RD Sharma - Mathematics

Book: RD Sharma - Mathematics

Chapter: 17. Understanding Shapes-III (Special Types of Quadrilaterales)

Subject: Maths - Class 8th

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Chapter Exercises

Exercise 17.1
Exercise 17.2
Exercise 17.3

Exercise 17.1

1

Given below is a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD = (ii) ∠DCB=


(iii) OC = (iv) ∠DAB+ ∠CDA =
2

The following figures are parallelograms. Find the degree values of the unknowns x, y, z.
3

Can the following figures be parallelograms? Justify your answer.
4

In the adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the geometrical truths you use to find them.
5

In the following figures GUNS and RUNS are parallelograms. Find x and y.
6

In the following figure RISK and CLUE are parallelograms. Find the measure of x.
7

Two opposite angles of a parallelogram are (3x-2)° and (50-x)°. Find the measure of each angle of the parallelogram.
8

If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.
9

The measure of one angle of a parallelogram is 70°. What are the measures of the remaining angles?
10

Two adjacent angles of a parallelogram are as 1 : 2. Find the measures of all the angles of the parallelogram.
11

In a parallelogram ABCD, ∠D= 135°, determine the measure of ∠A and∠B.
12

ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D.
13

The sum of two opposite angles of a parallelogram is 130°. Find all the angles of the parallelogram.
14

All the angles of a quadrilateral are equal to each other. Find the measure of each. Is the quadrilateral a parallelogram? What special type of parallelogram is it?
15

Two adjacent sides of a parallelogram are 4 cm and 3 cm respectively. Find its perimeter.
16

The perimeter of a parallelogram is 150 cm. One of its sides is greater than the other by 25 cm. Find the length of the sides of the parallelogram.
17

The shorter side of a parallelogram is 4.8 cm and the longer side is half as much again as the shorter side. Find the perimeter of the parallelogram.
18

Two adjacent angles of a parallelogram are (3x-4)° and (3x+10)°. Find the angles of the parallelogram.
19

In a parallelogram ABCD, the diagonals bisect each other at O. If ∠ABC =30°, ∠BDC= 10° and ∠CAB =70°. Find:

∠DAB, ∠ADC, ∠BCD, ∠AOD, ∠DOC, ∠BOC, ∠AOB, ∠ACD, ∠CAB, ∠ADB, ∠ACB, ∠DBC, and ∠DBA.
20

Find the angles marked with a question mark shown in Fig. 17.27
21

The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram.
22

In Fig. 17.28, ABCD and AEFG are parallelograms. If ∠C =55°, what is the measure of ∠F? Figure
23

In Fig. 17.29, BDEF and DCEF are each a parallelogram. Is it true that BD=DC? Why or why not?
24

In Fig. 17.29, suppose it is known that DE = DF. Then, is ΔABC isosceles? Why or why not?
25

Diagonals of parallelogram ABCD intersect at O as shown in Fig. 17.30. XY contains O, and x,Y are points on opposite sides of the parallelogram. Give reasons for each of the following:
26

In Fig. 17.31, ABCD is a parallelogram, CE bisects ∠C andAF bisects ∠A. In each of the following, if the statement is true, give a reason for the same.
27

Diagonals of a parallelogram ABCD intersect at O. AL and CM are drawn perpendiculars to BD such that L and M lie on BD. Is AL = CM? Why or why not?
28

Points E and F lie on diagonals AC of a parallelogram ABCD such that AE = CF. What type of quadrilateral is BFDE?
29

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

Exercise 17.2

1

Which of the following statements are true for a rhombus?

(i) It has two pairs of parallel sides.


(ii) It has two pairs of equal sides.


(iii) It has only two pairs of equal sides.


(iv) Two of its angles are at right angles.


(v) Its diagonals bisect each other at right angles.


(vi) Its diagonals are equal and perpendicular.


(vii) It has all its sides of equal lengths.


(viii) It is a parallelogram.


