RD Sharma 2020 solution class 9 chapter 13 Quadrilaterals VSAQS

VSAQS

Page-13.73

Question 1:

In a parallelogram ABCD, write the sum of angles A and B.

Answer 1:

In Parallelogram ABCD, and are adjacent angles.

Thus, .

Then, we have and as consecutive interior angles which must be supplementary.

Hence, the sum of and is .

Question 2:

In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.

Answer 2:

In Parallelogram ABCD , and are Adjacent angles.

We know that in a parallelogram, adjacent angles are supplementary.

$$\begin{gathered} \mathrm{Now}, \angle \mathrm{A}+\angle \mathrm{D}=180° \\ \Rightarrow \angle \mathrm{A}+115°=180° \\ \Rightarrow \angle \mathrm{A}=180°-115° \\ \Rightarrow \angle \mathrm{A}=65° \\ \mathrm{So}, \mathrm{measure} \mathrm{of} \angle \mathrm{A} \mathrm{is} 65°. \end{gathered}$$

 

Question 3:

PQRS is a square such that PR and SQ intersect at O. State the measure of ∠POQ.

Answer 3:

PQRS is a square given as:

Since the diagonals of a square intersect at right angle.

Therefore, the measure of is .

Question 4:

If PQRS is a square, then write the measure of ∠SRP.

Answer 4:

The square PQRS is given as:

Since PQRS is a square.

Therefore,

and

Now, in , we have

That is, (Angles opposite to equal sides are equal)

By angle sum property of a triangle.

()

Hence, the measure of is.

Question 5:

If ABCD is a rhombus with ∠ABC = 56°, find the measure of ∠ACD.

Answer 5:

The figure is given as follows:

ABCD is a rhombus.

Therefore,

ABCD is a parallelogram.

Thus,

[(Given)]

[]

Now in ,we have:

Hence the measure of is .

Question 6:

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

Answer 6:

Let the shorter side of the parallelogram be cm.

The longer side is given ascm.

Perimeter of the parallelogram is given as 22 cm

Therefore,

Hence, the measure of the shorter side is cm.

Question 7:

If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.

Answer 7:

We have, .

So, let ,

,

and

By angle sum property of a quadrilateral, we get:

Smallest angle is :

Hence, the smallest angle measures.

Question 8:

In a parallelogram ABCD, if ∠A = (3x − 20)°, ∠B = (y + 15)°, ∠C = (x + 40)°, then find the values of x and y.

Answer 8:

In parallelogram ABCD, and are opposite angles.

We know that in a parallelogram, the opposite angles are equal.

Therefore,

We have and

Therefore,

Therefore,

Similarly,

Also,

Therefore,

By angle sum property of a quadrilateral, we have:

Hence the required values for x and y are and respectively.

Question 9:

If measures opposite angles of a parallelogram are (60 − x)° and (3x − 4)°, then find the measures of angles of the parallelogram.

Answer 9:

Let ABCD be a parallelogram, with and .

We know that in a parallelogram, the opposite angles are equal.

Therefore,

Thus, the given angles become

Similarly,

Also, adjacent angles in a parallelogram form the consecutive interior angles of parallel lines, which must be supplementary.

Therefore,

Similarly,

Thus, the angles of a parallelogram are ,, and .

Question 10:

In a parallelogram ABCD, the bisector of ∠A also bisects BC at X. Find AB : AD.

Answer 10:

Parallelogram ABCD is given as follows:

We have AX bisects bisecting BC at X.

That is,

We need to find

Since, AX is the bisector

That is,

…… (i)

Also, ABCD is a parallelogram

Therefore, and AB intersects them

…… (ii)

In by angle sum property of a triangle:

From (i) and (ii), we get:

…… (iii)

From (i) and (iii),we get:

Sides opposite to equal angles are equal. Therefore,

As X is the mid point of BC. Therefore,

Also, ABCD is a parallelogram, then,

Thus,

Hence the ratio of is .

Question 11:

In the given figure, PQRS is an isosceles trapezium. Find x and y.

Answer 11:

Trapezium is given as follows:

We know that PQRS is a trapezium with

Therefore,

Hence, the required value for x is .

Question 12:

In the given figure, ABCD is a trapezium. Find the values of x and y.

Answer 12:

The figure is given as follows:

We know that ABCD is a trapezium with

Therefore,

It is given that and .

Similarly,

Hence, the required values for x and y is and respectively.

Question 13:

In the given figure, ABCD and AEFG are two parallelograms. If ∠C = 58°, find ∠F.

Answer 13:

ABCD and AEFG are two parallelograms as shown below:

Since ABCD is a parallelogram, with

We know that the opposite angles of a parallelogram are equal.

Therefore,

Similarly, AEFG is a parallelogram, with

We know that the opposite angles of a parallelogram are equal.

Therefore,

Hence, the required measure for is .

