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# R.D. Sharma solutions Mathematics for Class 9 by R D Sharma (2018-19 Session) chapter 14 Quadrilaterals

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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

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

∠COD = 1/2 (∠A+ ∠B).

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

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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 ∠A = 60°. If the bisectors of ∠A and ∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.

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

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.

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.

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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.

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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.

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°.

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.

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.

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

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