(ix) It is a quadrilateral.


(x) It can be a square.


(xi) It is a square.
2

Fill in the blanks, in each of the following, so as to make the statement true:

(i) A rhombus is a parallelogram in which _______.


(ii) A square is a rhombus in which _________.


(iii) A rhombus has all its sides of ______ length.


(iv) The diagonals of a rhombus _____ each other at ______ angles.


(v) If the diagonals of a parallelogram bisect each other at right angles, then it is a ______.
3

The diagonals of a parallelogram are not perpendicular. Is it a rhombus? Why or why not?
4

The diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? If your answer is ‘No’, draw a figure to justify your answer.
5

ABCD is a rhombus. If ∠ACB = 40°, find ∠ADB.
6

If the diagonals of a rhombus are 12 cm and 16 cm, find the length of each side.
7

Construct a rhombus whose diagonals are of length 10 cm and 6 cm.
8

Draw a rhombus, having each side of lengfth 3.5 cm and one of the angles as 40°.
9

One side of a rhombus is of length 4 cm and the length of an altitude is 3.2 cm. Draw the rhombus.
10

Draw a rhombus ABCD, if AB = 6 cm and AC = 5 cm.
11

ABCD is a rhombus and its diagonals intersect at O.

(i) Is ΔBOC≅ΔDOC? State the condruence condition used?


(ii) Also state, if ∠BCO =∠DCO.
12

Show that each diagonal of a rhombus bisects the angle through which it passes.
13

ABCD is a rhombus whose diagonals interesct at O. If AB=10 cm, diagonal BD = 16 cm, find the length of diagonal AC.
14

The diagonal of a quadrilateral are of lengths 6 cm and 8 cm. If the diagonals bisect each other at right angles, what is the length of each side of the quadrilateral?

Exercise 17.3

1

Which of the following statements are true for a rectangle?

(i) It has two pairs of equal sides.


(ii) It has all its sides of equal length.


(iii) Its diagonals are equal.


(iv) Its diagonals bisect each other.


(v) Its diagonals are perpendicular.


(vi) Its diagonals are perpendicular and bisect each other.


(vii) Its diagonals are equal and bisect each other.


(viii) Its diagonals are equal and perpendicular, and bisect each other.


(ix) All rectangles are squares.


(x) All rhombuses are parallelograms.


(xi) All squares are rhombuses and also rectangles.


(xii) All squares are not parallelograms.
2

Which of the following statements are true for a square?

(i) It is a rectangle.


(ii) It has all its sides of equal length.


(iii) Its diagonals bisect each other at right angle.


(v) Its diagonals are equal to its sides.
3

Fill in the blanks in each of the following, so as to make the statement true :

(i) A rectangle is a parallelogram in which ________.


(ii) A square is a rhombus in which __________.


(iii) A square is a rectangle in which ___________.
4

A window frame has one diagonal longer then the other. Is the window frame a rectangle? Why or why not?
5

In a rectangle ABCD, prove that ΔACB ≅ΔCAD.
6

The sides of a rectangle are in the ratio 2 : 3, and its perimeter is 20 cm. Draw the rectangle.
7

The sides of a rectangle are in the ratio 4 : 5. Find its sides if the perimeter is 90 cm.
8

Find the length of the diagonal of a rectangle whose sides are 12 cm and 5 cm.
9

Draw a rectangle whose one side measures 8 cm and the length of each of whose diagonals is 10 cm.
10

Draw a square whose each side measures 4.8 cm.
11

Identify all the quadrilaterals that have:
12

Explain how a square is

(i) a quadrilateral?


(ii) a parallelogram?


(iii) a rhombus?


(iv) a rectangle?
13

Name the quadrilaterals whose diagonals:

(i) bisect each other


(ii) are perpendicular bisector of each other


(iii) are equal.
14

ABC is a right angled triangle and O is the mid-point of the side opposite to the right angle. Explain why O is equidistant from A, B, and C.
15

A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?