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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.

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76. In figure, the side BC of ∆ABC is produced to D. The bisector of ∠A meets BC in L. Prove that ∠ABC + ∠ACD = 2 ∠ALC

Given: The side BC of ∆ABC is produced to D. The bisector of ∠A meets BC in L.
To Prove: ∠ABC + ∠ACD = 2 ∠ALC.
Proof: ∠ABC + ∠ACD
= ∠ABC + (∠ABC + ∠BAC)
| Exterior angle theorem
= 2 ∠ABC + ∠BAC
= 2 ∠ABC + 2 ∠BAL
| ∵ AL is the bisector of ∠A
= 2 (∠ABC + ∠BAL)
= 2 ∠ALC | Exterior angle theorem

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