Question 14:

Complete each of the following statements by means of one of those given in brackets against each:
(i) If one pair of opposite sides are equal and parallel, then the figure is ........................
(parallelogram, rectangle, trapezium)

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ................ (square, rectangle, trapezium)

(iii) A line drawn from the mid-point of one side of a triangle .............. another side intersects the third side at its mid-point. (perpendicular to parallel to, to meet)

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a .................
(rectangle, square, rhombus)

(v) Consecutive angles of a parallelogram are ...................
(supplementary, complementary)

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ...............
(rectangle, parallelogram, rhombus)

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a ....................
(parallelogram, rhombus, rectangle)

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a ..................
(kite, rhombus, square)

Answer 14:

(i) If one pair of opposite sides are equal and parallel, then the figure is parallelogram.

Reason:

In and,

(Given)

(Common)

(Because , Alternate interior angles are equal)

So, by SAS Congruence rule, we have

Also,

(Corresponding parts of congruent triangles are equal)

But, these are alternate interior angles, which are equal.

Thus, and

Hence, quadrilateral ABCD is parallelogram

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is trapezium.

(iii) A line drawn from the mid-point of one side of a triangle parallel to another side intersects the third side at its mid-point.

Reason:

This is a theorem.

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a rectangle.

Reason:

Let ABCD be the given parallelogram.

We have,

In a parallelogram, opposite angles are equal.

Therefore,

Similarly,

Also,

Thus, a parallelogram with all the angles being right angle and opposite sides being equal is a rectangle.

(v) Consecutive angles of a parallelogram are supplementary.

Reason:

Let ABCD be the given parallelogram.

Thus, .

Therefore,

Consecutive angles and are supplementary.

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which and .

We need to show that ABCD is a parallelogram.

In and , we have

(Common)

(Given)

(Given)

So, by SSS criterion of congruence, we have

By corresponding parts of congruent triangles property.

…… (i)

And

Now lines AC intersects AB and DC at A and C,such that

(From (i))

That is, alternate interior angles are equal.

Therefore, .

Similarly, .

Therefore, ABCD is a parallelogram.

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which and .

We need to show that ABCD is a parallelogram.

In quadrilateral ABCD, we have

Therefore,

…… (i)

Since sum of angles of a quadrilateral is

From equation (i), we get:

Similarly,

Now, line AB intersects AD and BC at A and B respectively

Such that

That is, sum of consecutive interior angles is supplementary.

Therefore, .

Similarly, we get .

Therefore, ABCD is a parallelogram.

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a rhombus.

We have ABCD, a parallelogram with .

Since ABCD is a parallelogram.

Thus,

And

But,

Therefore,all four sides of the parallelogram are equal, then it is a rhombus.

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Question 15:

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

Answer 15:

The quadrilateral can be drawn as follows:

We have AO and BO as the bisectors of angles and respectively.

In ,We have,

…… (1)

By angle sum property of a quadrilateral, we have:

Putting in equation (1):

……(2)

It is given that in equation (2), we get:

Hence, the sum of and is .

Question 16:

The diagonals of a rectangle ABCD meet at O, If ∠BOC = 44°, find ∠OAD.

Answer 16:

The rectangle ABCD is given as:

We have,

(Linear pair)

Since, diagonals of a rectangle are equal and they bisect each other. Therefore, in , we have

(Angles opposite to equal sides are equal.)

Therefore,

Now,in , we have

Since, each angle of a rectangle is a right angle.

Therefore,

Thus,

Hence, the measure of is.

Question 17:

If ABCD is a rectangle with ∠BAC = 32°, find the measure of ∠DBC.

Answer 17:

Figure is given as :

Suppose the diagonals AC and BD intersect at O.

Since, diagonals of a rectangle are equal and they bisect each other.

Therefore, in , we have

Angles opposite to equal sides are equal.

Therefore,

But,

Now,

Hence, the measure of is .

Question 18:

If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB, then find the value of k.

Answer 18:

The quadrilateral can be drawn as follows:

We have AO and BO as the bisectors of angles and respectively.

In ,We have,

…… (I)

By angle sum property of a quadrilateral, we have:

Putting in equation (I):

…… (II)

On comparing equation (II) with

We get .

Hence, the value for k is .

Question 19:

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.

Answer 19:

Rhombus PQRS is given.

Diagonal PR is produced to T.

Also, .

We know that in a rhombus, the diagonals bisect each other at right angle.

Therefore,

Now,

In , by angle sum property of a triangle, we get:

Or, (Because O lies on SQ)

We have, .Thus the alternate interior opposite angles must be equal.

Therefore,

In ,we have

Since opposite sides of a rhombus are equal.

Therefore,

Also,

Angles opposite to equal sides are equal.

Thus,

But

Thus,

Hence the required values for x,y and z are , and respectively.

Question 20:

In the given figure, ABCD is a rectangle in which diagonal AC is produced to E. If ∠ECD = 146°, find ∠AOB.

Answer 20:

ABCD is a rectangle

With diagonal AC produced to point E.

We have

(Linear pair)

We know that the diagonals of a parallelogram bisect each other.

Thus

Also, angles opposite to equal sides are equal.

Therefore,

By angle sum property of a traingle

Also, and are vertically opposite angles.

Therefore,

Hence, the required measure for is .