#### Chapters

Chapter 2: Exponents of Real Numbers

Chapter 3: Rationalisation

Chapter 4: Algebraic Identities

Chapter 5: Factorisation of Algebraic Expressions

Chapter 6: Factorisation of Polynomials

Chapter 7: Linear Equations in Two Variables

Chapter 8: Co-ordinate Geometry

Chapter 9: Introduction to Euclid’s Geometry

Chapter 10: Lines and Angles

Chapter 11: Triangle and its Angles

Chapter 12: Congruent Triangles

Chapter 13: Quadrilaterals

Chapter 14: Areas of Parallelograms and Triangles

Chapter 15: Circles

Chapter 16: Constructions

Chapter 17: Heron’s Formula

Chapter 18: Surface Areas and Volume of a Cuboid and Cube

Chapter 19: Surface Areas and Volume of a Circular Cylinder

Chapter 20: Surface Areas and Volume of A Right Circular Cone

Chapter 21: Surface Areas and Volume of a Sphere

Chapter 22: Tabular Representation of Statistical Data

Chapter 23: Graphical Representation of Statistical Data

Chapter 24: Measures of Central Tendency

Chapter 25: Probability

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

## Chapter 13: Quadrilaterals

#### Chapter 13: Quadrilaterals Exercise 13.10 solutions [Page 4]

Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle

In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angles of the quadrilateral

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

In a quadrilateral ABCD, CO and DO are the bisectors of `∠`C and ∠D respectively. Prove that

`∠`COD = `1/2` (`∠`A+ `∠`B).

#### Chapter 13: Quadrilaterals Exercise 13.20 solutions [Pages 19 - 20]

Two opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. Find the measure of each angle of the parallelogram .

If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram .

Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle

The perimeter of a parallelogram is 22 cm . If the longer side measures 6.5 cm what is the measure of the shorter side?

In a parallelogram ABCD, ∠D = 135°, determine the measures of ∠A and ∠B

ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D .

In Fig. below, ABCD is a parallelogram in which ∠DAB **= **75° and ∠DBC = 60°. Compute

∠CDB and ∠ADB.

**The following statement are true and false .**

In a parallelogram, the diagonals are equal

**The following statement are true and false. **

In a parallelogram, the diagonals bisect each other.

**The following statement are true and false .**

In a parallelogram, the diagonals intersect each other at right angles .

**The following statement are true and false .**

In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.

**The following statement are true and false .**

If all the angles of a quadrilateral are equal, it is a parallelogram .

**The following statement are true and false .**

If three sides of a quadrilateral are equal, it is a parallelogram .

**The following statement are true and false . **

If three angles of a quadrilateral are equal, it is a parallelogram .

**The following statement are true and false . **

If all the sides of a quadrilateral are equal it is a parallelogram.

In Fig., below, ABCD is a parallelogram in which ∠A = 60°. If the bisectors of ∠A and ∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.

In below fig. ABCD is a parallelogram and E is the mid-point of side B If DE and AB when produced meet at F, prove that AF = 2AB.

#### Chapter 13: Quadrilaterals Exercise 13.30 solutions [Pages 42 - 43]

In a parallelogram ABCD, determine the sum of angles ∠C and ∠D .

In a parallelogram ABCD, if `∠`B = 135°, determine the measures of its other angles .

ABCD is a square. AC and BD intersect at O. State the measure of ∠AOB.

ABCD is a rectangle with ∠ABD = 40°. Determine ∠DBC .

The sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a parallelogram.

P and Q are the points of trisection of the diagonal BD of a parallelogram AB Prove that CQ is parallel to AP. Prove also that AC bisects PQ.

ABCD is a square E, F, G and H are points on AB, BC, CD and DA respectively, such that AE = BF = CG = DH. Prove that EFGH is a square.

ABCD is a rhombus, EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles

ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC.

#### Chapter 13: Quadrilaterals Exercise 13.40 solutions [Pages 62 - 65]

In a ∆ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are 7 cm, 8 cm and 9 cm, respectively, find the perimeter of ∆DEF.

In a triangle ∠ABC, ∠A = 50°, ∠B = 60° and ∠C = 70°. Find the measures of the angles of

the triangle formed by joining the mid-points of the sides of this triangle.

In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC =

21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.

In a ΔABC median AD is produced to X such that AD = DX. Prove that ABXC is a

parallelogram.

In a ΔABC, E and F are the mid-points of AC and AB respectively. The altitude AP to BC

intersects FE at Q. Prove that AQ = QP.

In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing

through A. If L is the mid-point of BC, prove that ML = NL.

In Fig. below, triangle ABC is right-angled at B. Given that AB = 9 cm, AC = 15 cm and D,

E are the mid-points of the sides AB and AC respectively, calculate

(i) The length of BC (ii) The area of ΔADE.

