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Triangles

Short Answer Type

41. In figure, X and Y are two points on equal sides AB and AC of a ∆ABC such that AX = AY. Prove that XC = YB.

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42. Prove that the angles opposite to equal sides of a triangle are equal. Is the converse true?

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43. In figure, AB = AC, D is the point in the interior of ∆ABC such that ∠DBC = ∠DCB. Prove that AD bisects ∠BAC of ∆ABC.

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Long Answer Type

44.

In figure, ABCD is a square and ∠DEC is an equilateral triangle. Prove that

(i) ∆ADE ≅ ∆BCE
(ii) AE = BE
(iii) ∠DAE = 15°

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Short Answer Type

45. In ∆ABC, AB = AC, ∠A = 36°. If the internal bisector of ∠C meets AB at D, Prove that AD = DC.

Given: In ∆ABC, AB = AC, ∠A = 36°. The internal bisector of ∠C meets AB at D.
To Prove: AD = DC

Proof: In ∆ABC,
∵ AB = AC
∴ ∠ABC = ∠ACB ...(1)
| Angles opposite to equal sides of a triangle are equal
Also, ∠BAC + ∠ABC + ∠ACB = 180°
| Angle sum property of a triangle
⇒ 360° + ∠ABC + ∠ACB = 180°
⇒ ∠ABC + ∠ACB = 144° ...(2)
From (1) and (2),

$angle A B C equals angle A C B equals 1 half cross times 144 degree equals 72 degree$