Given: measure of angle D=~ measure of angle F ;
line GE bisects angle DEF
Prove: line DG=~ line FG
=~ is the congruent sign
To prove that line DG is congruent to line FG, we will use the Angle Bisector Theorem and the definition of congruent angles.
The Angle Bisector Theorem states that if a line bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides of the angle.
Given that line GE bisects angle DEF, we can use the Angle Bisector Theorem on triangle DEF as follows:
Let's label the segments on line DE as DG and GE, and the segments on line FE as FG and GE.
According to the Angle Bisector Theorem, we have:
(DG / GE) = (DF / FE)
Since the measure of angle D is congruent to the measure of angle F, we can say that DF is congruent to FE. Therefore, we can write:
(DG / GE) = (DF / FE) = 1
This means that DG is congruent to GE.
To prove that DG is congruent to FG, we can use the Transitive Property of Congruence. Since DG is congruent to GE and GE is congruent to FG (since GE bisects angle DEF), we can conclude that DG is congruent to FG.
Thus, by using the Angle Bisector Theorem and the Transitive Property of Congruence, we have proven that line DG is congruent to line FG.