https://file.aiquickdraw.com/mathbot/file/1727288997962tdj53nnk.jpg

The image shows a geometric diagram with a triangle, labeled points, and a request to write a three-column proof. The given information is:

**Given:**
1. \( RQ = TP \)
2. \( ZQ = ZP \)

**To Prove:**
\( RZ = TZ \)

### Step-by-step explanation for the proof:

#### Statement 1:
- **Statement:** \( RQ = TP \)
- **Reason:** Given

#### Statement 2:
- **Statement:** \( ZQ = ZP \)
- **Reason:** Given

#### Statement 3:
- **Statement:** \( ZQ + RZ = ZP + TZ \)
- **Reason:** Segment Addition Postulate

#### Statement 4:
- **Statement:** \( RQ + RZ = TP + TZ \)
- **Reason:** Substitution (from Statements 1 and 2)

#### Statement 5:
- **Statement:** \( RZ = TZ \)
- **Reason:** Subtraction Property of Equality

Thus, the three-column proof shows that \( RZ = TZ \), as required.

Would you like more details or further clarification on any part of this proof?

Here are 5 related questions for deeper understanding:
1. What is the segment addition postulate, and how is it applied in proofs?
2. Why is the substitution property crucial in geometric proofs?
3. Can this proof be adapted to other triangle configurations? How?
4. How does the concept of congruent segments apply to this proof?
5. How can we use coordinate geometry to verify the equality of segments like \( RZ \) and \( TZ \)?

**Tip:** In geometric proofs, always try to simplify equations using properties like substitution and segment addition. This helps in breaking down complex proofs step by step.

Write a three-column proof to show RZ = TZ, given that RQ = TP and ZQ = ZP in a geometric diagram with triangle and labeled points.

Explore a detailed three-column proof showing that RZ = TZ in a geometric diagram. The proof involves using the segment addition postulate, substitution, and subtraction properties of equality. Ideal for students in Grades 9-10.

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