Problem: Write a two-column proof for each problem.
We will solve
Problem 1 as an example and provide a detailed explanation of the solution. The format will be a two-column proof with statements and reasons.
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Problem 1:
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Given: \( EH \) bisects \( \overline{DG} \); \( \angle D \cong \angle G \)
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Prove: \( \triangle DEF \cong \triangle GHF \)
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Two-Column Proof:
|
Statements |
Reasons |
|-----------------------------------------------------|----------------------------------------------------------------------------|
| 1. \( EH \) bisects \( \overline{DG} \) | Given |
| 2. \( DE \cong GE \) | Definition of angle bisector (bisecting a segment creates two congruent segments) |
| 3. \( \angle D \cong \angle G \) | Given |
| 4. \( \angle DEF \cong \angle GHF \) | Vertical angles are congruent (angles formed by intersecting lines) |
| 5. \( EF \cong HF \) | Reflexive Property of Congruence (a segment is congruent to itself) |
| 6. \( \triangle DEF \cong \triangle GHF \) | Side-Angle-Side (SAS) Congruence Postulate (DE ≅ GE, ∠D ≅ ∠G, EF ≅ HF) |
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Explanation of the Solution:
1.
Step 1: Understand the given information.
- We are told that \( EH \) bisects \( \overline{DG} \). This means that \( EH \) divides \( \overline{DG} \) into two congruent segments, so \( DE \cong GE \).
- We are also given that \( \angle D \cong \angle G \).
2.
Step 2: Identify additional information from the diagram.
- From the diagram, we can see that \( \angle DEF \) and \( \angle GHF \) are vertical angles. Vertical angles are always congruent, so \( \angle DEF \cong \angle GHF \).
3.
Step 3: Use the reflexive property.
- The segment \( EF \) is common to both triangles \( \triangle DEF \) and \( \triangle GHF \). By the reflexive property of congruence, \( EF \cong HF \).
4.
Step 4: Apply the SAS Congruence Postulate.
- To prove that two triangles are congruent, we need to show that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle.
- Here, we have:
- \( DE \cong GE \) (from the bisector)
- \( \angle D \cong \angle G \) (given)
- \( EF \cong HF \) (reflexive property)
- These satisfy the Side-Angle-Side (SAS) Congruence Postulate, which states that if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
5.
Step 5: Conclude the proof.
- By the SAS Congruence Postulate, we can conclude that \( \triangle DEF \cong \triangle GHF \).
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Final Answer:
\[
\boxed{\triangle DEF \cong \triangle GHF}
\]
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If you need solutions for other problems, please let me know! I can provide similar detailed explanations for any specific problem.
Parent Tip: Review the logic above to help your child master the concept of geometry triangle proofs worksheet.
