Geometry Practice Test 7

ACQP is a cyclic quadrilateral (sum of two opposite angles in a cyclic quadrilateral is 180°.)
∠APQ + ∠ACQ = 180°
∠APQ = 180° - 75° = 105°
∠APQ + ∠BPQ = 180° (Linear pair)
∠ACQ + ∠RCQ = 180° (Linear pair)
∠APQ = 105, ∠ACQ = 75°
∠BPQ = 180° - 105° ∠RCQ = 180°-75°
∠BPQ = 75° ∠RCQ = 105°
In ΔQCR,
∠R + ∠QCR + ∠CQR = 180°
∠CQR = 180° - (30° + 105°)
= 180° - 135°
∠CQR = 45°
∠PQR = ∠CQR = 45° (vertically opposite angle)
In ΔPBQ,
∠PBQ + ∠BQP + ∠QPB = 180°
45° + 75° + ∠PBQ = 180°
∠PBQ = 180 - 120°
∠PBQ = 60°
∠CBA = 60°