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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.

1318 Views

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.

2598 Views

Short Answer Type

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

∵ In ∆ABC,
AB = AC
∴ ∠B = ∠C ...(1)
| Angles opposite to equal sides of a triangle are equal
In ∆ABC,
∠A + ∠B + ∠C = 180°
| Sum of all the angles of a triangle is 180°
⇒ 90° + ∠B + ∠C = 180°
| ∵ ∠A = 90° (given)
⇒ ∠B + ∠C = 90° ...(2)
From (1) and (2), we get
∠B = ∠C = 45°.

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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.

1965 Views

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.]

1214 Views

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.

2506 Views