The document Hots Questions- Quadrilaterals Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.

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**Q.1. If the bisectors of the angles of a quadrilateral enclose a rectangle, then show that it is a parallelogram.****Sol.**

Angle bisectors of the quadrilateral ABCD enclose a rectangle PQRS.

∴ ∠P = 90^{o}

⇒ In Δ PCD, ∠1+∠2 = 90^{o}

But, ∠1 and ∠2 are 1/2∠D and 1/2 ∠C respectively.

⇒ ∠D +∠C = 180^{0} [∵ 2∠1 + 2∠2 = 180^{o}]

⇒ ∠ D and ∠ C form a pair of co-interior supplementary angles AD || BC

Similarly, AB || DC ⇒ ABCD is a ||^{gm}.**Q.2. L, M, N, K are midpoints of sides BC, CD, DA and AB respectively of a square ABCD.****Prove that DL, DK, BM and BN enclose a rhombus.**

**Sol.**

BK = DM [halves of equal sides]

∴ BM || DK. Similarly, BN || DL

Also, ∆ ABN ≌ ∆ ADK [SAS congruency]

⇒ ∠ 1 = ∠ 2

Also, ∆PND ≌ ∆PKB [ASA congruency]

⇒ PB = PD

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

**Sol.**

We have ∠a + ∠ADC = 180° [linear pair]

Similarly, ∠b + ∠BCD = 180°

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

But x + x + ∠ADC + ∠BCD = 360° ...(2)

From (1) and (2)

x + x + ∠ADC + ∠BCD = a + b + ∠ADC + ∠BCD

⇒ x + x = a + b

⇒ 2x = a + b

⇒ x = ((a+b)/2)**Q.4. In the adjoining figure, ABCD is a || ^{gm}. Find the angles A, B, C and D.**

**Sol.**

In ∆ACD, 4x + 5x + 6x = 180°

⇒ 15x = 180°

⇒ x = (180^{o}/15) = 12°

∴ ∠D = 6 × 12° = 72°

⇒ B = 72° [∵ opposite angles of ||gm are equal]

∵ ∠ A + ∠ D = 180° [cointerior angles]

∴ ∠A = 180° – ∠D = 180° – 72° = 180°

⇒ ∠ C = 108°**Q.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.**

**Sol.**

In ∆SMR, SM = SR

⇒ ∠ 1 = ∠ 2 [Angles opposite to equal sides are equal]

∠1 = ∠3

[∵QR || PM, corresponding angles are equal]

Similarly, ∠2 = ∠4 [corresponding angles]

⇒ ∠ 3 = ∠ 4

⇒ in ∆ QRN, QN = QR

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