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Triangles

Short Answer Type

41. In figure, X and Y are two points on equal sides AB and AC of a ∆ABC such that AX = AY. Prove that XC = YB.

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42. Prove that the angles opposite to equal sides of a triangle are equal. Is the converse true?

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43. In figure, AB = AC, D is the point in the interior of ∆ABC such that ∠DBC = ∠DCB. Prove that AD bisects ∠BAC of ∆ABC.

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Long Answer Type

44.

In figure, ABCD is a square and ∠DEC is an equilateral triangle. Prove that

(i) ∆ADE ≅ ∆BCE
(ii) AE = BE
(iii) ∠DAE = 15°

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Short Answer Type

45. In ∆ABC, AB = AC, ∠A = 36°. If the internal bisector of ∠C meets AB at D, Prove that AD = DC.

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46. In figure, AB = BC, AD = EC. Prove that ∆ABE ≅ ∆CBD.

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47. ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC. Show that ∠B = ∠C.

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48. In an isosceles triangle ABC with AB = AC, BD and CE are two medians. Prove that BD = CE.

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49. In figure, if PS = PR, ∠TPS = ∠QPR then prove that PT = PQ.

Given: PS = PR, ∠TPS = ∠QPR
To Prove: PT = PQ
Proof: In ∆PSR,
∵ PS = PR    | Given
∴ ∠PSR = ∠PRS    ...(1)
| Angles opposite to equal sides of a triangle are equal
⇒ ∠TPS + ∠PTS = ∠QPR + ∠PQR
| An exterior angle of a triangle is equal to the sum of its two interior opposite angles
⇒ ∠PTS = ∠PQR    | Using (1)
⇒ ∠PTQ = ∠PQT ∴ PT = PQ
| Sides opposite to equal angles of a triangle are equal.

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50. In figure, ∠x = ∠y and PQ = PR. Prove that PE = RS.

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