# FRANK Solutions for Class 9 Maths Chapter 19 - Quadrilaterals

If you are given the measure of one angle of a parallelogram, can you find the measure of the remaining angles? You can by referring to our Frank Solutions for ICSE Class 9 Mathematics Chapter 19 Quadrilaterals during revision. Further, you can practise Maths questions and answers on how to prove that the given figure is a parallelogram or a rectangle.

The Frank textbook solutions at the TopperLearning portal even includes problems on providing proofs to show that the given quadrilateral is a rhombus. If you are looking for more such ICSE Class 9 Maths chapter solutions, check the Selina solutions and solved sample question papers.

## Chapter 19 - Quadrilaterals Exercise Ex. 19.1

In the following figures, find the remaining angles of the parallelogram

In the following figures, find the remaining angles of the parallelogram

In the following figures, find the remaining angles of the parallelogram

In the following figures, find the remaining angles of the parallelogram

In the following figures, find the remaining angles of the parallelogram

The consecutive angles of a parallelogram are in the ratio 3:6. Calculate the measures of all the angles of the parallelogram.

In the given figure, ABCD is a parallelogram, find the values of x and y.

In the given figure, MP is the bisector of ∠P and RN is the bisector of ∠R of parallelogram PQRS. Prove that PMRN is a parallelogram.

Construction: Join PR.

## Chapter 19 - Quadrilaterals Exercise Ex. 19.2

In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that

RN and RM trisect QS.

In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that

PMRN is a parallelogram.

In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that

MN bisects QS.

In the given figure, PQRS is a parallelogram in which PA = AB = Prove that:

SA ‖ QB and SA = QB.

Construction:

Join BS and AQ.

Join diagonal QS.

In the given figure, PQRS is a parallelogram in which PA = AB = Prove that:

SAQB is a parallelogram.

Construction:

Join BS and AQ.

Join diagonal QS.

In the given figure, PQRS is a trapezium in which PQ ‖ SR and PS = QR. Prove that:

∠PSR = ∠QRS and ∠SPQ = ∠RQP

Construction:

Draw SM ⊥ PQ and RN ⊥ PQ

In a parallelogram ABCD, E is the midpoint of AB and DE bisects angle D. Prove that:

- BC = BE.
- CE is the bisector of angle C and angle DEC is a right angle

Prove that if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.

Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right angle.

i.e. OA = OC, OB = OD

And, ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°

To prove ABCD a rhombus, we need to prove ABCD is a parallelogram and all sides of ABCD are equal.

Now, in ΔAOD and DCOD

OA = OC (Diagonal bisects each other)

∠AOD = ∠COD (Each 90°)

OD = OD (common)

∴ΔAOD ≅ΔCOD (By SAS congruence rule)

∴ AD = CD ….(i)

Similarly, we can prove that

AD = AB and CD = BC ….(ii)

From equations (i) and (ii), we can say that

AB = BC = CD = AD

Since opposite sides of quadrilateral ABCD are equal, so, we can say that ABCD is a parallelogram.

Since all sides of a parallelogram ABCD are equal, so, we can say that ABCD is a rhombus.

Prove that the diagonals of a kite intersect each other at right angles.

Consider ABCD is a kite.

Then, AB = AD and BC = DC

Prove that the diagonals of a square are equal and perpendicular to each other.

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