Question

In the diagram below,
←→
A
D
,
←→
B
E
, and
←→
C
F
intersect at point G.

Line A-D is perpendicular to line C-F. Angle A-G-C is divided into angles B-G-C and B-G-A measuring x plus 20. Angle D-G-F is divided into angles D-G-E and angle E-G-F measuring 2 times x plus 4.

What is the measure of
∠
F
G
E
?

A.
36
°

B.
48
°

C.
52
°

D.
108
°

Answer 1

Let ∠AGC = ∠BGA + ∠BGC = (x+20)+(x+20) = 2x+40.
Let ∠DGF = ∠DGE + ∠EGF = (2x+4)+(2x+4) = 4x+8.

∠AGC and ∠DGF are vertical angles (they are opposite at G), so they are congruent:
2x + 40 = 4x + 8 → 32 = 2x → x = 16.

Thus ∠FGE = 2x + 4 = 2(16) + 4 = 36°.

Answer: A (36°).