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Triangles

Short Answer Type

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

Given: AE bisects ∠DAC and ∠B = ∠C
To Prove: AE || BC
Proof: In ∆ABC,
Ext. ∠DAC = ∠ABC + ∠ACB ...(1)
| An exterior angle of a triangle is equal to the sum of its two interior opposite angles
⇒ ∠DAC = ∠ACB + ∠ACB
| ∵ ∠B = ∠C (Given)
⇒ ∠DAC = 2∠ACB
⇒ 2∠CAE = 2∠ACB

$left enclose table row cell because space AE space bisects space angle DAC end cell row cell therefore space angle CAE equals angle DAE equals 1 half angle DAC end cell end table end enclose$

⇒ ∠CAE = ∠ACB
But these angles form a pair of equal alternate interior angles
∴ 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.

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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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