Question

AB and CD are parallel
Find the measure of angle R

Answer 1

Answer:

Given:

AB || CD

XY is a tranversal passing through AB and CD.

∠APR = 15°

∠CQY = 75°

∠SQD = 90°

Solution:

∠APY = ∠CQY = 75° (Corresponding Angles)

∠APY = ∠APR + ∠RPQ

75° = 15° + ∠RPQ

⇒∠RPQ = 75° - 15° = 60°

∠CQS = ∠CQY + ∠YQS = 75° + ∠YQS ...(i)

∠CQS + ∠DQS = 180° (Linear Pair)

∠CQY + ∠YQS + ∠DQS = 180° ...from (i)

75° + ∠YQS + 90° = 180°

∠YQS + 165° = 180°

⇒∠YQS = 180° - 165° = 15°

∠ RQP = ∠YQS = 15° (Vertically Opposite Angles)

In ∆PQR,

∠RPQ + ∠PRQ + ∠RQP = 180° (Angle Sum Property)

60° + ∠PRQ + 15° = 180°

75° + ∠PRQ = 180°

∠PRQ = 180° - 75° = 105°

⇒∠PRQ = 105°