Given: PQ¯¯¯¯¯¯¯¯⊥QS¯¯¯¯¯¯¯, RS¯¯¯¯¯¯¯≅SQ¯¯¯¯¯¯¯

Question

, T is the midpoint of PR¯¯¯¯¯¯¯¯


Prove: ΔPQT ≅ ΔRST

(5 points)
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Statement Reason
1. PQ¯¯¯¯¯¯¯¯⊥QS¯¯¯¯¯¯¯, RS¯¯¯¯¯¯¯≅SQ¯¯¯¯¯¯¯
1. Given
2. Response area 2. All right angles are congruent
3. T is the midpoint of PR¯¯¯¯¯¯¯¯
3. Given
4. Response area 4. Def. of midpoint
5. Response area 5. Vertical Angles
6. △PQT≅
Response area 6. Response area



Angles
Segments
Triangles
Congruency Reason

Answer 1

Here's the completed proof in the specified format:

| Statement | Reason | |--------------------------------------------------|---------------------------------------------| | 1. PQ¯¯¯¯¯¯¯¯⊥QS¯¯¯¯¯¯¯¯, RS¯¯¯¯¯¯¯≅SQ¯¯¯¯¯¯¯ | 1. Given | | 2. ∠PQT ≅ ∠RST | 2. All right angles are congruent | | 3. T is the midpoint of PR¯¯¯¯¯¯¯¯ | 3. Given | | 4. PT¯¯¯¯¯¯¯≅TR¯¯¯¯¯¯¯ | 4. Def. of midpoint | | 5. ∠QTP ≅ ∠RST | 5. Vertical Angles | | 6. ΔPQT ≅ ΔRST | 6. AAS (Angle-Angle-Side) Congruence Theorem |

Here's a brief explanation of each step:

1. States the conditions given in the problem.
2. Identifies that PQ and QS form right angles.
3. States that T is the midpoint as per the problem.
4. Concludes that PT and TR are congruent because T is the midpoint.
5. Uses the property of vertical angles to conclude the angles are congruent.
6. Final conclusion using the AAS congruence criterion, proving the triangles are congruent.