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Areas of Parallelograms and Triangles
Question
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Angles A, B and C of a triangle ABC are equal to each other. Prove that ∆ABC is equilateral.

Answer

∵ ∠A = ∠B
∴ BC = CA    ...(1)
| Sides opposite to equal angles of ∆ABC
∵ ∠B = ∠C
∴ CA = AB    ...(2)
| Sides opposite to equal angles of ∆ABC
∵ ∠C = ∠A
∴ AB = BC    ...(3)
| Sides opposite to equal angles of ∆ABC
From (1), (2) and (3), we have
AB = BC = CA
⇒ ∆ABC is equilateral.

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Some More Questions From Areas of Parallelograms and Triangles Chapter

In figure, AP and BQ are perpendiculars to the line-segment AB and AP = BQ. Prove that O is the midpoint of line segments AB and PQ.

In figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB = AD and CB = CD.

AB is a line-segment. AX and BY are two equal line-segments drawn on opposite sides of line AB such that AX || BY. If AB and XY intersect each other at P. Prove that:
(i)    ∆APX ≅ ∆BPY
(ii)    AB and XY bisect each other at P.

In figure, ∠QPR = ∠PQR and M and N are respectively points on sides QR and PR of ∆PQR, such that QM = PN. Prove that OP = OQ, where O is the point of intersecting of PM and QN.

In figure, ∠B = ∠.E, BD = CE and ∠1 = ∠2. Show ∆ABC ≅ ∆AED.

In figure given below, AD is the median of ∆ABC.
BE ⊥ AD, CF ⊥ AD. Prove that BE = CF.

In the given figure, if AB = FE, BC = ED, AB ⊥ BD and FE ⊥ EC, then prove that AD = FC.

In figure, OA = OB and OD = OC. Show that:
(i) ∆AOD ≅ ∆BOC and (ii) AD = BC.

AB is a line segment and line l is its perpendicular bisector. If a point P lies on I, show that P is equidistant from A and B.