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Lines and Angles

Short Answer Type

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

Given: The sides BA and DC of a quadrilateral ABCD are produced.
To Prove: ∠x + ∠y = ∠a + ∠b
Construction: Join BD

Proof: In ∆BCD,

∠a = ∠2 + ∠4 ...(1)
| Exterior angle theorem
In ∆ADB,
∠b = ∠1 + ∠3 ...(2)
| Exterior angle theorem
Adding (1) and (2), we get
∠a + ∠b = (∠1 + ∠2) + (∠3 + ∠4)
⇒ ∠a + ∠b - ∠x + ∠y

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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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84.

In figure, the sides AB and AC of ∆ABC are produced to points E and D respectively. If bisectors BO and CO of ∠CBE and ∠BCD respectively meet at point O, then prove that $angle BOC equals 90 degree minus 1 half angle BAC.$