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Quadrilaterals

Short Answer Type

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

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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

6.
Diagonal AC of a parallelogram ABCD bisects ∠A (see figure). Show that:
(i) it bisects ∠C also
(ii) ABCD is a rhombus.

Given: Diagonal AC of a parallelogram ABCD bisects ∠A.
To Prove: (i) it bisects ∠C also.
(ii) ABCD is a rhombus.
Proof: (i) In ∆ADC and ∆CBA,
AD = CB
| Opp. sides of || gm ABCD
CA = CA    | Common
DC = BA
| Opp. sides of || gm ABCD
∴ ∆ADC ≅ ∆CBA
| SSS Congruence Rule
∴ ∠ACD = ∠CAB    | C.RC.T.
and    ∠DAC = ∠BCA    | C.RC.T.
But ∠CAB = ∠DAC    | Given
∴ ∠ACD = ∠BCA
∴ AC bisects ∠C also.
(ii) From above,
∠ACD = ∠CAD
∴ AD = CD
| Sides opposite to equal angles of a triangle are equal
∴ AB = BC = CD = DA
| ∵ ABCD is a || gm
∴ 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.