In Fig. below, M, N and P are the mid-points of AB, AC and BC respectively. If MN = 3 cm, NP = 3.5 cm and MP = 2.5 cm, calculate BC, AB and AC.

In Fig. below, AB = AC and CP || BA and AP is the bisector of exterior ∠CAD of ΔABC.

Prove that (i) ∠PAC = ∠BCA (ii) ABCP is a parallelogram

ABCD is a kite having AB = AD and BC = CD. Prove that the figure formed by joining the

mid-points of the sides, in order, is a rectangle.

Let Abc Be an Isosceles Triangle in Which Ab = Ac. If D, E, F Be the Mid-points of the Sides Bc, Ca and a B Respectively, Show that the Segment Ad and Ef Bisect Each Other at Right Angles.

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral

bisect each other.

**Fill in the blank to make the following statement correct**

The triangle formed by joining the mid-points of the sides of an isosceles triangle is __ __

**Fill in the blank to make the following statement correct:**

The triangle formed by joining the mid-points of the sides of a right triangle is __ __

**Fill in the blank to make the following statement correct:**

The figure formed by joining the mid-points of consecutive sides of a quadrilateral is __ __

ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively

intersecting at P, Q and R. Prove that the perimeter of ΔPQR is double the perimeter of

ΔABC

In Fig. below, BE ⊥ AC. AD is any line from A to BC intersecting BE in H. P, Q and R are

respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°.

ABC is a triang D is a point on AB such that AD = `1/4` AB and E is a point on AC such that AE = `1/4` AC. Prove that DE = `1/4` BC.

In below Fig, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = `1/4` AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.

In the below Fig, ABCD and PQRC are rectangles and Q is the mid-point of Prove thaT

i) DP = PC (ii) PR = `1/2` AC

ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH

BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If

L is the mid-point of BC, prove that LM = LN.

#### Chapter 13: Quadrilaterals solutions [Pages 68 - 70]

In a parallelogram ABCD, write the sum of angles A and B.

In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.

PQRS is a square such that PR and SQ intersect at O. State the measure of ∠POQ.

If PQRS is a square, then write the measure of ∠SRP.

If ABCD is a rhombus with ∠ABC = 56°, find the measure of ∠ACD.

The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm, what is the measure of shorter side?

If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.

In a parallelogram ABCD, if ∠A = (3x − 20)°, ∠B = (y + 15)°, ∠C = (x + 40)°, then find the values of xand y.

If measures opposite angles of a parallelogram are (60 − x)° and (3x − 4)°, then find the measures of angles of the parallelogram.

In a parallelogram ABCD, the bisector of ∠A also bisects BC at X. Find AB : AD.

In the given figure, PQRS is an isosceles trapezium. Find x and y.

In the given figure, ABCD is a trapezium. Find the values of x and y.

In the given figure, ABCD and AEFG are two parallelograms. If ∠C = 58°, find ∠F.

Complete the following statement by means of one of those given in brackets against each:

If one pair of opposite sides are equal and parallel, then the figure is ........................

parallelogram

rectangle

trapezium

Complete the following statement by means of one of those given in brackets against each:

If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ................

square

rectangle

trapezium

Complete the following statement by means of one of those given in brackets against each:

A line drawn from the mid-point of one side of a triangle .............. another side intersects the third side at its mid-point.

perpendicular to parallel to

to meet

Complete the following statement by means of one of those given in brackets against each:

If one angle of a parallelogram is a right angle, then it is necessarily a .................

rectangle

square

rhombus

Complete the following statement by means of one of those given in brackets against each:

Consecutive angles of a parallelogram are ...................

supplementary

complementary

Complete the following statement by means of one of those given in brackets against each:

If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ...............

rectangle

parallelogram

rhombus

Complete the following statement by means of one of those given in brackets against each:

If opposite angles of a quadrilateral are equal, then it is necessarily a ....................

parallelogram

rhombus

rectangle

Complete the following statement by means of one of those given in brackets against each:

f consecutive sides of a parallelogram are equal, then it is necessarily a ..................

kite

rhombus

square

In a quadrilateral ABCD, bisectors of angles A and B intersect at O such that ∠AOB = 75°, then write the value of ∠C + ∠D.

The diagonals of a rectangle ABCD meet at O, If ∠BOC = 44°, find ∠OAD.

If ABCD is a rectangle with ∠BAC = 32°, find the measure of ∠DBC.

If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB, then find the value of k.

In the given figure, PQRS is a rhombus in which the diagonal PR is produced to T. If ∠SRT = 152°, find x, y and z.

In the given figure, ABCD is a rectangle in which diagonal AC is produced to E. If ∠ECD = 146°, find ∠AOB.

