Long Answer TypeIn ∆ABC and ∆DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, Band C are joined to vertices D, E and F respectively (see figure). Show that:
(i) quadrilateral ABED is a parallelogram
(ii) quadrilateral BEFC is a parallelogram
(iii) AD || CF and AD = CF
(iv) quadrilateral ACFD is a parallelogram
(v) AC = DF
(vi) ∆ABC ≅ ∆DEF. [CBSE 2012
Short Answer TypeABCD is a trapezium in which AB || CD and AD = BC (see figure): Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ∆ABC = ∆BAD
(iv) diagonal AC = diagonal BD.
[Hint. Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
Long Answer TypeShort Answer TypeGiven: ABCD is a parallelogram and line segments AX, CY bisect the angles A and C respectively.
To Prove: AX || CY.
Proof: ∵ ABCD is a parallelogram.
∴ ∠A = ∠C | Opposite ∠s
| ∵ Halves of equals are equal
⇒ ∠1 = ∠2 ....(1)
| ∵ AX is the bisector of ∠A and CY is the bisector of ∠C
Now, AB || DC and CY intersects them
∴ ∠2 = ∠3 ...(2)
| Alternate interior ∠s
From (1) and (2), we get
∠1 = ∠3
But these form a pair of equal corresponding angles
∴ AX || CY.
Long Answer TypeShort Answer TypeGiven ∆ABC, lines are drawn through A, B and C parallel respectively to the sides
BC, CA and AB forming ∆PQR. Show that BC = 
Long Answer TypeABCD is a trapezium in which AB || CD and AD = BC. Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ∆ABC ≅ ∆BAD.
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