Geo Unit 4 HW 6 Questions 4 through 6 - Free Printable

Let's solve Problem 5 step by step and complete the proof that:

> Given:
> - $ \angle RPQ \cong \angle RTS $
> - $ R $ is the midpoint of $ \overline{QS} $
>
> Prove: $ \triangle PQR \cong \triangle TSR $

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Step-by-step Proof (Using Two-Column Format)

We are given:

1. $ \angle RPQ \cong \angle RTS $
2. $ R $ is the midpoint of $ \overline{QS} $

We need to prove: $ \triangle PQR \cong \triangle TSR $

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#### Statements | Reasons

| Statement | Reason |
|--------|--------|
| 1. $ \angle RPQ \cong \angle RTS $ | 1. Given |
| 2. $ R $ is the midpoint of $ \overline{QS} $ | 2. Given |
| 3. $ \overline{QR} \cong \overline{RS} $ | 3. Definition of Midpoint |
| 4. $ \angle PRQ \cong \angle SRT $ | 4. Vertical Angles Theorem (since $ \angle PRQ $ and $ \angle SRT $ are vertical angles) |
| 5. $ \triangle PQR \cong \triangle TSR $ | 5. ASA (Angle-Side-Angle) Congruence Postulate |

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Explanation:

Let’s go through each step:

1. Given: $ \angle RPQ \cong \angle RTS $ — This is one pair of congruent angles.

2. Given: $ R $ is the midpoint of $ \overline{QS} $ — So, $ QR = RS $, meaning the segment from $ Q $ to $ R $ is equal in length to $ R $ to $ S $.

3. From this, we conclude: $ \overline{QR} \cong \overline{RS} $ — This is a side between the two triangles.

4. Now, look at the angles at point $ R $. $ \angle PRQ $ and $ \angle SRT $ are vertical angles, formed by the intersection of lines $ PQ $ and $ TS $ at point $ R $. Vertical angles are always congruent.

So: $ \angle PRQ \cong \angle SRT $

5. Now we have:
- $ \angle RPQ \cong \angle RTS $ (given)
- $ \overline{QR} \cong \overline{RS} $ (from midpoint)
- $ \angle PRQ \cong \angle SRT $ (vertical angles)

These form ASA (Angle-Side-Angle), where the side is included between the two angles.

✔ Therefore, by ASA Congruence Postulate, $ \triangle PQR \cong \triangle TSR $

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Final Answer:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle RPQ \cong \angle RTS $ | 1. Given |
| 2. $ R $ is the midpoint of $ \overline{QS} $ | 2. Given |
| 3. $ \overline{QR} \cong \overline{RS} $ | 3. Definition of Midpoint |
| 4. $ \angle PRQ \cong \angle SRT $ | 4. Vertical Angles Theorem |
| 5. $ \triangle PQR \cong \triangle TSR $ | 5. ASA Congruence Postulate |

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✔ Proof Complete!

Let me know if you'd like help with Problem 6 (HL) as well!