Question

in the figure below, lines a and b are parallel. using the figure above, determine if each statement is true or false. circle your answers. 1. ∠3 and ∠5 are same - side interior angles. true false 2. ∠1 and ∠7 are alternate interior angles. true false 3. the measures of ∠2 and ∠5 add up to 180°. true false 4. ∠2 and ∠4 are congruent. true false 5. ∠2 and ∠5 are alternate interior angles. true false 6. the measure of ∠6 is 46°. true false each statement below is false. using the figure above, rewrite each statement to make it true by updating the underlined part. some of the statements can be correctly updated in more than one way. 7. ∠1 and ∠6 are corresponding angles. 8. ∠3 and ∠5 are congruent. 9. ∠3 and ∠4 are supplementary. 10. the measure of ∠4 is 46°.

Explanation:

Step1: Recall same - side interior angles

Same - side interior angles are between the parallel lines and on the same side of the transversal. ∠3 and ∠5 are between lines a and b and on the same side of transversal t, so the statement "∠3 and ∠5 are same - side interior angles" is True.

Step2: Recall alternate interior angles

Alternate interior angles are between the parallel lines and on opposite sides of the transversal. ∠1 and ∠7 are not between the parallel lines, so the statement "∠1 and ∠7 are alternate interior angles" is False.

Step3: Recall angle - sum property for parallel lines

∠2 and ∠5 are same - side interior angles. For parallel lines, same - side interior angles are supplementary (add up to 180°), so the statement "The measures of ∠2 and ∠5 add up to 180°" is True.

Step4: Recall vertical angles

Vertical angles are congruent. ∠2 and ∠4 are vertical angles, so the statement "∠2 and ∠4 are congruent" is True.

Step5: Recall alternate interior angles

∠2 and ∠5 are same - side interior angles, not alternate interior angles, so the statement "∠2 and ∠5 are alternate interior angles" is False.

Step6: Recall corresponding angles

∠1 and ∠5 are corresponding angles. ∠1 = 46°, and ∠6 and ∠5 are supplementary. ∠6=180 - 46=134°, so the statement "The measure of ∠6 is 46°" is False.

Step7: Rewrite corresponding - angles statement

∠1 and ∠5 are corresponding angles. So we can rewrite "∠1 and ∠6 are corresponding angles" as "∠1 and ∠5 are corresponding angles".

Step8: Rewrite congruent - angles statement

∠3 and ∠5 are same - side interior angles, not congruent. ∠3 and ∠7 are congruent (alternate exterior angles). So we can rewrite "∠3 and ∠5 are congruent" as "∠3 and ∠7 are congruent".

Step9: Rewrite supplementary - angles statement

∠3 and ∠4 are not supplementary. ∠3 and ∠2 are supplementary (linear - pair). So we can rewrite "∠3 and ∠4 are supplementary" as "∠3 and ∠2 are supplementary".

Step10: Rewrite angle - measure statement

∠4 and ∠1 are alternate exterior angles. ∠1 = 46°, so ∠4 = 46°. The original statement is True, but if we want to rewrite it based on the false - statement requirement, we can say "The measure of ∠4 is equal to the measure of ∠1".

Answer:

1. True
2. False
3. True
4. True
5. False
6. False
7. ∠1 and ∠5 are corresponding angles.
8. ∠3 and ∠7 are congruent.
9. ∠3 and ∠2 are supplementary.
10. The measure of ∠4 is equal to the measure of ∠1.