Question

developing proof use the given information to determine which lines, if any, are parallel. justify each conclusion with a theorem or postulate.

1. ∠11 is supplementary to ∠10.
2. ∠6≅∠9
3. ∠13 is supplementary to ∠14.
4. ∠13≅∠15
5. ∠12 is supplementary to ∠3.
6. ∠2≅∠13

Explanation:

Step1: Recall parallel - line postulates and theorems

If two lines are cut by a transversal, corresponding angles are congruent when the lines are parallel, alternate - interior angles are congruent, and same - side interior angles are supplementary.

Step15: For ∠11 is supplementary to ∠10

If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel. Let the transversal be the line that forms ∠10 and ∠11. If ∠10 and ∠11 are same - side interior angles, then the lines cut by this transversal are parallel.

Step17: For ∠13 is supplementary to ∠14

If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel. Let the transversal be the line that forms ∠13 and ∠14. If ∠13 and ∠14 are same - side interior angles, then the lines cut by this transversal are parallel.

Step19: For ∠12 is supplementary to ∠3

If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel. Let the transversal be the line that forms ∠3 and ∠12. If ∠3 and ∠12 are same - side interior angles, then the lines cut by this transversal are parallel.

Step16: For ∠6≅∠9

If two lines are cut by a transversal and alternate - interior angles are congruent, then the lines are parallel. Let the transversal be the line that forms ∠6 and ∠9. Since ∠6 and ∠9 are alternate - interior angles, the lines cut by this transversal are parallel.

Step18: For ∠13≅∠15

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. Let the transversal be the line that forms ∠13 and ∠15. Since ∠13 and ∠15 are corresponding angles, the lines cut by this transversal are parallel.

Step20: For ∠2≅∠13

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. Let the transversal be the line that forms ∠2 and ∠13. Since ∠2 and ∠13 are corresponding angles, the lines cut by this transversal are parallel.

1. If ∠11 is supplementary to ∠10, then the lines cut by the transversal that forms these angles are parallel (Same - Side Interior Angles Supplementary Theorem).
2. If ∠6≅∠9, then the lines cut by the transversal that forms these angles are parallel (Alternate - Interior Angles Congruent Theorem).
3. If ∠13 is supplementary to ∠14, then the lines cut by the transversal that forms these angles are parallel (Same - Side Interior Angles Supplementary Theorem).
4. If ∠13≅∠15, then the lines cut by the transversal that forms these angles are parallel (Corresponding Angles Congruent Postulate).
5. If ∠12 is supplementary to ∠3, then the lines cut by the transversal that forms these angles are parallel (Same - Side Interior Angles Supplementary Theorem).
6. If ∠2≅∠13, then the lines cut by the transversal that forms these angles are parallel (Corresponding Angles Congruent Postulate).

Answer:

1. The lines cut by the transversal forming ∠10 and ∠11 are parallel by the Same - Side Interior Angles Supplementary Theorem.
2. The lines cut by the transversal forming ∠6 and ∠9 are parallel by the Alternate - Interior Angles Congruent Theorem.
3. The lines cut by the transversal forming ∠13 and ∠14 are parallel by the Same - Side Interior Angles Supplementary Theorem.
4. The lines cut by the transversal forming ∠13 and ∠15 are parallel by the Corresponding Angles Congruent Postulate.
5. The lines cut by the transversal forming ∠12 and ∠3 are parallel by the Same - Side Interior Angles Supplementary Theorem.
6. The lines cut by the transversal forming ∠2 and ∠13 are parallel by the Corresponding Angles Congruent Postulate.