Question

$\overline{rs} \parallel \overline{tu}$ and $\overline{tu} \cong \overline{rs}$. complete the proof that $\overline{st} \cong \overline{ru}$.
(image of a quadrilateral with vertices s, t, u, r and triangle markings)

	statement	reason
1	$\overline{rs} \parallel \overline{tu}$
2	$\overline{tu} \cong \overline{rs}$
3	$\angle rtu \cong \angle srt$
4	$\overline{rt} \cong \overline{rt}$
5	$\triangle rst \cong \triangle tur$
6	$\overline{st} \cong \overline{ru}$

Explanation:

Step1: Identify Given

Given \(\overline{RS} \parallel \overline{TU}\) (from problem statement).

Step2: Alternate Interior Angles

\(\angle RTU \cong \angle SRT\) because alternate interior angles are congruent when lines are parallel (\(\overline{RS} \parallel \overline{TU}\), transversal \(\overline{RT}\)).

Step3: Reflexive Property

\(\overline{RT} \cong \overline{RT}\) by reflexive property (a segment is congruent to itself).

Step4: SAS Congruence

\(\triangle RST \cong \triangle TUR\) by SAS (side - angle - side): \(\overline{TU} \cong \overline{RS}\) (given), \(\angle RTU \cong \angle SRT\) (alternate interior angles), \(\overline{RT} \cong \overline{RT}\) (reflexive).

Step5: Corresponding Parts

\(\overline{ST} \cong \overline{RU}\) because corresponding parts of congruent triangles are congruent (CPCTC).

Filling in the reasons:

1. Reason for \(\overline{RS} \parallel \overline{TU}\): Given (from problem statement)
2. Reason for \(\overline{TU} \cong \overline{RS}\): Given (from problem statement)
3. Reason for \(\angle RTU \cong \angle SRT\): Alternate Interior Angles Theorem (since \(\overline{RS} \parallel \overline{TU}\) and \(\overline{RT}\) is transversal)
4. Reason for \(\overline{RT} \cong \overline{RT}\): Reflexive Property of Congruence
5. Reason for \(\triangle RST \cong \triangle TUR\): SAS (Side - Angle - Side) Congruence Postulate
6. Reason for \(\overline{ST} \cong \overline{RU}\): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Alternate Interior Angles Theorem
4. Reason: Reflexive Property of Congruence
5. Reason: SAS Congruence Postulate
6. Reason: CPCTC