 109^\circ \\
d = 12^\circ, e = 24^\circ, f = 144^\circ & g = 24.5^\circ \\
h = 43^\circ, i = 94^\circ & j = 64^\circ \\
k = 67^\circ, l = 81^\circ, m = 32^\circ & n = 130^\circ \\
p = 64^\circ & q = 38^\circ \\
r = 58^\circ & s = 69^\circ, t = 111^\circ \\
\end{array}
}
\]