Chapter 8 – QUADRILATERALS Exercise 8.1

(Q 1) The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

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(Q 2) If the diagonals of a parallelogram are equal, then show that it is a rectangle.

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(Q 3) Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

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(Q 4) Show that the diagonals of a square are equal and bisect each other at right angles.

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(Q 5) Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

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(Q 6) Diagonal AC of a parallelogram ABCD bisects
Ð A (see Fig. 8.19). Show that
v (i) it bisects ∠ C also,
(ii) ABCD is a rhombus.

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(Q 7) ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

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(Q 8) ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:
i. ABCD is a square
ii. diagonal BD bisects ∠ B as well as ∠D

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(Q 9) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ
(see Fig. 8.20). Show that:
(i)∆ APD ≅ ∆ CQB
(ii) AP = CQ
(iii) ∆ AQB ≅ ∆ CPD
(iv) AQ = CP
(v) APCQ is a parallelogram.

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(Q 10) ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see
Fig. 8.21). Show that
(i)∆APB≅ ∆CQD
(ii)AP=CQ

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(Q 11) In ∆ABC and ∆DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. 8.22). 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

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(Q 12) ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). 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.]

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