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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.

Given: AP and DP are bisectors of two adjacent angles A and D of a quadrilateral ABCD.
To Prove: 2 ∠APD = ∠B + ∠C
Proof: We know that the sum of all the angles of a quadrilateral is 360°. So,
∠A + ∠B + ∠C + ∠D = 360°
⇒ ∠A + ∠D = 360° - (∠B + ∠C) ...(1)
Now, in ∆PAD,
∠APD + ∠PAD + ∠PDA = 180°
| Angle sum property of a triangle

$rightwards double arrow space space space angle APD plus 1 half angle straight A plus 1 half angle straight D equals 180 degree$

∵ AP and DP are the bisectors of two adjacent angles A and D of quadrilateral ABCD
⇒ 2 ∠APD + ∠A + ∠D = 360°
⇒ 2 ∠APD = 360° - (∠A + ∠D)
⇒ 2∠APD = ∠B + ∠C

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