Step-by-step explanation:
Use one of the triangle congruence theorems (SAS, SSS, ASA, AAS, or HL) to prove that △ABC≅△EDC
.
It is given that ∠FAB≅∠GED
. ∠BAC is the supplement of ∠FAB, and ∠DEC is the supplementary of ∠GED
by the definition of supplementary angles.
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)
