RD Sharma solutions for Mathematics [English] Class 9 chapter 13 - Quadrilaterals [Latest edition]

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).

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.

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.

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.

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

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 ........................

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 ................ 

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. 

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 .................

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

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

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 ...............

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 ....................

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 ..................

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

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.

Mark the correct alternative in each of the following:
The opposite sides of a quadrilateral have

The consecutive sides of a quadrilateral have

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

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

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?

Diagonals necessarily bisect opposite angles in a

The two diagonals are equal in a

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

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

The bisectors of the angle of a parallelogram enclose a

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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 =

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

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

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 =

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

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

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