Hots Questions- Quadrilaterals Class 9 Notes | EduRev

Mathematics (Maths) Class 9

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Class 9 : Hots Questions- Quadrilaterals Class 9 Notes | EduRev

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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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.  

∴ ∠P = 90o

⇒ In Δ PCD,  ∠1+∠2 = 90o

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

⇒ ∠D +∠C = 1800 [∵ 2∠1 + 2∠2 = 180o]

 

Hots Questions- Quadrilaterals Class 9 Notes | EduRev

⇒ ∠ D and ∠ C form a pair of cointerior supplementary angles AD || BC
Similarly, AB || DC ⇒ 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.

Hots Questions- Quadrilaterals Class 9 Notes | EduRev

Hint: 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.

 

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

Hots Questions- Quadrilaterals Class 9 Notes | EduRev

Hint: 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)

 

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

Hots Questions- Quadrilaterals Class 9 Notes | EduRev

Hint: In ∆ACD, 4x + 5x + 6x = 180°
⇒ 15x = 180°
⇒ x = (180o/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°

 

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

Hots Questions- Quadrilaterals Class 9 Notes | EduRev

Hint: In ∆SMR, SM = SR
⇒ ∠ 1 = ∠ 2                 [Angles opposite to equal sides are equal]
∠1 = ∠3                
[∵QR || PM, corres. angles are equal]
Similarly, ∠2 = ∠4                    [corres. angles]
⇒ ∠ 3 = ∠ 4
⇒ in ∆ QRN, QN = QR

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