You've uploaded a geometry proof problem with a completed two-column proof. Let’s
verify and explain the solution step by step to ensure it’s logically sound.
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Given:
- Points A, B, C, D are collinear in that order (as shown in the diagram).
- Segment AB is congruent to segment CD:
→ $\overline{AB} \cong \overline{CD}$
Prove:
- Segment AC is congruent to segment BD:
→ $\overline{AC} \cong \overline{BD}$
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##
✔ Step-by-Step Explanation of the Proof:
| Statement | Reason | Explanation |
|----------|--------|-----------|
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1. $\overline{AB} \cong \overline{CD}$ | Given | This is our starting assumption. |
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2. $AB = CD$ | Definition of Congruence | Congruent segments have equal lengths. So if segments are congruent, their measures are equal. |
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3. $AC + CD = AD$ | Segment Addition Postulate | Since points A, C, D are collinear and C is between A and D, the whole segment AD equals the sum of parts AC and CD. |
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4. $AB + BD = AD$ | Segment Addition Postulate | Similarly, since A, B, D are collinear and B is between A and D, AD = AB + BD. |
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5. $CD + BD = AD$ | Substitution (from statement 2) | Since AB = CD (statement 2), we can substitute CD for AB in statement 4. So AB + BD becomes CD + BD. |
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6. $AC + CD = CD + BD$ | Transitive Property | Both expressions equal AD (from statements 3 and 5), so they equal each other. |
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7. $AC = BD$ | Subtraction Property of Equality | Subtract CD from both sides of the equation $AC + CD = CD + BD$, leaving $AC = BD$. |
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8. $\overline{AC} \cong \overline{BD}$ | Definition of Congruence | If two segments have equal lengths, then they are congruent. |
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## 🎯 Final Answer:
✔ The proof is
correct and complete.
We started with the given congruence $\overline{AB} \cong \overline{CD}$, used properties of segment addition and equality, substituted values, and concluded that $\overline{AC} \cong \overline{BD}$ — which was what we were asked to prove.
This is a classic example of using the
Segment Addition Postulate and
algebraic properties to prove segment congruence in geometry.
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## 💡 Tip for Understanding:
Think visually:
- AC = AB + BC
- BD = BC + CD
Since AB = CD (given), then AC = AB + BC = CD + BC = BD.
So AC and BD are made up of the same "parts" — one equal segment plus the common middle segment BC. That’s why they’re equal!
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Let me know if you’d like to see a diagram-based explanation or alternative proofs!
Parent Tip: Review the logic above to help your child master the concept of proofs worksheet.
