Given IE is congruent to GH, EF is congruent to HF. F is the midpoint of GÌ.
Prove the triangles EFI is congruent to HFG.
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Can you please help me to understand how to do this problem? I don't understand how to find a proof. Thank you so much for your help.
Pair up the sides. You will find that the three sides of the two triangles match up, so since the two triangles have matching sides, they are congruent: SSS
Steve, thank you so much for your help.
To prove that triangles EFI and HFG are congruent, we need to use the given information and apply congruence criteria.
Given: IE is congruent to GH, EF is congruent to HF, and F is the midpoint of GI.
To prove: EFI is congruent to HFG.
Proof:
Step 1: Given information.
IE is congruent to GH (Given)
EF is congruent to HF (Given)
F is the midpoint of GI (Given)
Step 2: F is the midpoint of GÌ implies GI is congruent to IF.
Since F is the midpoint of GI, by definition of midpoint, GI is congruent to IF.
Step 3: Prove triangles EFI is congruent to HFG using the Side-Side-Side (SSS) congruence criterion.
Using the given and proven information:
- EF is congruent to HF (Given)
- EI is congruent to GH (Given)
- GI is congruent to IF (Proven in Step 2)
Therefore, we have all three pairs of corresponding sides of the triangles congruent. Hence, by the SSS criterion for congruence, triangles EFI and HFG are congruent.
Thus, we have proved that triangles EFI and HFG are congruent using the given information and the SSS congruence criterion.
To prove that triangles EFI and HFG are congruent, we need to show that their corresponding sides and angles are congruent. Here's a step-by-step explanation of how to prove this:
1. Given: IE ≅ GH, EF ≅ HF, F is the midpoint of GI
2. To prove: Triangle EFI ≅ Triangle HFG
Proof:
Step 1: Given information
- We are given that IE is congruent to GH (IE ≅ GH)
- We are given that EF is congruent to HF (EF ≅ HF)
- We are given that F is the midpoint of GI (F is the midpoint of GI)
Step 2: Identify corresponding parts
- Corresponding parts of triangles are sides or angles that are in the same position or have the same measurement in each triangle.
Step 3: Corresponding sides (SSS - Side-Side-Side)
- Since IE ≅ GH, EF ≅ HF, and F is the midpoint of GI, we have the following congruent sides:
- Segment EF ≅ Segment HF (by the given information)
- Segment IE ≅ Segment GH (by the given information)
- Segment FI ≅ Segment FG (since F is the midpoint of GI)
Step 4: Corresponding angles (AAA - Angle-Angle-Angle)
- Triangles EFI and HFG have two angles in common:
- Angle E is congruent to angle H (vertical angles)
- Angle F is congruent to angle F (common angle)
- Since corresponding angles are congruent, we can say ∠EFI ≅ ∠HFG.
Step 5: Conclusion
- We have shown that all corresponding sides and angles of triangles EFI and HFG are congruent. Therefore, using the SSS and AAA postulates, we conclude that Triangle EFI is congruent to Triangle HFG.
I hope this explanation helps you understand how to prove the congruence of triangles EFI and HFG. If you have any further questions, feel free to ask!