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Triangles

Short Answer Type

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.

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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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Short Answer Type

39. Suppose line segments AB and CD intersect at O in such a way that AO = OD and OB = OC. Prove that AC = BD but AC may not be parallel to BD.

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40. In the figure, D and E are points on the base BC of a ∆ABC such that AD = AE and ∠BAD = ∠CAE. Prove that AB = AC.

Given: D and E are points on the base BC of a ∆ABC such that AD = AE and ∠BAD = ∠CAE.
To Prove: AB = AC
Proof: In ∆ADE,
∵ AD = AE | Given
∴ ∠ADE = ∠AED ...(1)
| Angles opposite to equal sides of a triangle are equal
In ∆ABD,
Ext. ∠ADE = ∠BAD + ∠ABD ...(2)
| An exterior angle of a triangle is equal to the sum of its two interior opposite angles
In ∆AEC,
Ext. ∠AED = ∠CAE + ∠ACE .. .(3)
| An exterior angle of a triangle is equal to the sum of its two interior opposite angles
From (1), (2) and (3),
∠BAD + ∠ABD = ∠CAE + ∠ACE
⇒ ∠ABD = ∠ACE
| ∵ ∠BAD = ∠CAE (Given)
⇒ ∠ABC = ∠ACB
∴ AB = AC
| Sides opposite to equal angles of a triangle are equal

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