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31. ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see figure). Show that:

(i) ∆ABE ≅ ∆ACF
(ii) AB = AC, i.e., ∆ABC is an isosceles triangle.

Given: ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal.
To Prove: (i) ∆ABE = ∆ACF
(ii) AB = AC, i.e., ∆ABC is an isosceles triangle.
Proof: (i) In ∆ABE and ∆ACF
BE = CF    | Given
∠BAE = ∠CAF    | Common
∠AEB = ∠AFC    | Each = 90°
∴ ∆ABE ≅ ∆ACF    | By AAS Rule
(ii) ∆ABE ≅ ∆ACF | Proved in (i) above
∴ AB = AC    | C.P.C.T.
∴ ∆ABC is an isosceles triangle.

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32. ABC and DBC are two isosceles triangles on the same base BC (see figure). Show that ∠ABD = ∠ACD.

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Long Answer Type

33.

∆ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see figure). Show that ∆BCD is a right angle.

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Short Answer Type

34. ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

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35. Show that the angles of an equilateral triangle are 60° each.

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36. Angles A, B and C of a triangle ABC are equal to each other. Prove that ∆ABC is equilateral.

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37. BD and CE are the bisectors of ∠B and ∠C of an isosceles triangle ABC with AB = AC. Prove that BD = CE.

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Long Answer Type

38.

ABC is a triangle in which ∠B = 2∠C. D is a point on side BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.
[Hint. Take a point P on AC such that BP bisects ∠B. Join P and D.]

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