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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.

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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.

In ∆ABC,
∵ ∠A = ∠B | Given
∴ ∠CAB = ∠CBA (= 85°)
| Angles opposite to equal sides of a triangle are equal
Also, x + ∠CAB + ∠CBA = 180°
| Angle sum property of a triangle
⇒ x + 85° + 85° = 180°
⇒ x = 10°
In ∆BCD,
∵ BD = CD
∴ x = z = 10°
| Angles opposite to equal sides of a triangle are equal
Also, z + x + y = 180°
| Angle sum property of a triangle
⇒ 10° + 10° + y = 180°
⇒ y = 160°

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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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