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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

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Long Answer Type

77. Prove that the sum of the angles of a hexagon is 720°.

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Short Answer Type

78. In figure, PS is the bisector of ∠QPR and PT ⊥ QR. Show that

angle T P S equals 1 half left parenthesis angle straight Q minus angle straight R right parenthesis.

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79. ABC is a triangle right angled at A. AL is drawn perpendicular to BC. Show that ∠BAL = ∠ACB.

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80. Prove that the sum of all the angles of a quadrilateral is 360°.

Given: ABCD is a quadrilateral
To Prove: ∠A + ∠B + ∠C + ∠D = 360°
Construction: Join AC.
Proof: In ∆ABC,

∠1 + ∠B + ∠3 = 180°    ...(1)
| Angle sum property of a triangle
In ∆ADC,
∠2 + ∠D + ∠4 = 180°    ...(2)
| Angle sum property of a triangle
Adding (1) and (2), we get
(∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360°
∠    ∠A + ∠B + ∠C + ∠D = 360°

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