Question

describing congruent angle relationships

use the diagram to fill in the missing angle that makes the sentence true.

1 and \\( \angle 5 \\) are corresponding angles.
2 \\( \angle 7 \\) and are corresponding angles.
3 \\( \angle 8 \\) and are corresponding angles.
4 and \\( \angle 6 \\) are corresponding angles.
5 and \\( \angle 6 \\) are alternate interior angles.
6 \\( \angle 3 \\) and are alternate interior angles.
7 \\( \angle 2 \\) and are alternate exterior angles.
8 and \\( \angle 7 \\) are alternate exterior angles.

identify the pair of congruent angles in each diagram.

9 \\( \cong \angle 3 \\)
10 \\( \cong \\)
11 \\( \cong \\)

Explanation:

Step1: Recall corresponding angles definition

Corresponding angles are in the same relative position. For \( \angle5 \), \( \angle1 \) is in the same relative position (upper - left of the intersection of transversal and parallel lines).

Step2: Find \( \angle7 \) corresponding angle

For \( \angle7 \), \( \angle3 \) is in the same relative position (lower - right of the intersection of transversal and parallel lines).

Step3: Determine \( \angle8 \) corresponding angle

For \( \angle8 \), \( \angle4 \) is in the same relative position (lower - left of the intersection of transversal and parallel lines).

Step4: Identify \( \angle6 \) corresponding angle

For \( \angle6 \), \( \angle2 \) is in the same relative position (upper - right of the intersection of transversal and parallel lines).

Step5: Recall alternate interior angles definition

Alternate interior angles are between the parallel lines and on opposite sides of the transversal. For \( \angle6 \), \( \angle4 \) is the alternate interior angle.

Step6: Find \( \angle3 \) alternate interior angle

For \( \angle3 \), \( \angle5 \) is the alternate interior angle.

Step7: Recall alternate exterior angles definition

Alternate exterior angles are outside the parallel lines and on opposite sides of the transversal. For \( \angle2 \), \( \angle8 \) is the alternate exterior angle.

Step8: Determine \( \angle7 \) alternate exterior angle

For \( \angle7 \), \( \angle1 \) is the alternate exterior angle.

Step9: Recall vertical angles (congruent)

In the first sub - diagram for question 9, \( \angle1\cong\angle3 \) (vertical angles).

Step10: In the second sub - diagram for question 10

\( \angle4\cong\angle6 \) (vertical angles).

Step11: In the third sub - diagram for question 11

\( \angle7\cong\angle9 \) (vertical angles).

Answer:

1. \( \angle1 \)
2. \( \angle3 \)
3. \( \angle4 \)
4. \( \angle2 \)
5. \( \angle4 \)
6. \( \angle5 \)
7. \( \angle8 \)
8. \( \angle1 \)
9. \( \angle1 \)
10. \( \angle4\cong\angle6 \)
11. \( \angle7\cong\angle9 \)