Educational worksheet on angle pairs formed by a transversal, including naming relationships and solving for unknown angles.

Problem Analysis:

The problem involves understanding the relationships between angles formed by a transversal intersecting two parallel lines. We will solve both parts step by step.

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Part A: Naming the Relation Between Angles

When a transversal intersects two parallel lines, several types of angle pairs are formed. These include:
1. Corresponding Angles: Angles that are in the same relative position at each intersection.
2. Alternate Interior Angles: Angles that lie on opposite sides of the transversal and between the two lines.
3. Alternate Exterior Angles: Angles that lie on opposite sides of the transversal and outside the two lines.
4. Consecutive Interior Angles (or Same-Side Interior Angles): Angles that lie on the same side of the transversal and between the two lines.
5. Vertical Angles: Angles opposite each other when two lines cross.

Using these definitions, we can identify the relationships for each pair of angles:

#### 1. ∠1 and ∠8
- ∠1 and ∠8 are Alternate Exterior Angles because they are on opposite sides of the transversal and outside the two lines.

#### 2. ∠4 and ∠6
- ∠4 and ∠6 are Consecutive Interior Angles (or Same-Side Interior Angles) because they are on the same side of the transversal and between the two lines.

#### 3. ∠4 and ∠5
- ∠4 and ∠5 are Linear Pair or Supplementary Angles because they form a straight line and add up to 180°.

#### 4. ∠2 and ∠6
- ∠2 and ∠6 are Corresponding Angles because they are in the same relative position at each intersection.

#### 5. ∠3 and ∠5
- ∠3 and ∠5 are Alternate Interior Angles because they are on opposite sides of the transversal and between the two lines.

#### 6. ∠3 and ∠6
- ∠3 and ∠6 are Consecutive Interior Angles (or Same-Side Interior Angles) because they are on the same side of the transversal and between the two lines.

#### 7. ∠1 and ∠5
- ∠1 and ∠5 are Corresponding Angles because they are in the same relative position at each intersection.

#### 8. ∠2 and ∠7
- ∠2 and ∠7 are Alternate Exterior Angles because they are on opposite sides of the transversal and outside the two lines.

---

Part B: Finding the Measures of Specified Angles

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

#### Figure ①
- Given: ∠x is adjacent to a 30° angle.
- Since ∠x and the 30° angle form a linear pair, their sum is 180°:
\[
∠x + 30° = 180° \implies ∠x = 180° - 30° = 150°
\]
- ∠y is a corresponding angle to the 30° angle, so:
\[
∠y = 30°
\]

#### Figure ②
- Given: One angle is 127°.
- ∠x is an alternate interior angle to the 127° angle, so:
\[
∠x = 127°
\]
- ∠y is a consecutive interior angle to the 127° angle, so their sum is 180°:
\[
∠y + 127° = 180° \implies ∠y = 180° - 127° = 53°
\]

#### Figure ③
- Given: One angle is 64°.
- ∠x is a consecutive interior angle to the 64° angle, so their sum is 180°:
\[
∠x + 64° = 180° \implies ∠x = 180° - 64° = 116°
\]
- ∠y is a corresponding angle to the 64° angle, so:
\[
∠y = 64°
\]

#### Figure ④
- Given: One angle is 73°.
- ∠x is an alternate interior angle to the 73° angle, so:
\[
∠x = 73°
\]
- ∠y is a vertical angle to the 73° angle, so:
\[
∠y = 73°
\]

---

Final Answers:

#### Part A:
\[
\begin{aligned}
&\text{∠1 and ∠8: } \text{Alternate Exterior Angles} \\
&\text{∠4 and ∠6: } \text{Consecutive Interior Angles} \\
&\text{∠4 and ∠5: } \text{Linear Pair/Supplementary Angles} \\
&\text{∠2 and ∠6: } \text{Corresponding Angles} \\
&\text{∠3 and ∠5: } \text{Alternate Interior Angles} \\
&\text{∠3 and ∠6: } \text{Consecutive Interior Angles} \\
&\text{∠1 and ∠5: } \text{Corresponding Angles} \\
&\text{∠2 and ∠7: } \text{Alternate Exterior Angles}
\end{aligned}
\]

#### Part B:
\[
\begin{aligned}
&\text{Figure ①: } ∠x = 150°, ∠y = 30° \\
&\text{Figure ②: } ∠x = 127°, ∠y = 53° \\
&\text{Figure ③: } ∠x = 116°, ∠y = 64° \\
&\text{Figure ④: } ∠x = 73°, ∠y = 73°
\end{aligned}
\]

Boxed Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Part A: } \\
&\text{∠1 and ∠8: Alternate Exterior Angles} \\
&\text{∠4 and ∠6: Consecutive Interior Angles} \\
&\text{∠4 and ∠5: Linear Pair/Supplementary Angles} \\
&\text{∠2 and ∠6: Corresponding Angles} \\
&\text{∠3 and ∠5: Alternate Interior Angles} \\
&\text{∠3 and ∠6: Consecutive Interior Angles} \\
&\text{∠1 and ∠5: Corresponding Angles} \\
&\text{∠2 and ∠7: Alternate Exterior Angles} \\
&\text{Part B: } \\
&\text{Figure ①: } ∠x = 150°, ∠y = 30° \\
&\text{Figure ②: } ∠x = 127°, ∠y = 53° \\
&\text{Figure ③: } ∠x = 116°, ∠y = 64° \\
&\text{Figure ④: } ∠x = 73°, ∠y = 73°
\end{aligned}
}
\]