#### Chapter 13: Quadrilaterals solutions [Pages 70 - 73]

Mark the correct alternative in each of the following:

The opposite sides of a quadrilateral have

no common point

one common point

two common points

infinitely many common points

The consecutive sides of a quadrilateral have

no common point

one common point

two common points

infinitely many common points

PQRS is a quadrilateral, PR and QS intersect each other at O. In which of the following case, PQRS is a parallelogram?

∠P = 100°, ∠Q = 80°, ∠R = 95°

PQRS is a quadrilateral, PR and QS intersect each other at O. In which of the following case, PQRS is a parallelogram?

∠P =85°, ∠Q = 85°, ∠R = 95°

PQRS is a quadrilateral, PR and QS intersect each other at O. In which of the following case, PQRS is a parallelogram?

PQ = 7 cm, QR = 7 cm, RS = 8 cm, SP = 8 cm

OP = 6.5 cm, OQ = 6.5 cm, OR = 5.2 cm, OS = 5.2 cm

Which of the following quadrilateral is not a rhombus?

All four sides are equal

Diagonals bisect each other

Diagonals bisect opposite angles

One angle between the diagonals is 60°

Diagonals necessarily bisect opposite angles in a

rectangle

parallelogram

isosceles trapezium

square

The two diagonals are equal in a

parallelogram

rhombus

rectangle

trapezium

We get a rhombus by joining the mid-points of the sides of a

parallelogram

rhombus

rectangle

triangle

The bisectors of any two adjacent angles of a parallelogram intersect at

30°

45°

60°

90°

The bisectors of the angle of a parallelogram enclose a

parallelogram

rhombus

rectangle

^{square}

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

parallelogram

rectangle

square

rhombus

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a

square

rhombus

trapezium

none of these

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

square

rectangle

trapezium

none of these

The figure formed by joining the mid-points of the adjacent sides of a square is a

rhombus

square

rectangle

parallelogram

The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a

rectangle

parallelogram

rhombus

square

If one angle of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is

176°

68°

112°

102°

In a parallelogram ABCD, if ∠DAB = 75° and ∠DBC = 60°, then ∠BDC =

75°

60°

45°

55°

ABCD is a parallelogram and E and F are the centroids of triangles ABD and BCDrespectively, then EF =

AE

BE

CE

DE

ABCD is a parallelogram, M is the mid-point of BD and BM bisects ∠B. Then ∠AMB =

45°

60°

90°

75°

If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is

108°

54°

72°

81°

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?

140°

150°

168°

180°

If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to

16 cm

15 cm

20 cm

17 cm

ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC = 35°, then ∠ABC =

70°

110°

90°

120°

In a rhombus ABCD, if ∠ACB = 40°, then ∠ADB =

70°

45°

50°

60°

In ΔABC, ∠A = 30°, ∠B = 40° and ∠C = 110°. The angles of the triangle formed by joining the mid-points of the sides of this triangle are

70°, 70°, 40°

60°, 40°, 80°

30°, 40°, 110°

60°, 70°, 50°

The diagonals of a parallelogram ABCD intersect at O. If ∠BOC = 90° and ∠BDC = 50°, then ∠OAB =

40°

50°

10°

90°

ABCD is a trapezium in which AB || DC. M and N are the mid-points of AD and the respectively. If AB = 12 cm, MN = 14 cm, then CD =

10 cm

12 cm

14 cm

16 cm

Diagonals of a quadrilateral ABCD bisect each other. If ∠A= 45°, then ∠B =

115°

120°

125°

135°

P is the mid-point of side BC of a parallelogram ABCD such that ∠BAP = ∠DAP. If AD = 10 cm, then CD =

5 cm

6 cm

8 cm

10 cm

In ΔABC, E is the mid-point of median AD such that BE produced meets AC at F. IF AC = 10.5 cm, then AF =

3 cm

3.5 cm

2.5 cm

5 cm

ABCD is a parallelogram and E is the mid-point of BC. DE and AB when produced meet at F. Then, AF =

\[\frac{3}{2}AB\]

2 AB

3 AB

\[\frac{5}{4}AB\]

In a quadrilateral ABCD, ∠A + ∠C is 2 times ∠B + ∠D. If ∠A = 140° and ∠D = 60°, then ∠B=

60°

80°

120°

80°

None of these

The diagonals AC and BD of a rectangle ABCD intersect each other at P. If ∠ABD = 50°, then ∠DPC =

70°

90°

80°

100°

## Chapter 13: Quadrilaterals

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

#### Textbook solutions for Class 9

## RD Sharma solutions for Class 9 Mathematics chapter 13 - Quadrilaterals

RD Sharma solutions for Class 9 Maths chapter 13 (Quadrilaterals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 by R D Sharma (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 9 Mathematics chapter 13 Quadrilaterals are The Mid-point Theorem, Another Condition for a Quadrilateral to Be a Parallelogram, Properties of a Parallelogram, Types of Quadrilaterals, Angle Sum Property of a Quadrilateral, Concept of Quadrilaterals.

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