#### Chapters

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equation In Two Variables

Chapter 5: Introduction To Euclid's Geometry

Chapter 6: Lines & Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Areas of Parallelograms & Triangles

Chapter 10: Circles

Chapter 11: Construction

Chapter 12: Heron's Formula

Chapter 13: Surface Area & Volumes

Chapter 14: Statistics & Probability

## Chapter 8: Quadrilaterals

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 8 Quadrilaterals Exercise 8.1 [Pages 73 - 75]

#### Choose the correct alternative:

Three angles of a quadrilateral are 75º, 90º and 75º. The fourth angle is ______.

90º

95º

105º

120º

A diagonal of a rectangle is inclined to one side of the rectangle at 25º. The acute angle between the diagonals is ______.

55º

50º

40º

25º

ABCD is a rhombus such that ∠ACB = 40º. Then ∠ADB is ______.

40º

45º

50º

60º

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.

PQRS is a rectangle

PQRS is a parallelogram

Diagonals of PQRS are perpendicular

Diagonals of PQRS are equal

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if ______.

PQRS is a rhombus

PQRS is a parallelogram

Diagonals of PQRS are perpendicular

Diagonals of PQRS are equal

If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3:7:6:4, then ABCD is a ______.

Rhombus

Parallelogram

Trapezium

Kite

If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠C and ∠D at R and of ∠D and ∠A at S, then PQRS is a ______.

Rectangle

Rhombus

Parallelogram

Quadrilateral whose opposite angles are supplementary

If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form ______.

A square

A rhombus

A rectangle

Any other parallelogram

The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.

A rhombus

A rectangle

A square

Any parallelogram

D and E are the mid-points of the sides AB and AC of ∆ABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is ______.

A square

A rectangle

A rhombus

A parallelogram

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if, ______.

ABCD is a rhombus

Diagonals of ABCD are equal

Diagonals of ABCD are equal and perpendicular

Diagonals of ABCD are perpendicular

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠DAC = 32º and ∠AOB = 70º, then ∠DBC is equal to ______.

24º

86º

38º

32º

Which of the following is not true for a parallelogram?

Opposite sides are equal

Opposite angles are equal

Opposite angles are bisected by the diagonals

Diagonals bisect each other

D and E are the mid-points of the sides AB and AC respectively of ∆ABC. DE is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is ______.

∠DAE = ∠EFC

AE = EF

DE = EF

∠ADE = ∠ECF

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 8 Quadrilaterals Exercise 8.2 [Pages 75 - 76]

Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3 cm and OD = 2 cm, determine the lengths of AC and BD.

Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reason for your answer.

Can the angles 110º, 80º, 70º and 95º be the angles of a quadrilateral? Why or why not?

In quadrilateral ABCD, ∠A + ∠D = 180º. What special name can be given to this quadrilateral?

All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?

Diagonals of a rectangle are equal and perpendicular. Is this statement true? Give reason for your answer.

Can all the four angles of a quadrilateral be obtuse angles? Give reason for your answer.

In ∆ABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE.

In figure, it is given that BDEF and FDCE are parallelograms. Can you say that BD = CD? Why or why not?

In figure, ABCD and AEFG are two parallelograms. If ∠C = 55º, determine ∠F.

Can all the angles of a quadrilateral be acute angles? Give reason for your answer

Can all the angles of a quadrilateral be right angles? Give reason for your answer.

Diagonals of a quadrilateral ABCD bisect each other. If ∠A = 35º, determine ∠B.

Opposite angles of a quadrilateral ABCD are equal. If AB = 4 cm, determine CD.

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 8 Quadrilaterals Exercise 8.3 [Pages 78 - 79]

One angle of a quadrilateral is of 108º and the remaining three angles are equal. Find each of the three equal angles.

ABCD is a trapezium in which AB || DC and ∠A = ∠B = 45º. Find angles C and D of the trapezium.

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

ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus.

E and F are points on diagonal AC of a parallelogram ABCD such that AE = CF. Show that BFDE is a parallelogram.

E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC.

Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a ∆ABC as shown in figure. Show that BC = `1/2` QR.

D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆DEF is also an equilateral triangle.

Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (figure). Show that AC and PQ bisect each other.

In figure, P is the mid-point of side BC of a parallelogram ABCD such that ∠BAP = ∠DAP. Prove that AD = 2CD.

### NCERT solutions for Mathematics Exemplar Class 9 Chapter 8 Quadrilaterals Exercise 8.4 [Pages 82 - 83]

A square is inscribed 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 a parallelogram ABCD, AB = 10 cm and AD = 6 cm. The bisector of ∠A meets DC in E. AE and BC produced meet at F. Find the length of CF.

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.

P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ⊥ BD. Prove that PQRS is a square.

A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.

P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ∠A = ∠B and ∠C = ∠D.

In figure, AB || DE, AB = DE, AC || DF and AC = DF. Prove that BC || EF and BC = EF.

E is the mid-point of a median AD of ∆ABC and BE is produced to meet AC at F. Show that AF = `1/3`AC

Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = `1/2` (AB + CD).

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

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

ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square

D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.

## Chapter 8: Quadrilaterals

## NCERT solutions for Mathematics Exemplar Class 9 chapter 8 - Quadrilaterals

NCERT solutions for Mathematics Exemplar Class 9 chapter 8 (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 Exemplar Class 9 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 9 chapter 8 Quadrilaterals are Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles, Angle Sum Property of a Quadrilateral, Types of Quadrilaterals, Another Condition for a Quadrilateral to Be a Parallelogram, Theorem of Midpoints of Two Sides of a Triangle, Property: The Opposite Sides of a Parallelogram Are of Equal Length., Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles., Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram., Property: The Opposite Angles of a Parallelogram Are of Equal Measure., Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram., Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection), Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram.

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