smartpugteaching on X: Parallel Lines Cut By A Transversal, lines ... - Free Printable

Part A: Finding the Measure of Specified Angles

We will solve each problem step by step using properties of parallel lines cut by a transversal, such as corresponding angles, alternate interior angles, and supplementary angles.

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

Solution:
- Since \( AB \parallel EH \), \( \angle CBG \) and \( \angle HFC \) are corresponding angles. Therefore, \( \angle HFC = 75^\circ \).
- \( \angle HFG \) is a linear pair with \( \angle HFC \), so they are supplementary. 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 \( \angle DBC \)

Solution:
- Since \( AB \parallel EH \), \( \angle CBG \) and \( \angle HFC \) are corresponding angles. Therefore, \( \angle HFC = 38^\circ \).
- \( \angle DBC \) is an alternate interior angle to \( \angle CBG \). 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 \( \angle EFG \)

Solution:
- Since \( AB \parallel EH \), \( \angle DBC \) is an alternate interior angle to \( \angle CBG \). Therefore, \( \angle DBC = 36^\circ \).
- \( \angle EFG \) is a corresponding angle to \( \angle CBG \). 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 \( \angle DFG \)

Solution:
- Since \( AC \parallel BD \), \( \angle AEH \) is a corresponding angle to \( \angle GEF \). Therefore, \( \angle AEH = 62^\circ \).
- \( \angle DFG \) is an alternate interior angle to \( \angle GEF \). 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 properties of angles formed by parallel lines and a transversal.

#### 1) \( \angle 1 \) and \( \angle 2 \) are vertically opposite angles.
- Analysis: Vertically opposite angles are formed by the intersection of two lines and are always equal. Here, \( \angle 1 \) and \( \angle 2 \) are not vertically opposite angles; they are adjacent angles.
- Answer: F

#### 2) \( \angle 1 \) and \( \angle 5 \) are corresponding angles.
- Analysis: 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.
- Answer: T

#### 3) \( \angle 2 \) and \( \angle 5 \) are alternate exterior angles.
- Analysis: Alternate exterior angles are on opposite sides of the transversal and outside the two lines. Here, \( \angle 2 \) and \( \angle 5 \) are not alternate exterior angles; they are consecutive interior angles.
- Answer: F

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

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

#### 6) \( \angle 3 \) is congruent to \( \angle 8 \).
- Analysis: \( \angle 3 \) and \( \angle 8 \) are corresponding angles. Corresponding angles are congruent when the lines are parallel. Here, \( \angle 3 \) and \( \angle 8 \) are corresponding angles and are congruent.
- Answer: T

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. F
2. T
3. F
4. T
5. T
6. T

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