Given that ∠FAB≅∠GED and C is the midpoint of AE¯¯¯¯¯, which of the following proves that △ABC≅△EDC?

The figure shows two triangles A B C and C D E. The triangles have a common vertex C. Points F, A, C, E, G lie on one line in order from left to right. Points B, C, D lie on one line.

1
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)

2
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)

3
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)

4
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)

The figure shows the same triangles A B C and C D E as in the beginning of the task. Angle B A C is highlighted in green. Angle D E C is highlighted in blue.

Notice that ∠BAC

and ∠DEC are supplemental to congruent angles ∠FAB and ∠GED respectively. The Congruent Supplements Theorem states that if two angles are supplementary to two congruent angles, then the two angles are congruent. Therefore, ∠BAC≅∠DEC

by the Congruent Supplements Theorem.

The figure shows the same triangles A B C and C D E as in the previous figure. Angles B A C and D E C are congruent and highlighted in blue.

Notice that ∠ACB

and ∠ECD are vertical angles. The Vertical Angles Theorem states that vertical angles are congruent.

Therefore, ∠ACB≅∠ECD

by the Vertical Angles Theorem.

The figure shows the same triangles A B C and C D E as in the previous figure. Angles B C A and D C E are congruent and highlighted in red.

It is also given that C

is the midpoint of AE⎯⎯⎯⎯⎯

.

By the definition of midpoint, AC⎯⎯⎯⎯⎯≅EC⎯⎯⎯⎯⎯

.

The figure shows the same triangles A B C and C D E as in the previous figure. Sides A C and C E are congruent and highlighted in red.

So, two pairs of corresponding angles in △ABC

and △EDC are congruent, and the included sides in △ABC and △EDC

are congruent. The Angle-Side-Angle (ASA) Congruence Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Therefore, △ABC≅△EDC

by ASA.

Translate these seven statements and reasons into a 2-column proof.

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)

There you go

(from Lesson)