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Lines and Angles

Short Answer Type

71. In figure, find the value of x.

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72. In figure, if QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°, find the value of x and y.

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73. In figure, if line segment AB intersects CD at O such that ∠OAD = 80°, ∠ODA = 50° and ∠OCB = 40°, then find ∠OBC.

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74. In figure, AP and DP are bisectors of two adjacent angles A and D of a quadrilateral ABCD. Prove that 2 ∠APD = ∠B + ∠C.

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75. In figure, AB II DE, ∠ABC = 75°, ∠CDE = 145°, find ∠BCD.

Construction: Extend ED, AB and CB.
∠BHF = ∠CBA
| Corresponding angles
∵ AB || FE and BH intersects them
= 75°    ...(1)
∠BHF + ∠BHD = 180° | Linear pair axiom
⇒ 75° + ∠BHD = 180°
⇒ ∠BHD = 105°    ...(2)
∠CDH + ∠CDE = 180° | Linear pair axiom
⇒ ∠CDH + 145° = 180°
⇒    ∠CDH = 35°    ...(3)
In ∆CHD,
∠CDH + ∠CHD + ∠BCD = 180°
| Angle sum property of a triangle
⇒ 35° + 105° + ∠BCD = 180°
⇒    ∠BCD = 40°