Practice worksheet for solving angle relationships using geometric diagrams and algebraic expressions.

To solve the problems involving angle relationships, we need to use basic geometric principles such as:

1. Complementary Angles: Two angles are complementary if their measures add up to 90°.
2. Supplementary Angles: Two angles are supplementary if their measures add up to 180°.
3. Vertical Angles: Vertical angles are congruent (equal in measure).
4. Adjacent Angles: Adjacent angles share a common vertex and a common side but do not overlap.

Let's solve each problem step by step.

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Problem 1:

Given:
- \( m\angle CPM = \)
- \( m\angle QTM = 62^\circ \)

Since \( \angle CPM \) and \( \angle QTM \) are vertical angles, they are congruent:
\[ m\angle CPM = m\angle QTM = 62^\circ \]

Answer:
\[ \boxed{62^\circ} \]

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Problem 2:

Given:
- \( m\angle XFY = \)
- \( m\angle TFY = 24^\circ \)

Since \( \angle XFY \) and \( \angle TFY \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle XFY + m\angle TFY = 180^\circ \]
\[ m\angle XFY + 24^\circ = 180^\circ \]
\[ m\angle XFY = 180^\circ - 24^\circ \]
\[ m\angle XFY = 156^\circ \]

Answer:
\[ \boxed{156^\circ} \]

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Problem 3:

Given:
- \( m\angle CXZ = \)
- \( m\angle BXZ = 52^\circ \)

Since \( \angle CXZ \) and \( \angle BXZ \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle CXZ + m\angle BXZ = 180^\circ \]
\[ m\angle CXZ + 52^\circ = 180^\circ \]
\[ m\angle CXZ = 180^\circ - 52^\circ \]
\[ m\angle CXZ = 128^\circ \]

Answer:
\[ \boxed{128^\circ} \]

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Problem 4:

Given:
- \( m\angle ZMP = \)
- \( m\angle ZMQ = 82^\circ \)

Since \( \angle ZMP \) and \( \angle ZMQ \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle ZMP + m\angle ZMQ = 180^\circ \]
\[ m\angle ZMP + 82^\circ = 180^\circ \]
\[ m\angle ZMP = 180^\circ - 82^\circ \]
\[ m\angle ZMP = 98^\circ \]

Answer:
\[ \boxed{98^\circ} \]

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Problem 5:

Given:
- \( m\angle XFD = \)
- \( m\angle EFD = 53^\circ \)

Since \( \angle XFD \) and \( \angle EFD \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle XFD + m\angle EFD = 180^\circ \]
\[ m\angle XFD + 53^\circ = 180^\circ \]
\[ m\angle XFD = 180^\circ - 53^\circ \]
\[ m\angle XFD = 127^\circ \]

Answer:
\[ \boxed{127^\circ} \]

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Problem 6:

Given:
- \( m\angle CRU = \)
- \( m\angle BRU = 69^\circ \)

Since \( \angle CRU \) and \( \angle BRU \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle CRU + m\angle BRU = 180^\circ \]
\[ m\angle CRU + 69^\circ = 180^\circ \]
\[ m\angle CRU = 180^\circ - 69^\circ \]
\[ m\angle CRU = 111^\circ \]

Answer:
\[ \boxed{111^\circ} \]

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Problem 7:

Given:
- \( m\angle USM = \)
- \( m\angle KSM = 92^\circ \)

Since \( \angle USM \) and \( \angle KSM \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle USM + m\angle KSM = 180^\circ \]
\[ m\angle USM + 92^\circ = 180^\circ \]
\[ m\angle USM = 180^\circ - 92^\circ \]
\[ m\angle USM = 88^\circ \]

Answer:
\[ \boxed{88^\circ} \]

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Problem 8:

Given:
- \( m\angle KMR = \)
- \( m\angle QMR = 133^\circ \)

Since \( \angle KMR \) and \( \angle QMR \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle KMR + m\angle QMR = 180^\circ \]
\[ m\angle KMR + 133^\circ = 180^\circ \]
\[ m\angle KMR = 180^\circ - 133^\circ \]
\[ m\angle KMR = 47^\circ \]

Answer:
\[ \boxed{47^\circ} \]

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Problem 9:

Given:
- \( m\angle HMR = \)
- \( m\angle ZMR = 115^\circ \)

Since \( \angle HMR \) and \( \angle ZMR \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle HMR + m\angle ZMR = 180^\circ \]
\[ m\angle HMR + 115^\circ = 180^\circ \]
\[ m\angle HMR = 180^\circ - 115^\circ \]
\[ m\angle HMR = 65^\circ \]

Answer:
\[ \boxed{65^\circ} \]

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Problem 10:

Given:
- \( m\angle EXP = \)
- \( m\angle GXP = 74^\circ \)

Since \( \angle EXP \) and \( \angle GXP \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle EXP + m\angle GXP = 180^\circ \]
\[ m\angle EXP + 74^\circ = 180^\circ \]
\[ m\angle EXP = 180^\circ - 74^\circ \]
\[ m\angle EXP = 106^\circ \]

Answer:
\[ \boxed{106^\circ} \]

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Problem 11:

Given:
- \( m\angle CZQ = \)
- \( m\angle GZQ = 11^\circ \)

Since \( \angle CZQ \) and \( \angle GZQ \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle CZQ + m\angle GZQ = 180^\circ \]
\[ m\angle CZQ + 11^\circ = 180^\circ \]
\[ m\angle CZQ = 180^\circ - 11^\circ \]
\[ m\angle CZQ = 169^\circ \]

Answer:
\[ \boxed{169^\circ} \]

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Problem 12:

Given:
- \( m\angle FTR = \)
- \( m\angle VTR = 129^\circ \)

Since \( \angle FTR \) and \( \angle VTR \) are adjacent angles forming a straight line, they are supplementary:
\[ m\angle FTR + m\angle VTR = 180^\circ \]
\[ m\angle FTR + 129^\circ = 180^\circ \]
\[ m\angle FTR = 180^\circ - 129^\circ \]
\[ m\angle FTR = 51^\circ \]

Answer:
\[ \boxed{51^\circ} \]

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Final Answers:

\[
\boxed{
62^\circ, 156^\circ, 128^\circ, 98^\circ, 127^\circ, 111^\circ, 88^\circ, 47^\circ, 65^\circ, 106^\circ, 169^\circ, 51^\circ
}
\]