Question 1169211: Given that ∠FAB≅∠GED and C is the midpoint of AE¯¯¯¯¯, which of the following proves that △ABC≅△EDC?
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A.
1. ∠FAB≅∠GED (Given)2. ∠BAC is the supp. of ∠FAB; ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3. ∠BAC≅∠DEC (≅ Supp. Thm.)4. ∠ACB≅∠DCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6. AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)7. △ABC≅△EDC (by ASA Steps 3, 6, 4)
B.
1. ∠FAB≅∠GED (Given)2. ∠BAC is the supp. of ∠FAB; ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3. ∠BAC≅∠DEC (≅ Supp. Thm.)4. ∠ACB≅∠DCE (Adj. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6. AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)7. △ABC≅△EDC (by ASA Steps 3, 6, 4)
C.
1. ∠FAB≅∠GED (Given)2. ∠BAC is the supp. of ∠FAB; ∠DEC is thesupp. of ∠DEG (Def. of Supp. ∠s)3. BC¯¯¯¯¯≅CD¯¯¯¯¯ (≅ Supp. Thm.)4. ∠ACB≅∠BCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6. BC¯¯¯¯¯≅CD¯¯¯¯¯ (Def. of mdpt.)7. △ABC≅△EDC (by SAS Steps 3, 1, 6)
D.
1. ∠FAB≅∠GED (Given)2. ∠BAC is the supp. of ∠FAB; ∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s)3. BC¯¯¯¯¯≅CD¯¯¯¯¯ (≅ Supp. Thm.)4. ∠ACB≅∠BCE (Vert. ∠s Thm.)5. C is the midpoint of AE¯¯¯¯¯ (Given)6. BC¯¯¯¯¯≅CD¯¯¯¯¯ (Def. of mdpt.)7. △ABC≅△EDC (by SAS Steps 3, 1, 6)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze each option to determine which one correctly proves that △ABC≅△EDC.
**Understanding the Diagram and Given Information**
* We are given that ∠FAB≅∠GED.
* C is the midpoint of AE¯¯¯¯¯. This means AC¯¯¯¯¯≅EC¯¯¯¯¯.
* We need to use the given information to prove that △ABC≅△EDC.
**Analyzing Each Option**
* **A.**
* 1. ∠FAB≅∠GED (Given)
* 2. ∠BAC is the supp. of ∠FAB; ∠DEC is the supp. of ∠GED (Def. of Supp. ∠s)
* 3. ∠BAC≅∠DEC (≅ Supp. Thm.)
* 4. ∠ACB≅∠DCE (Vert. ∠s Thm.)
* 5. C is the midpoint of AE¯¯¯¯¯ (Given)
* 6. AC¯¯¯¯¯≅EC¯¯¯¯¯ (Def. of mdpt.)
* 7. △ABC≅△EDC (by ASA Steps 3, 6, 4)
* This option correctly uses the Angle-Side-Angle (ASA) congruence theorem.
* **B.**
* The only difference is that step 4 states ∠ACB≅∠DCE (Adj. ∠s Thm.). Adjacent angles are not necessarily congruent. Vertical angles are. Therefore, this option has an error.
* **C.**
* This option introduces BC¯¯¯¯¯≅CD¯¯¯¯¯ in steps 3 and 6, which is not supported by the given information. Also it tries to use SAS with angles that are not in the correct order.
* Also step 4 says ∠ACB≅∠BCE. Which is incorrect. ∠ACB and ∠DCE are vertical angles.
* **D.**
* This option has the same errors as option C.
**Conclusion**
Option A correctly uses the given information and the ASA congruence theorem to prove that △ABC≅△EDC.
**Final Answer**
The correct answer is A.
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