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**Question 1. If the bisectors of the angles of a quadrilateral enclose a rectangle, then show that it is a parallelogram. Hint: **Angle bisectors of the quadrilateral ABCD enclose a rectangle PQRS.

âˆ´ âˆ P = 90^{o}

â‡’ In Î” PCD, âˆ 1+âˆ 2 = 90^{o}

But , âˆ 1 and âˆ 2 are 1/2âˆ D and 1/2 âˆ C respectively

â‡’ âˆ D +âˆ C = 180^{0} [âˆµ 2âˆ 1 + 2âˆ 2 = 180^{o}]

â‡’ âˆ D and âˆ C form a pair of cointerior supplementary angles AD || BC

Similarly, AB || DC â‡’ ABCD is a ||^{gm}.

**Question 2. L,M,N,K are mid points of sides BC, CD, DA and AB respectively of a square ABCD. Prove that DL, DK, BM and BN enclose a rhombus.**

**Hint: **BK = DM [halves of equal sides]

âˆ´ BM || DK. Similarly, BN || DL

Also, âˆ† ABN â‰Œ âˆ† ADK [SAS congruency]

â‡’ âˆ 1 = âˆ 2

Also, âˆ†PND â‰Œ âˆ†PKB [ASA congruency]

â‡’ PB = PD

â‡’ DQBP is a rhombus.

**Question 3. The sides AD and BC of a quadrilateral are produced as shown in the given figure. Prove that x = ((a+b)/2)**

**Hint:** We have âˆ a + âˆ ADC = 180Â° [linear pair]

Similarly, âˆ b + âˆ BCD = 180Â°

Adding (a + b) + âˆ ADC + âˆ BCD = 360Â° ...(1)

But x + x + âˆ ADC + âˆ BCD = 360Â° ...(2)

From (1) and (2) x + x + âˆ ADC + âˆ BCD

= a + b + âˆ ADC + âˆ BCD

â‡’ x + x = a + b

â‡’ 2x = a + b

â‡’ **x = ((a+b)/2)**

**Question 4. In the adjoining figure, ABCD is a || ^{gm}. Find the angles A, B, C and D.**

Hint: In âˆ†ACD, 4x + 5x + 6x = 180Â°

â‡’ 15x = 180Â°

â‡’ x = (180^{o}/15) = 12Â°

âˆ´ âˆ D = 6 Ã— 12Â° = 72Â°

â‡’ B = 72Â° [âˆµ opposite angles of ||gm are equal]

âˆµ âˆ A + âˆ D = 180Â° [cointerior angles]

âˆ´ âˆ A = 180Â° â€“ âˆ D = 180Â° â€“ 72Â° = 180Â°

â‡’ âˆ C = 108Â°

**Question 5. PQRS is a || ^{gm}. PS is produced to M, so that SM = SR and MR produced meet PQ produced at N. Prove that QN = QR**

**Hint:** In âˆ†SMR, SM = SR

â‡’ âˆ 1 = âˆ 2 [Angles opposite to equal sides are equal]

âˆ 1 = âˆ 3

[âˆµQR || PM, corres. angles are equal]

Similarly, âˆ 2 = âˆ 4 [corres. angles]

â‡’ âˆ 3 = âˆ 4

â‡’ in âˆ† QRN, QN = QR

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