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

Let ABCD be a quadrilateral in which
∠A : ∠B : ∠C : ∠D = 3 : 5 : 9 : 13
Sum of the ratios = 3 + 5 + 9+ 13 = 30
Also, ∠A + ∠B + ∠C + ∠D = 360°
Sum of all the angles of a quadrilateral is 360°

$therefore space space space space space space space space space space space space space space space space angle straight A equals 3 over 30 cross times 360 degree equals 36 degree space space space space space space space space space space space space space space space space space space space angle straight A equals 5 over 30 cross times 360 degree equals 60 degree space space space space space space space space space space space space space space space space space space space angle straight C equals 9 over 30 cross times 360 degree equals 108 degree and space space space space space space space space space space space space space space space angle straight D equals 13 over 30 cross times 360 degree equals 156 degree space space space space space space space space space space space space space space space space space space space$

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

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

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Long Answer Type

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

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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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Short Answer Type

Diagonal AC of a parallelogram ABCD bisects ∠A (see figure). Show that:
(i) it bisects ∠C also
(ii) ABCD is a rhombus.

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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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Long Answer Type

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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9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see figure). Show that:

(i) ∆APD ≅ ∆CQB
(ii) AP = CQ
(iii) ∆AQB ≅ ∆CPD
(iv) AQ = CP
(v) APCQ is a parallelogram.

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Short Answer Type

10.

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

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