R s aggarwal solutions mathematics class 9 Solutions for Chapter 10 quadrilaterals

Three angles of a quadrilateral are 75, 90 and 75. Find the measure of the fourth angle.

The angles of a quadrilateral are in the ratio 2: 4: 5: 7. Find the angles.

In the adjoining figure, ABCD is a trapezium in which AB || DC. If ∠ A = 55 and ∠ B = 70, find ∠C and ∠ D.

In the adjoining figure, ABCD is a square and △ EDC is an equilateral triangle. Prove that

(i) AE = BE,

(ii) ∠ DAE = 15

In the adjoining figure, BM ⊥ AC and DN ⊥ AC. If BM = DN, prove that AC bisects BD.

In the given figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that

(i) AC bisects ∠ A and ∠ C,

(ii) BE = DE,

(iii) ∠ ABC = ∠ ADC

In the given figure, ABCD is a square and ∠ PQR = 90. If PB = QC = DR, prove that

(i) QB = RC,

(ii) PQ = QR,

(iii) ∠ QPR = 45

If O is a point within a quadrilateral ABCD, show that OA + OB + OC + OD > AC + BD.

In the adjoining figure, ABCD is a quadrilateral and AC is one of its diagonals. Prove that

(i) AB + BC + CD + DA > 2AC

(ii) AB + BC + CD > DA

(iii) AB + BC + CD + DA > AC + BD.

Prove that the sum of all the angles of a quadrilateral is 360

In the adjoining figure, ABCD is a parallelogram in which ∠ A = 72. Calculate ∠ B, ∠ C and ∠ D.

In the adjoining figure, ABCD is a parallelogram in which ∠ DAB = 80 and ∠ DBC = 60. Calculate ∠CDB and ∠ ADB.

In the adjoining figure, M is the midpoint of side BC of a parallelogram ABCD such that ∠ BAM = ∠ DAM. Prove that AD = 2CD.

In the adjoining figure, ABCD is a parallelogram in which ∠ A = 60o. If the bisectors of ∠ A and ∠ B

meet DC at P, prove that

(i) ∠ APB = 90,

(ii) AD = DP and PB = PC = BC,

(iii) DC = 2AD

In the adjoining figure, ABCD is a parallelogram in which ∠ BAO = 35, ∠ DAO = 40 and ∠ COD =105. Calculate

(i) ∠ ABO,

(ii) ∠ ODC,

(iii) ∠ ACB

(iv) ∠ CBD.

In a parallelogram ABCD, if ∠ A = (2x + 25) o and ∠ B = (3x - 5) o, find the value of x and the measure of each angle of the parallelogram.

If an angle of a parallelogram is four fifths of its adjacent angle, find the angles of the parallelogram.

Find the measure of each angle of a parallelogram, if one of its angles is 30o less than twice the smallest angle.

ABCD is a parallelogram in which AB = 9.5 cm and its perimeter is 30 cm. Find the length of each side of the parallelogram.

The lengths of the diagonals of a rhombus are 24cm and 18 cm respectively. Find the length of each side of the rhombus.

Each side of a rhombus is 10cm long and one of its diagonals measures 16cm. Find the length of the other diagonal and hence find the area of the rhombus.

1. In each of the figures given below, ABCD is a rectangle. Find the values of x and y in each case.

(i)

In a rhombus ABCD, the altitude from D to the side B bisects AB. Find the angles of the rhombus.

In the adjoining figure, ABCD is a square. A line segment CX cuts AB at X and the diagonal BD at O such that ∠ COD = 80 and ∠ OXA = x. Find the value of x.

In a rhombus ABCD show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM = CN. Show that AC and MN bisect each other.

In the adjoining figure, ABCD is a parallelogram. If P and Q are points on AD and BC respectively such that AP = 1/3 AD and CQ = 1/3 BC, prove that AQCP is a parallelogram.

In the adjoining figure, ABCD is a parallelogram whose diagonals intersect each other at O. A line segment EOF is drawn to meet AB at E and DC at F. Prove that OE = OF.

The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60. Find the angles of the parallelogram.

ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that

(i) ABCD is a square,

(ii) Diagonal BD bisects ∠ B as well as ∠ D.

In the adjoining figure, ABCD is a parallelogram in which AB is produced to E so that BE = AB. Prove that ED bisects BC.

In the adjoining figure, ABCD is a parallelogram and E is the midpoint of side BC. If DE and AB when produced to meet at F, prove that AF = 2AB.

Two parallel lines l and m are intersected by a transversal t. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

K, L, M and N are points on the sides AB, BC, CD and DA respectively of a square ABCD such that AK = BL = CM = DN. Prove that KLMN is a square.

A △ ABC is given. If lines are drawn through A, B, C, parallel respectively to the sides BC, CA and AB, forming △ PQR, as shown in the adjoining figure, show that BC = ½ QR.

In the adjoining figure, △ ABC is a triangle and through A, B, C lines are drawn, parallel respectively to BC, CA and AB, intersecting at P, Q and R. Prove that the perimeter of △ PQR is double the perimeter of △ ABC.

P, Q, R and S are respectively the midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD. Show that

(i) PQ || AC and PQ = ½ AC

(ii) PQ || SR

(iii) PQRS is a parallelogram.

A square is inscribed in an isosceles in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

In the adjoining figure, ABCD is a parallelogram in which E and F are the midpoints of AB and CD respectively. If GH is a line segment that cuts AD, EF and BC at G, P and H respectively, prove that GP = PH.

M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. Show that MN is bisected at O.

In the adjoining figure, PQRS is a trapezium in which PQ || SR and M is the midpoint of PS. A line segment MN || PQ meets QR at N. Show that N is the midpoint of QR.

In a parallelogram PQRS, PQ = 12 cm and PS = 9 cm. The bisector of ∠ P meets SR in M. PM and QR both when produced meet at T. Find the length of RT.

In the adjoining figure, ABCD is a trapezium in which AB || DC and P, Q are the midpoints of AD and BC respectively. DQ and AB when produced to meet at E. Also, AC and PQ intersect at R.

Prove that

(i) DQ = QE,

(ii) PR || AB and

(iii) AR = RC.

In the adjoining figure, AD is a median of △ ABC and DE || BA. Show that BE is also a median of △ ABC.

In the adjoining figure, AD and BE are the medians of △ ABC and DF || BE. Show that CF = ¼ AC.

Prove that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.

In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively, of △ABC. Show that ∠ EDF = ∠ A, ∠ DEF = ∠ B and ∠ DFE = ∠ C.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rectangle is a rhombus.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a square is a square.

Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

The diagonals of a quadrilateral ABCD are equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus.

The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral formed by joining the midpoints of its sides is a rectangle.

The midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD are joined to form a quadrilateral. If AC = BD and AC ⊥ BD then prove that the quadrilateral formed is a square.