Given: measured angle D =~ measured angle F; GE bisects DEF
prove: DG =~ FG
Huh? "bisects" means it divides DF into two equal parts: DG and GF
To prove that DG is congruent to FG, we will use the Angle Bisector Theorem and the Vertical Angles Theorem.
Steps to prove DG =~ FG:
1. Given that measured angle D is approximately equal to measured angle F, and GE bisects angle DEF.
2. By the Angle Bisector Theorem, we know that the line GE divides angle DEF into two congruent angles.
3. Let's call the measure of angle DEG as "x", so the measure of angle GEF is also "x".
4. Since GE bisects angle DEF, we can say that the measure of angle DEG + measure of angle GEF is equal to the measure of angle DEF.
5. Therefore, we have angle DEG + angle GEF = angle DEF.
6. By substituting our values, we can say that x + x = measure of angle DEF.
7. Simplifying, we get 2x = measure of angle DEF.
8. Now, let's consider triangle DEF. Since angle DEG is congruent to angle GEF, we can conclude that triangle DEG is an isosceles triangle.
9. In an isosceles triangle, the lengths of the sides opposite the congruent angles are equal. Therefore, DE = GE and EF = FG.
10. Now, let's look at triangle DGE. By the transitive property, we can say that DE = GE = GE.
11. Using the reflexive property, we can say that GE = GE = EG.
12. By substitution, we have DE = EG = EG.
13. Now, considering triangle FGE, we have EF = FG and GE = GE.
14. By adding the segments from triangle DGE to triangle FGE, we can say that DE + EG = EF + FG.
15. Substituting the equivalent values, we get EG + EG = EF + FG.
16. Simplifying, we have 2EG = EF + FG.
17. By subtracting EG from both sides, we get EG = FG.
18. Therefore, we have shown that DG is congruent to FG, as desired.