Question

qr ≅ pq and pr ⊥ qs. complete the proof that ps ≅ rs.

statement\treason
1 \tqr ≅ pq\tgiven
2 \tpr ⊥ qs\tgiven
3 \t∠qpr ≅ ∠prq\tisosceles triangle theorem
4 \t∠ptq ≅ ∠qtr\t
5 \tδpqt ≅ δrqt\t
6 \t∠pqs ≅ ∠rqs\t
7 \tqs ≅ qs\t
8 \tδpqs ≅ δrqs\t
9 \tps ≅ rs\t

Explanation:

Step 1: Analyze ∠PTQ and ∠QTR

Since \( \overline{PR} \perp \overline{QS} \), the angles \( \angle PTQ \) and \( \angle QTR \) are right angles. By the definition of perpendicular lines, all right angles are congruent. So \( \angle PTQ \cong \angle QTR \) because they are both right angles (Reason: All right angles are congruent).

Step 2: Prove \( \triangle PQT \cong \triangle RQT \)

We know \( \overline{QR} \cong \overline{PQ} \) (Given), \( \angle QPR \cong \angle PRQ \) (Isosceles Triangle Theorem), and \( \overline{QT} \cong \overline{QT} \) (Reflexive Property). Wait, actually, from step 1, we have \( \angle PTQ \cong \angle QTR \), \( \overline{PQ} \cong \overline{RQ} \) (Given), and \( \overline{QT} \cong \overline{QT} \) (Reflexive). So by SAS (Side - Angle - Side) Congruence Criterion, \( \triangle PQT \cong \triangle RQT \) (Reason: SAS Congruence).

Step 3: Analyze \( \angle PQS \) and \( \angle RQS \)

Since \( \triangle PQT \cong \triangle RQT \), corresponding parts of congruent triangles are congruent (CPCTC). So \( \angle PQS \cong \angle RQS \) (Reason: CPCTC).

Step 4: Prove \( \overline{QS} \cong \overline{QS} \)

This is the Reflexive Property of Congruence (Reason: Reflexive Property of Congruence).

Step 5: Prove \( \triangle PQS \cong \triangle RQS \)

We have \( \overline{PQ} \cong \overline{RQ} \) (Given), \( \angle PQS \cong \angle RQS \) (from step 3), and \( \overline{QS} \cong \overline{QS} \) (from step 4). By SAS Congruence Criterion, \( \triangle PQS \cong \triangle RQS \) (Reason: SAS Congruence).

Step 6: Prove \( \overline{PS} \cong \overline{RS} \)

Since \( \triangle PQS \cong \triangle RQS \), corresponding parts of congruent triangles are congruent (CPCTC). So \( \overline{PS} \cong \overline{RS} \) (Reason: CPCTC).

Now, filling in the reasons:

• Step 4 (Statement 4: \( \angle PTQ \cong \angle QTR \)): All right angles are congruent.
• Step 5 (Statement 5: \( \triangle PQT \cong \triangle RQT \)): SAS (Side - Angle - Side) Congruence Criterion (using \( \overline{PQ} \cong \overline{RQ} \), \( \angle PTQ \cong \angle QTR \), \( \overline{QT} \cong \overline{QT} \))
• Step 6 (Statement 6: \( \angle PQS \cong \angle RQS \)): CPCTC (Corresponding Parts of Congruent Triangles are Congruent) from \( \triangle PQT \cong \triangle RQT \)
• Step 7 (Statement 7: \( \overline{QS} \cong \overline{QS} \)): Reflexive Property of Congruence
• Step 8 (Statement 8: \( \triangle PQS \cong \triangle RQS \)): SAS (Side - Angle - Side) Congruence Criterion (using \( \overline{PQ} \cong \overline{RQ} \), \( \angle PQS \cong \angle RQS \), \( \overline{QS} \cong \overline{QS} \))
• Step 9 (Statement 9: \( \overline{PS} \cong \overline{RS} \)): CPCTC (Corresponding Parts of Congruent Triangles are Congruent) from \( \triangle PQS \cong \triangle RQS \)

Answer:

• Statement 4 Reason: All right angles are congruent.
• Statement 5 Reason: SAS Congruence Criterion.
• Statement 6 Reason: CPCTC.
• Statement 7 Reason: Reflexive Property of Congruence.
• Statement 8 Reason: SAS Congruence Criterion.
• Statement 9 Reason: CPCTC.