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Sponsor Area

Areas Of Parallelograms And Triangles

Question

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ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see figure). Prove that:

(i) ∆ABD ≅ ∆BAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.

Solution

Given: ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA.
To Prove: (i) ∆ABD ≅ ∆BAC
(ii)    BD = AC
(iii)    ∠ABD = ∠BAC.
Proof: (i) In ∆ABD and ∆BAC,
AD = BC    | Given
AB = BA    | Common
∠DAB = ∠CBA    | Given
∴ ∆ABD ≅ ∠BAC    | SAS Rule
(ii)    ∵ ∆ABD ≅ ∆BAC    | Proved in (i)
∴ BD = AC    | C.P.C.T.
(iii)    ∵ ∆ABD ≅ ∠BAC    | Proved in (i)
∴ ∠ABD = ∠BAC.    | C.P.C.T.

Sponsor Area

Some More Questions From Areas of Parallelograms and Triangles Chapter

In figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠BAD = ∠ABE and ∠EPA = ∠DPB (see figure). Show that.

(i) ∆DAP ≅ ∆EBP
(ii) AD = BE.

In figure, PS = QR and ∠SPQ = ∠RQP. Prove that PR = QS and ∠QPR = ∠PQS.

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.

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