u is the midpoint of $\overline{qs}$ and $\overline{rt}$. complete the proof that $\overline{qr} \parallel \overline{st}$.

| | statement | reason |
| --- | --- | --- |
| 1 | u is the midpoint of $\overline{qs}$ | given |
| 2 | u is the midpoint of $\overline{rt}$ | given |
| 3 | $\overline{qu} \cong \overline{su}$ | definition of midpoint |
| 4 | $\overline{ru} \cong \overline{tu}$ | definition of midpoint |
| 5 | $\angle qur \cong \angle sut$ | vertical angle theorem |
| 6 | $\triangle qru \cong \triangle stu$ | sas |
| 7 | $\angle qru \cong \angle stu$ | cpctc |
| 8 | $\overline{qr} \parallel \overline{st}$ | |

Question

u is the midpoint of $\overline{qs}$ and $\overline{rt}$. complete the proof that $\overline{qr} \parallel \overline{st}$.

| | statement | reason |
| --- | --- | --- |
| 1 | u is the midpoint of $\overline{qs}$ | given |
| 2 | u is the midpoint of $\overline{rt}$ | given |
| 3 | $\overline{qu} \cong \overline{su}$ | definition of midpoint |
| 4 | $\overline{ru} \cong \overline{tu}$ | definition of midpoint |
| 5 | $\angle qur \cong \angle sut$ | vertical angle theorem |
| 6 | $\triangle qru \cong \triangle stu$ | sas |
| 7 | $\angle qru \cong \angle stu$ | cpctc |
| 8 | $\overline{qr} \parallel \overline{st}$ |  |

Accepted Answer

# Explanation:
## Step1: Recall the theorem for parallel lines
We know that if alternate interior angles are congruent, then the two lines are parallel. In this case, we have $\angle QRU \cong \angle STU$ (from step 7, CPCTC), and these are alternate interior angles formed by the transversal $RT$ intersecting $QR$ and $ST$.

## Step2: Apply the Alternate Interior Angles Theorem
The Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. Here, since $\angle QRU$ and $\angle STU$ are alternate interior angles and they are congruent, we can conclude that $\overline{QR} \parallel \overline{ST}$ by the Alternate Interior Angles Theorem.

# Answer:
Alternate Interior Angles Theorem

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the theorem for parallel lines

Step2: Apply the Alternate Interior Angles Theorem

Answer:

	statement	reason
2	u is the midpoint of \\(\overline{rt}\\)	given
3	\\(\overline{qu} \cong \overline{su}\\)	definition of midpoint
4	\\(\overline{ru} \cong \overline{tu}\\)	definition of midpoint
5	\\(\angle qur \cong \angle sut\\)	vertical angle theorem
6	\\(\triangle qru \cong \triangle stu\\)	sas
7	\\(\angle qru \cong \angle stu\\)	cpctc
8	\\(\overline{qr} \parallel \overline{st}\\)