Parallel Lines and Transversals Worksheets - Math Monks - Free Printable

Part A: Finding the Measure of Specified Angles

We will solve each problem step by step using properties of parallel lines and transversals, such as corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

#### 1)
- Given: \( \angle CBG = 75^\circ \)
- To find: \( \angle HFC \) and \( \angle HFG \)

Since \( AB \parallel EH \) and \( CD \) is a transversal:
- \( \angle CBG \) and \( \angle HFC \) are corresponding angles. Therefore, \( \angle HFC = 75^\circ \).
- \( \angle HFG \) is the supplementary angle to \( \angle HFC \) because they form a linear pair. Thus, \( \angle HFG = 180^\circ - 75^\circ = 105^\circ \).

Answer: \( \angle HFC = 75^\circ \), \( \angle HFG = 105^\circ \)

#### 2)
- Given: \( \angle CBG = 38^\circ \)
- To find: \( \angle HFC \) and \( \angle DBC \)

Since \( AB \parallel EH \) and \( CD \) is a transversal:
- \( \angle CBG \) and \( \angle HFC \) are corresponding angles. Therefore, \( \angle HFC = 38^\circ \).
- \( \angle DBC \) and \( \angle CBG \) are alternate interior angles. Therefore, \( \angle DBC = 38^\circ \).

Answer: \( \angle HFC = 38^\circ \), \( \angle DBC = 38^\circ \)

#### 3)
- Given: \( \angle CBG = 36^\circ \)
- To find: \( \angle DBC \) and \( \angle EFG \)

Since \( AB \parallel EH \) and \( CD \) is a transversal:
- \( \angle DBC \) and \( \angle CBG \) are alternate interior angles. Therefore, \( \angle DBC = 36^\circ \).
- \( \angle EFG \) and \( \angle CBG \) are corresponding angles. Therefore, \( \angle EFG = 36^\circ \).

Answer: \( \angle DBC = 36^\circ \), \( \angle EFG = 36^\circ \)

#### 4)
- Given: \( \angle GEF = 62^\circ \)
- To find: \( \angle AEH \) and \( \angle DFG \)

Since \( AC \parallel BD \) and \( EF \) is a transversal:
- \( \angle AEH \) and \( \angle GEF \) are corresponding angles. Therefore, \( \angle AEH = 62^\circ \).
- \( \angle DFG \) and \( \angle GEF \) are alternate interior angles. Therefore, \( \angle DFG = 62^\circ \).

Answer: \( \angle AEH = 62^\circ \), \( \angle DFG = 62^\circ \)

Part B: Deciding Whether Each Statement is True or False

We will analyze each statement based on the definitions of angle relationships formed by parallel lines and a transversal.

#### 1) \( \angle 1 \) and \( \angle 2 \) are vertically opposite angles.
- True: Vertically opposite angles are formed by the intersection of two lines and are always equal. Here, \( \angle 1 \) and \( \angle 2 \) are vertically opposite angles.

#### 2) \( \angle 1 \) and \( \angle 5 \) are corresponding angles.
- True: Corresponding angles are in the same relative position at each intersection where a straight line crosses two others. Here, \( \angle 1 \) and \( \angle 5 \) are corresponding angles.

#### 3) \( \angle 2 \) and \( \angle 5 \) are alternate exterior angles.
- False: Alternate exterior angles are on opposite sides of the transversal and outside the two parallel lines. Here, \( \angle 2 \) and \( \angle 5 \) are not alternate exterior angles; they are corresponding angles.

#### 4) \( \angle 4 \) and \( \angle 6 \) are alternate interior angles.
- True: Alternate interior angles are on opposite sides of the transversal and between the two parallel lines. Here, \( \angle 4 \) and \( \angle 6 \) are alternate interior angles.

#### 5) \( \angle 3 \) and \( \angle 6 \) are consecutive interior angles.
- True: Consecutive interior angles are on the same side of the transversal and between the two parallel lines. Here, \( \angle 3 \) and \( \angle 6 \) are consecutive interior angles.

#### 6) \( \angle 3 \) is congruent to \( \angle 8 \).
- True: \( \angle 3 \) and \( \angle 8 \) are corresponding angles, and corresponding angles are always congruent when the lines are parallel.

Final Answers

#### Part A:
1. \( \angle HFC = 75^\circ \), \( \angle HFG = 105^\circ \)
2. \( \angle HFC = 38^\circ \), \( \angle DBC = 38^\circ \)
3. \( \angle DBC = 36^\circ \), \( \angle EFG = 36^\circ \)
4. \( \angle AEH = 62^\circ \), \( \angle DFG = 62^\circ \)

#### Part B:
1. T
2. T
3. F
4. T
5. T
6. T

Boxed Answer:
\[
\boxed{
\begin{array}{ll}
\text{Part A:} & \\
1. & \angle HFC = 75^\circ, \angle HFG = 105^\circ \\
2. & \angle HFC = 38^\circ, \angle DBC = 38^\circ \\
3. & \angle DBC = 36^\circ, \angle EFG = 36^\circ \\
4. & \angle AEH = 62^\circ, \angle DFG = 62^\circ \\
\text{Part B:} & \\
1. & T \\
2. & T \\
3. & F \\
4. & T \\
5. & T \\
6. & T \\
\end{array}
}
\]