**Chapter 8 – QUADRILATERALS Exercise 8.1**

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## (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.

## Ans

## (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

## Ans

## (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.]

## Ans

**Chapter 8 – QUADRILATERALS Exercise 8.1**

Ð A (see Fig. 8.19). Show that

v (i) it bisects ∠ C also,

(ii) ABCD is a rhombus.

i. ABCD is a square

ii. diagonal BD bisects ∠ B as well as ∠D

(see Fig. 8.20). Show that:

(i)∆ APD ≅ ∆ CQB

(ii) AP = CQ

(iii) ∆ AQB ≅ ∆ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram.

Fig. 8.21). Show that

(i)∆APB≅ ∆CQD

(ii)AP=CQ

(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

(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.]