Question
Find x if PQ || RS and ∠1: ∠2 = 3: 5.
- 60°
- 75°
- 37.5°
- 135°
Hint:
Find the angles of triangle OSR and apply the ratio and Angle Sum property of triangle to get the equation upon solving we get x.
The correct answer is: 37.5°
PQ || RS ⇒ ∠PQO = ∠OSR (Alternate Interior Angles)
⇒ ∠1 = x ——— (i)
Also, ∠POQ = ∠SOR (Vertically Opposite Angles)
⇒ ∠SOR = 80°
In ∆ SOR, ∠OSR + ∠SRO + ∠ROS = 180° (Angle Sum property of triangle)
⇒ x + ∠2 + 80° = 180°
⇒ ∠1 + ∠2 = 100°
⇒ ∠1 + (5/3) ∠1 = 100° (∠1: ∠2 = 3: 5)
⇒ (8/3) ∠1 = 100°
⇒ ∠1 = 37.5°
From equation (i), x = ∠1 = 37.5°.
⇒ ∠1 = x ——— (i)
Also, ∠POQ = ∠SOR (Vertically Opposite Angles)
⇒ ∠SOR = 80°
In ∆ SOR, ∠OSR + ∠SRO + ∠ROS = 180° (Angle Sum property of triangle)
⇒ x + ∠2 + 80° = 180°
⇒ ∠1 + ∠2 = 100°
⇒ ∠1 + (5/3) ∠1 = 100° (∠1: ∠2 = 3: 5)
⇒ (8/3) ∠1 = 100°
⇒ ∠1 = 37.5°
From equation (i), x = ∠1 = 37.5°.
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