Question

$\overline{tv} \cong \overline{su}$. complete the proof that $\overline{st} \perp \overline{tu}$.

1. $\overline{tu} \parallel \overline{sv}$: given
2. $\overline{uv} \parallel \overline{st}$: given
3. $\overline{tv} \cong \overline{su}$: given
4. $\overline{st} \cong \overline{uv}$: parallelograms have congruent opposite sides
5. $\overline{tu} \cong \overline{tu}$: reflexive property of congruence
6. $\triangle stu \cong \triangle vut$: sss
7. $\angle stu \cong \angle tuv$: cpctc
8. $m\angle stu + m\angle tuv = 180^{\circ}$: dropdown
9. $m\angle stu + m\angle stu = 180^{\circ}$: dropdown
10. $m\angle stu = 90^{\circ}$: properties of addition, subtraction, multiplication, and division
11. $\overline{st} \perp \overline{tu}$: definition of perpendicular lines

Explanation:

Step1: Identify reason for Step8

Consecutive angles of a parallelogram are supplementary (they form same-side interior angles between parallel lines, so their measures sum to $180^\circ$).

Step2: Identify reason for Step9

Substitute $\angle TUV$ with $\angle STU$ using the congruence from Step7 (CPCTC result: $\angle STU \cong \angle TUV$, so $m\angle STU = m\angle TUV$).

Answer:

For Step8 Reason: Consecutive parallelogram angles are supplementary (or Same-side interior angles are supplementary)
For Step9 Reason: Substitution Property of Equality (using $\angle STU \cong \angle TUV$)