Question 724078: I am in need of help with proving the following:
GIVEN:
Line segment EG bisects angle DEF
Angle EDG is congruent to angle EFG
Prove: Triangle DGF is isosceles
Starting with the two given statments the next statements and reasons I have are:
Step 1
Line segment EG bisects angle DEF; Reason: Given
Step 2 Angle EDG is congruent to angle EFG; Reason: Given
Step 3
Triangle EDG is congruent to Triangle EFG;
Reason: ASA
Step 4 Line segment ED is congruent to line segment EF;
Reason: CPCTC
Step 5 Line segment EG is congruent to line segment EG;
Reason: Reflexive
Step 6
Triangle ABC is Isosceles;
Reason: Definition of Isosceles
Am I on the right track? Any help would be great. Thanks in advance!
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A drawing is always needed, otherwise, you get your letters all mixed up, so here is my drawing:
Proving that triangles EDG and EFG are congruent needs to be part of the proof, but I see two big problems with your proof:
A) You lost track of the goal of proving that "Triangle DGF is isosceles"
and ended up trying to prove that "Triangle ABC is Isosceles". That's not what was asked, and I do not know where ABC came from, because we were not using that part of the alphabet at all.
B) I disagree with
"Step 3
Triangle EDG is congruent to Triangle EFG;
Reason: ASA"
You have two pairs of congruent angles (given, or almost given), but no pair of congruent sides. (It is almost given that angles GED and GEF are congruent, because EG bisects angle DEF).
To invoke ASA congruency, you need two pairs of congruent angles (the AA part of ASA), and the pair of sides between those angles (the middle S in ASA) have to be congruent too. You do not have proof of congruent pairs of sides.
It would not help (not yet) saying that
"Line segment EG is congruent to line segment EG;
Reason: Reflexive"
because side EG is not between the angle pairs proven to be congruent, and there is no SSA congruency (You probably know why the letters are not listed in the equivalent, A-S-S order).
However, it is easy to prove that the third angles in triangles EGD and EGF (angles EGD and EGF) are congruent, because you already know the other two pairs of angles are congruent.
Once angles EGD and EGF are proven congruent, you can invoke ASA congruency to show that triangles GDE and GFE are congruent, using EG congruent to EG as the S part is ASA, with congruent angle pair EGD and EGF as one of the A's and congruent angle pair GED and GEF as the other A.
Since triangles GDE and GFE are congruent, by CPCTC side GD is congruent to GF,
and that makes DGF isosceles. (You can also use CPCTC to show that DE and FE are congruent, and say that triangle DEF is isosceles too).
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