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Triangles

Short Answer Type

51. In the given figure, AE bisects ∠DAC and ∠B = ∠C, prove that AE || BC.

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52. If the bisector of the vertical angle of a triangle bisects the base of the triangle, then prove that the triangle is isosceles.

Given: A ∆ABC in which the bisector of the vertical angle ∠BAC bisects the base BC, i.e., BD = CD

To Prove: ∆ABC is isosceles

Given: A ∆ABC in which the bisector of the vertical angle ∠BAC bi

Construction: Produce AD to E such that AD = DE. Join EC.
Proof: In ∆ADB and ∆EDC,
BD = CD    | Given
AD = ED    | By construction
∠ADB = ∠EDC
| Vertically opposite angles
∴ ∆ADB ≅ ∆EDC
| SAS congruence rule ∴ AB = EC    ...(1) | CPCT
and    ∠BAD = ∠CED    | CPCT
But ∠BAD = ∠CAD    | Given
∴ ∠CAD = ∠CED
∴ AC = CE    ...(2)
| Sides opposite to equal angles of a triangle are equal
From (1) and (2),
AB = AC
∴ ∆ABC is isosceles.

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53. In figure, ∠BAC = 85°, ∠A - ∠B and BD = CD. Find the measure of ∠x, ∠y and ∠z. Give reasons to support your answer.

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54. In ∆ABC, the bisector AD of ∠A is perpendicular to side BC (see figure). Show that AB = AC and ∆ABC is isosceles.

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55. E and F are respectively the mid-points of equal sides AB and AC of ∆ABC (see figure). Show that BF = CE.

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56. In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD (see figure). Show that AD = AE.

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57. AD and BE are respectively the altitudes of an isosceles triangle ABC. in which AC = BC. Prove that AE = BD.

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58. PS is an altitude of an isosceles triangle PQR in which PQ = PR. Show that PS bisects ∠P.

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59. In ∆ABC, AB = AC. D is a point in side ∆ABC such that BD = DC. Prove that ∠ABD = ∠ACD.

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60. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that AD is also the median of the triangle.

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