Question

qt || rs and qt ≅ rs. complete the proof that δqtu ≅ δsru.
image of triangles qtu and sru with u as the intersection of their sides
statement | reason

1. qt || rs |
2. qt ≅ rs |
3. ∠qut ≅ ∠rus |
4. ∠t ≅ ∠r |
5. δqtu ≅ δsru |

Explanation:

Step1: Identify Given Information

We are given that \(\overline{QT} \parallel \overline{RS}\) (Statement 1) and \(\overline{QT} \cong \overline{RS}\) (Statement 2).

Step2: Analyze Vertical Angles

For \(\angle QUT \cong \angle RUS\) (Statement 3), the reason is "Vertical Angles Theorem" (vertical angles are congruent).

Step3: Analyze Alternate Interior Angles

Since \(\overline{QT} \parallel \overline{RS}\), \(\angle T \cong \angle R\) (Statement 4) by the "Alternate Interior Angles Theorem" (alternate interior angles are congruent when lines are parallel).

Step4: Prove Triangle Congruence

We have \(\overline{QT} \cong \overline{RS}\) (Statement 2), \(\angle T \cong \angle R\) (Statement 4), and \(\angle QUT \cong \angle RUS\) (Statement 3). So by the "ASA (Angle - Side - Angle) Congruence Criterion", \(\triangle QTU \cong \triangle SRU\) (Statement 5).

Now, filling in the reasons:

1. Reason for \(\overline{QT} \parallel \overline{RS}\): Given
2. Reason for \(\overline{QT} \cong \overline{RS}\): Given
3. Reason for \(\angle QUT \cong \angle RUS\): Vertical Angles Theorem
4. Reason for \(\angle T \cong \angle R\): Alternate Interior Angles Theorem (because \(\overline{QT} \parallel \overline{RS}\))
5. Reason for \(\triangle QTU \cong \triangle SRU\): ASA (Angle - Side - Angle) Congruence Criterion (we have two angles and the included side congruent: \(\angle T \cong \angle R\), \(\overline{QT} \cong \overline{RS}\), \(\angle QUT \cong \angle RUS\))

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Vertical Angles Theorem
4. Reason: Alternate Interior Angles Theorem
5. Reason: ASA (Angle - Side - Angle) Congruence Criterion