Question

In the diagram below, quadrilateral D, E, F, GDEFG is inscribed in circle HH. Solve for xx and yy.
H
D
E
F
G
107°
82°
(2y -67)°
(x -87)°
Answer
Attempt 1 out of 2

Answer 1

Because D, E, F, G are concyclic, opposite interior angles are supplementary.

Let angles be:
- D = 107°
- E = 82°
- F = 2y − 67
- G = x − 87

So
107 + (2y − 67) = 180 → 40 + 2y = 180 → 2y = 140 → y = 70.

82 + (x − 87) = 180 → x − 5 = 180 → x = 185.

Check: F = 2(70) − 67 = 73° and G = 185 − 87 = 98°, and 107 + 73 = 180, 82 + 98 = 180.

Answer: x = 185, y = 70.