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81. In figure, the side BC of a ∆ABC is produced to D. The bisector of ∠BAC intersects the side BC at E. Prove that ∠ABC + ∠ACD = 2 ∠AEC.

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82. The sides BA and DC of a quadrilateral ABCD are produced as shown in figure. Show that ∠x + ∠y = ∠a + ∠b

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83. The degree measures of three angles of a triangle are x, y and z. If

straight z equals fraction numerator straight x plus straight y over denominator 2 end fraction comma

then find the value of z.

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Solution not provided.

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85. The angles of a triangle are 2x + 35, 3x - 5 and 5x - 50 degree. Find x.

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86. The sum of two angles of a triangle is 80° and the difference is 20°. Find all the angles of the triangle.

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87. In the given figure, if l || m, then find the value of x.

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88. In the figure, ∠X = 72°, ∠X ZY = 46°. If YO and ∠O are bisectors of ∠XYZ and ∠XZY respectively of ∆XYZ, find ∠OYZ and ∠YOZ.

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89. In figure, find the value of x.

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90. In figure, AB || DC, ∠BDC = 30° and ∠BAD = 80°, find ∠x, ∠y and ∠z.

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