A Name the relation between the angles formed | StudyX - Free Printable

Problem Analysis and Solution

#### Part A: Naming the Relation Between Angles Formed by the Transversal

We are given a figure where a transversal intersects two parallel lines. The angles formed can be classified into specific pairs based on their positions relative to the transversal and the parallel lines. Let's identify the relationships for each pair of angles:

1. ∠1 and ∠8:
- These angles are alternate exterior angles. They are located on opposite sides of the transversal and outside the two parallel lines.

2. ∠4 and ∠6:
- These angles are consecutive interior angles (also called same-side interior angles). They are located on the same side of the transversal and between the two parallel lines.

3. ∠4 and ∠5:
- These angles are adjacent angles. They share a common vertex and a common side but do not overlap.

4. ∠2 and ∠6:
- These angles are corresponding angles. They are in the same relative position at each intersection of the transversal with the parallel lines.

5. ∠3 and ∠5:
- These angles are vertical angles. They are opposite each other at the intersection of two lines.

6. ∠3 and ∠6:
- These angles are consecutive interior angles (also called same-side interior angles). They are located on the same side of the transversal and between the two parallel lines.

7. ∠1 and ∠5:
- These angles are corresponding angles. They are in the same relative position at each intersection of the transversal with the parallel lines.

8. ∠2 and ∠7:
- These angles are alternate interior angles. They are located on opposite sides of the transversal and between the two parallel lines.

Final Answers for Part A:

\[
\begin{aligned}
&\text{∠1 and ∠8 = alternate exterior angles} \\
&\text{∠4 and ∠6 = consecutive interior angles} \\
&\text{∠4 and ∠5 = adjacent angles} \\
&\text{∠2 and ∠6 = corresponding angles} \\
&\text{∠3 and ∠5 = vertical angles} \\
&\text{∠3 and ∠6 = consecutive interior angles} \\
&\text{∠1 and ∠5 = corresponding angles} \\
&\text{∠2 and ∠7 = alternate interior angles}
\end{aligned}
\]

#### Part B: Finding the Measures of Specified Angles

We will use the properties of angle pairs (such as corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles) to find the measures of the specified angles.

##### Figure ①:
- Given: ∠x is an alternate interior angle to the 30° angle.
- Since alternate interior angles are congruent, ∠x = 30°.
- ∠y is a linear pair with the 30° angle, so ∠y = 180° - 30° = 150°.

##### Figure ②:
- Given: ∠x is a corresponding angle to the 127° angle.
- Since corresponding angles are congruent, ∠x = 127°.
- ∠y is a vertical angle to the 127° angle, so ∠y = 127°.

##### Figure ③:
- Given: ∠x is a consecutive interior angle to the 64° angle.
- Since consecutive interior angles are supplementary, ∠x = 180° - 64° = 116°.
- ∠y is a vertical angle to the 64° angle, so ∠y = 64°.

##### Figure ④:
- Given: ∠x is an alternate interior angle to the 73° angle.
- Since alternate interior angles are congruent, ∠x = 73°.
- ∠y is a corresponding angle to the 73° angle, so ∠y = 73°.

Final Answers for Part B:

\[
\begin{aligned}
&\text{Figure ①: } \angle x = 30^\circ, \angle y = 150^\circ \\
&\text{Figure ②: } \angle x = 127^\circ, \angle y = 127^\circ \\
&\text{Figure ③: } \angle x = 116^\circ, \angle y = 64^\circ \\
&\text{Figure ④: } \angle x = 73^\circ, \angle y = 73^\circ
\end{aligned}
\]

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Part A:} \\
\text{∠1 and ∠8 = alternate exterior angles} \\
\text{∠4 and ∠6 = consecutive interior angles} \\
\text{∠4 and ∠5 = adjacent angles} \\
\text{∠2 and ∠6 = corresponding angles} \\
\text{∠3 and ∠5 = vertical angles} \\
\text{∠3 and ∠6 = consecutive interior angles} \\
\text{∠1 and ∠5 = corresponding angles} \\
\text{∠2 and ∠7 = alternate interior angles} \\
\\
\text{Part B:} \\
\text{Figure ①: } \angle x = 30^\circ, \angle y = 150^\circ \\
\text{Figure ②: } \angle x = 127^\circ, \angle y = 127^\circ \\
\text{Figure ③: } \angle x = 116^\circ, \angle y = 64^\circ \\
\text{Figure ④: } \angle x = 73^\circ, \angle y = 73^\circ
\end{array}
}
\]