Question

a) name the relation between the angles formed by the transversal in the figure below.
∠1 and ∠8 =
∠4 and ∠6 =
∠4 and ∠5 =
∠2 and ∠6 =
∠3 and ∠5 =
∠3 and ∠6 =
∠1 and ∠5 =
∠2 and ∠7 =
b) use your knowledge of angle pairs to find the measures of the specified angles in the following.
①
∠x = ∠y =
②
∠x = ∠y =
③
∠x = ∠y =
④
∠x = ∠y =

Explanation:

Step1: Recall angle - pair relationships

For two parallel lines cut by a transversal:

• $\angle1$ and $\angle8$ are alternate - exterior angles.
• $\angle4$ and $\angle6$ are alternate - interior angles.
• $\angle4$ and $\angle5$ are same - side interior angles.
• $\angle2$ and $\angle6$ are corresponding angles.
• $\angle3$ and $\angle5$ are same - side interior angles.
• $\angle3$ and $\angle6$ are alternate - interior angles.
• $\angle1$ and $\angle5$ are corresponding angles.
• $\angle2$ and $\angle7$ are alternate - exterior angles.

Step2: Use angle - pair properties to find angle measures

1. $\angle x = 30^{\circ}$ (corresponding angles), $\angle y=150^{\circ}$ (linear - pair with $\angle x$).
2. $\angle x = 127^{\circ}$ (corresponding angles), $\angle y = 53^{\circ}$ (linear - pair with $\angle x$).
3. $\angle x = 64^{\circ}$ (corresponding angles), $\angle y = 116^{\circ}$ (linear - pair with $\angle x$).
4. $\angle x = 73^{\circ}$ (corresponding angles), $\angle y = 107^{\circ}$ (linear - pair with $\angle x$).

Answer:

A)
$\angle1$ and $\angle8$: Alternate - exterior angles
$\angle4$ and $\angle6$: Alternate - interior angles
$\angle4$ and $\angle5$: Same - side interior angles
$\angle2$ and $\angle6$: Corresponding angles
$\angle3$ and $\angle5$: Same - side interior angles
$\angle3$ and $\angle6$: Alternate - interior angles
$\angle1$ and $\angle5$: Corresponding angles
$\angle2$ and $\angle7$: Alternate - exterior angles
B)

1. $\angle x = 30^{\circ}$, $\angle y = 150^{\circ}$
2. $\angle x = 127^{\circ}$, $\angle y = 53^{\circ}$
3. $\angle x = 64^{\circ}$, $\angle y = 116^{\circ}$
4. $\angle x = 73^{\circ}$, $\angle y = 107^{\circ}$