move equations and properties into the table to prove that opposite sides of parallelogram rstu are congruent.
$\overline{rs}\cong\overline{tu}$ $\overline{rt}\cong\overline{tr}$ $\overline{ru}\cong\overline{sr}$ $\overline{ru}\cong\overline{ts}$ $\overline{st}\cong\overline{tu}$
$\overline{ru}\parallel\overline{ts}$ $\angle u\cong\angle s$ $\angle urt\cong\angle str$ $\angle utr\cong\angle srt$
cpctc definition of a parallelogram definition of a quadrilateral reflexive property

|statement|reason|
|----|----|
|rstu is a parallelogram.|given|
| and
______ ______|alternate interior angles are congruent.|
|$\triangle rtu\cong\triangle trs$|angle - side - angle congruence|
| and
______ ______|

Question

move equations and properties into the table to prove that opposite sides of parallelogram rstu are congruent.
$\overline{rs}\cong\overline{tu}$  $\overline{rt}\cong\overline{tr}$  $\overline{ru}\cong\overline{sr}$  $\overline{ru}\cong\overline{ts}$  $\overline{st}\cong\overline{tu}$
$\overline{ru}\parallel\overline{ts}$  $\angle u\cong\angle s$  $\angle urt\cong\angle str$  $\angle utr\cong\angle srt$
cpctc  definition of a parallelogram  definition of a quadrilateral  reflexive property

|statement|reason|
|----|----|
|rstu is a parallelogram.|given|
|  and
______  ______|alternate interior angles are congruent.|
|$\triangle rtu\cong\triangle trs$|angle - side - angle congruence|
|  and
______  ______|

Accepted Answer

# Explanation:
## Step1: Identify parallel - sides property
Since $RSTU$ is a parallelogram, $\overline{RU}\parallel\overline{TS}$ and $\overline{RS}\parallel\overline{TU}$. By the property of alternate - interior angles for parallel lines, $\angle URT\cong\angle STR$ and $\angle UTR\cong\angle SRT$.
Statement: $\angle URT\cong\angle STR$, $\angle UTR\cong\angle SRT$; Reason: Alternate interior angles are congruent.
## Step2: Identify the common side
$\overline{RT}\cong\overline{TR}$ by the reflexive property.
Statement: $\overline{RT}\cong\overline{TR}$; Reason: Reflexive property.
## Step3: Prove triangle congruence
We have two pairs of congruent angles ($\angle URT\cong\angle STR$, $\angle UTR\cong\angle SRT$) and a common side ($\overline{RT}\cong\overline{TR}$), so $\triangle RTU\cong\triangle TRS$ by the Angle - Side - Angle (ASA) congruence criterion.
Statement: $\triangle RTU\cong\triangle TRS$; Reason: Angle - Side - Angle congruence.
## Step4: Prove opposite - side congruence
By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), $\overline{RU}\cong\overline{TS}$ and $\overline{RS}\cong\overline{TU}$.
Statement: $\overline{RU}\cong\overline{TS}$, $\overline{RS}\cong\overline{TU}$; Reason: CPCTC.

# Answer:
| Statement | Reason |
|--|--|
| $RSTU$ is a parallelogram. | Given |
| $\angle URT\cong\angle STR$, $\angle UTR\cong\angle SRT$ | Alternate interior angles are congruent |
| $\overline{RT}\cong\overline{TR}$ | Reflexive property |
| $\triangle RTU\cong\triangle TRS$ | Angle - Side - Angle congruence |
| $\overline{RU}\cong\overline{TS}$, $\overline{RS}\cong\overline{TU}$ | CPCTC |

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QUESTION IMAGE

Question

Explanation:

Step1: Identify parallel - sides property

Step2: Identify the common side

Step3: Prove triangle congruence

Step4: Prove opposite - side congruence

Answer:

Statement	Reason
\(\angle URT\cong\angle STR\), \(\angle UTR\cong\angle SRT\)	Alternate interior angles are congruent
\(\overline{RT}\cong\overline{TR}\)	Reflexive property
\(\triangle RTU\cong\triangle TRS\)	Angle - Side - Angle congruence
\(\overline{RU}\cong\overline{TS}\), \(\overline{RS}\cong\overline{TU}\)	CPCTC