Problem Analysis:
The task involves solving geometric proofs. There are two problems presented, each with a given diagram, statements, and reasons. The goal is to complete the proofs by filling in the missing steps and justifications.
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Problem 1:
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Given:
- \( \triangle ABC \cong \triangle DEF \)
- \( \overline{AB} \parallel \overline{DE} \)
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Prove:
- \( \overline{AC} \parallel \overline{DF} \)
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Solution:
1.
Statement: \( \triangle ABC \cong \triangle DEF \)
Reason: Given
2.
Statement: \( \angle BAC \cong \angle EDF \)
Reason: Corresponding parts of congruent triangles are congruent (CPCTC).
3.
Statement: \( \overline{AB} \parallel \overline{DE} \)
Reason: Given
4.
Statement: \( \angle BAC \cong \angle ADE \)
Reason: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
5.
Statement: \( \angle EDF \cong \angle ADE \)
Reason: Transitive property of congruence (from steps 2 and 4).
6.
Statement: \( \overline{AC} \parallel \overline{DF} \)
Reason: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
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Final Answer for Problem 1:
\[
\boxed{\overline{AC} \parallel \overline{DF}}
\]
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Problem 2:
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Given:
- \( \triangle GHI \cong \triangle JKL \)
- \( m\angle GHI = m\angle JKL \)
- \( m\angle GIH = m\angle JLK \)
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Prove:
- \( \overline{GH} \cong \overline{JK} \)
####
Solution:
1.
Statement: \( \triangle GHI \cong \triangle JKL \)
Reason: Given
2.
Statement: \( m\angle GHI = m\angle JKL \)
Reason: Given
3.
Statement: \( m\angle GIH = m\angle JLK \)
Reason: Given
4.
Statement: \( \angle GHI \cong \angle JKL \)
Reason: Definition of congruent angles (since their measures are equal).
5.
Statement: \( \angle GIH \cong \angle JLK \)
Reason: Definition of congruent angles (since their measures are equal).
6.
Statement: \( \overline{GH} \cong \overline{JK} \)
Reason: Corresponding parts of congruent triangles are congruent (CPCTC).
####
Final Answer for Problem 2:
\[
\boxed{\overline{GH} \cong \overline{JK}}
\]
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Summary of Solutions:
1. For Problem 1:
\[
\boxed{\overline{AC} \parallel \overline{DF}}
\]
2. For Problem 2:
\[
\boxed{\overline{GH} \cong \overline{JK}}
\]
Parent Tip: Review the logic above to help your child master the concept of geometric proofs worksheet with answers.
