SOLUTION: Given that ∠FAB≅∠GED and C is the midpoint of AE¯¯¯¯¯, which of the following proves that △ABC≅△EDC? https://thinkwell.cachefly.net/questionbank/95001-96000/956

Question 1169211: Given that ∠FAB≅∠GED and C is the midpoint of AE¯¯¯¯¯, which of the following proves that △ABC≅△EDC?
https://thinkwell.cachefly.net/questionbank/95001-96000/95608/img/95608a.svg
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.

RELATED QUESTIONS
How would you write the two column proof for a problem that states the following: Given:... (answered by drk)

If C is the midpoint of AE and (answered by JBarnum)
given: congruent triangle abc and congruetnt triangle edc, c is the midpoint of line bd... (answered by richwmiller)
given: triangle ABC and triangle EDC, C is the midpoint of line BD and line AE Prove:... (answered by drk)
Which of the following terms correctly describes abc and... (answered by Alan3354)
Given: C is the midpoint of BD and AE... (answered by Fombitz)
Given that A(0, 0), B(4, 0), C(0, 2), D(0, 0), E(12, 0), and F(0, 6), which of the... (answered by Fombitz,MathLover1)
how do you do a proof that has a given:triangle ABC and triangle EDC, C is the midpoint... (answered by richard1234)
I am having a really hard time with proofs in geometry. This problem comes from the... (answered by jim_thompson5910)