What is m∠DFE if m∠DEG = 18°?

Question

Use DEF.

Answer 1

The big clue here is the fact that DE = FE = c. That means that ΔDEF is isosceles with base angles D and F. m∠GEF = 18º since it is marked as congruent to ∠DEG, so putting them together we get m∠DEF = 36º. Since the angles in the triangle need to add to 180º and angles D and F are congruent, we could write the equation 2x + 36 = 180 where x stands for ∠F. Solving this equation, we would get that m∠DFE = 72º. Hope this helps!

Answer 2

Notice that line segment b bisects the angle E into two congruent line segments (c).

Assume the endpoints at G and E represent an x-axis where G is the origin (x=0). Also, assume the points forming angles D and F represent the y-axis. If the two angles are congruent or equivalent, then the distance to the y-axis must be equivalent.

Since the line segments (c) connecting at angle E are equivalent, it must follow that m< DEG = m<FEG. Otherwise, the line segments would not meet exactly at the origin (because a change in slope (or angle) would result in a change in distance traveled along the x-axis).

This implies that angle D is at coordinate (0,y) and angle F is at coordinate (0,-y). Therefore, the distance between endpoints G and F is equivalent to line segment a which then implies (via substitution) that m<DFE = m<FDE or m<DFE + m<FDE = 2*m<DFE (*).

Adding the congruent angles yields ....m<DEG + m<FEG = 2*18 = 36°.

Using the previous relation, the following can be derived... m<DFE + m<FDE + 36 = 180°.

(*) --> 2*m<DFE + 36 = 180° --> m<DFE = (180-36)/2 = 72°