Pairs of Angles Worksheets - Free Printable

Problem Analysis:

The worksheet involves identifying and solving problems related to pairs of angles, including complementary angles, supplementary angles, adjacent angles, and linear pairs. Let's solve each part step by step.

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Problem 1:

#### a) Name two angles that are complementary.
- Definition of Complementary Angles: Two angles are complementary if their measures add up to 90°.
- From the diagram:
- The given angles are \( \angle AOB = 45^\circ \) and \( \angle BOC = 45^\circ \).
- \( \angle AOB + \angle BOC = 45^\circ + 45^\circ = 90^\circ \).

Thus, the two complementary angles are:
\[ \boxed{\angle AOB \text{ and } \angle BOC} \]

#### b) Name the angle supplementary to \( \angle BOA \).
- Definition of Supplementary Angles: Two angles are supplementary if their measures add up to 180°.
- From the diagram:
- \( \angle BOA = 45^\circ \).
- The angle supplementary to \( \angle BOA \) must be \( 180^\circ - 45^\circ = 135^\circ \).
- The angle \( \angle COD \) is given as \( 3x \), and since \( x = 45^\circ \), \( \angle COD = 3 \times 45^\circ = 135^\circ \).

Thus, the angle supplementary to \( \angle BOA \) is:
\[ \boxed{\angle COD} \]

#### c) \( \angle AOE \) and \( \angle DOE \) are supplementary. Find \( m\angle AOE \).
- Given: \( \angle AOE \) and \( \angle DOE \) are supplementary.
- From the diagram:
- \( \angle DOE = 3x \).
- Since \( x = 45^\circ \), \( \angle DOE = 3 \times 45^\circ = 135^\circ \).
- Since \( \angle AOE \) and \( \angle DOE \) are supplementary, their measures add up to 180°:
\[
m\angle AOE + m\angle DOE = 180^\circ
\]
\[
m\angle AOE + 135^\circ = 180^\circ
\]
\[
m\angle AOE = 180^\circ - 135^\circ = 45^\circ
\]

Thus, the measure of \( \angle AOE \) is:
\[ \boxed{45^\circ} \]

#### d) Name the angles adjacent to \( \angle BOC \).
- Definition of Adjacent Angles: Two angles are adjacent if they share a common vertex and a common side but do not overlap.
- From the diagram:
- The angles adjacent to \( \angle BOC \) are \( \angle AOB \) and \( \angle COD \).

Thus, the angles adjacent to \( \angle BOC \) are:
\[ \boxed{\angle AOB \text{ and } \angle COD} \]

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Problem 2:

#### a) Name any one pair of adjacent angles.
- Definition of Adjacent Angles: Two angles are adjacent if they share a common vertex and a common side but do not overlap.
- From the diagram:
- One pair of adjacent angles is \( \angle POQ \) and \( \angle QOR \).

Thus, one pair of adjacent angles is:
\[ \boxed{\angle POQ \text{ and } \angle QOR} \]

#### b) Find \( m\angle QOR \).
- From the diagram:
- \( \angle QOR \) is given as \( 52^\circ \).

Thus, the measure of \( \angle QOR \) is:
\[ \boxed{52^\circ} \]

#### c) Name the angle complementary to \( \angle ROS \).
- Definition of Complementary Angles: Two angles are complementary if their measures add up to 90°.
- From the diagram:
- \( \angle ROS = 52^\circ \).
- The angle complementary to \( \angle ROS \) must be \( 90^\circ - 52^\circ = 38^\circ \).
- The angle \( \angle POQ \) is given as \( 38^\circ \).

Thus, the angle complementary to \( \angle ROS \) is:
\[ \boxed{\angle POQ} \]

#### d) Which angle forms a linear pair with \( \angle POQ \)?
- Definition of Linear Pair: Two angles form a linear pair if they are adjacent and their non-common sides form a straight line, meaning their measures add up to 180°.
- From the diagram:
- The angle adjacent to \( \angle POQ \) is \( \angle QOR \).
- Since \( \angle POQ \) and \( \angle QOR \) are adjacent and their non-common sides form a straight line, they form a linear pair.

Thus, the angle that forms a linear pair with \( \angle POQ \) is:
\[ \boxed{\angle QOR} \]

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Final Answers:

1. a) \( \boxed{\angle AOB \text{ and } \angle BOC} \)
b) \( \boxed{\angle COD} \)
c) \( \boxed{45^\circ} \)
d) \( \boxed{\angle AOB \text{ and } \angle COD} \)

2. a) \( \boxed{\angle POQ \text{ and } \angle QOR} \)
b) \( \boxed{52^\circ} \)
c) \( \boxed{\angle POQ} \)
d) \( \boxed{\angle QOR} \)

Final Boxed Answer:
\[
\boxed{
\begin{array}{l}
1. \text{a) } \angle AOB \text{ and } \angle BOC \\
\text{b) } \angle COD \\
\text{c) } 45^\circ \\
\text{d) } \angle AOB \text{ and } \angle COD \\
2. \text{a) } \angle POQ \text{ and } \angle QOR \\
\text{b) } 52^\circ \\
\text{c) } \angle POQ \\
\text{d) } \angle QOR \\
\end{array}
}
\]