Rd Sharma Solutions for Class 8 Math Chapter 17 Understanding Shapes Iii Special Types Of Quadrilaterals are provided here with simple step-by-step explanations. These solutions for Understanding Shapes Iii Special Types Of Quadrilaterals are extremely popular among Class 8 students for Math Understanding Shapes Iii Special Types Of Quadrilaterals Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Book of Class 8 Math Chapter 17 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Solutions. All Rd Sharma Solutions for class Class 8 Math are prepared by experts and are 100% accurate.

#### Question 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.

#### Question 5:

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

#### Question 6:

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

#### Question 7:

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

#### Question 8:

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

#### Question 9:

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

#### Question 10:

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

#### Question 11:

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

#### Question 12:

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

#### Question 13:

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

#### Question 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?

#### Question 15:

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

#### Question 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.

#### Question 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.

#### Question 18:

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

#### Question 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.

#### Question 20:

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

#### Question 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.

#### Question 22:

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

#### Question 23:

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

#### Question 24:

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

#### Question 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:
(i) OB = OD
(ii) ∠OBY = ∠ODX
(iii) ∠BOY = ∠DOX
(iv) ∆BOY ≅ ∆DOX
Now, state if XY is bisected at O.

(i) Diagonals of a parallelogram bisect each other.
(ii) Alternate angles
(iii) Vertically opposite angles
(iv)

ASA congruence:
XO = YO (c.p.c.t)
So, XY is bisected at O.

#### Question 26:

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

(i) ∠A = ∠C
(ii) $\angle FAB=\frac{1}{2}\angle A$
(iii) $\angle DCE=\frac{1}{2}\angle C$
(iv) $\angle CEB=\angle FAB$
(v) CE || AF

(i) True, since opposite angles of a parallelogram are equal.
(ii) True, as AF is the bisector of $\angle$A.
(iii) True, as CE is the bisector of $\angle$C.
(iv) True
$\angle$CEB = $\angle$DCE........(i)  (alternate angles)
$\angle$DCE= $\angle$ FAB.........(ii)    (opposite angles of a parallelogram are equal)

From equations (i) and (ii):
$\angle$CEB = $\angle$FAB

(v) True, as corresponding angles are equal ($\angle$CEB = $\angle$FAB).

#### Question 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?

#### Question 28:

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

#### Question 29:

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

#### Question 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.
(x) It can be a square.
(xi) It is a square.

(i) True
(ii) True
(iii) True
(iv) False
(v) True
(vi) False

Diagonals of a rhombus are perpendicular, but not equal.

(vii) True
(viii) True

It is a parallelogram because it has two pairs of parallel sides.

(ix) True

It is a quadrilateral because it has four sides.

(x) True

It can be a square if each of the angle is a right angle.

(xi) False

It is not a square because each of the angle is a right angle in a square.

#### Question 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 ......

(i) A rhombus is a parallelogram in which adjacent sides are equal.
(ii) A square is a rhombus in which all angles are right angled.
(iii) A rhombus has all its sides of equal length.
(iv) The diagonals of a rhombus bisect each other at right angles.
(v) If the diagonals of a parallelogram bisect each other at right angles, then it is a rhombus.

#### Question 3:

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

No, it is not a rhombus. This is because diagonals of a rhombus must be perpendicular.

#### Question 4:

No, it is not so.
Diagonals of a rhombus are perpendicular and bisect each other. Along with this, all of its sides are equal. In the figure given below, the diagonals are perpendicular to each other, but do not bisect each other.

#### Question 5:

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

#### Question 6:

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

#### Question 7:

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

1. Draw AC equal to 10 cm.
2. Draw XY, the right bisector of AC, meeting it at O.
3. With O as centre and radius equal to half of the length of the other diagonal,
i.e. 3 cm, cut OB = OD = 3 cm.
4. Join AB, AD and CB, CD.

#### Question 8:

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

1. Draw a line segment AB of 3.5 cm.
2. Draw $\angle$BAX equal to 40$°$.
3. With A as centre and the radius equal to AB, cut AD at 3.5 cm.
4. With D as centre, cut an arc of radius 3.5 cm.
5. With B as centre, cut an arc of radius 3.5 cm. This arc cuts the arc of step 4 at C.
6. Join DC and BC.

#### Question 9:

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

1. Draw a line segment AB of 4 cm.
2. Draw a perpendicular XY on AB, which intersects AB at P.
3. With P as centre, cut PE at 3.2 cm.
4. Draw a line WZ that passes through E. This line should be parallel to AB.
5. With A as centre, draw an arc of radius 4 cm that cuts WZ at D.
6. With D as centre and radius 4 cm, cut line DZ. Label it as point C.

#### Question 10:

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

1. Draw a line segment AC of 5 cm.
2. With A as centre, draw an arc of radius 6 cm on each side of AC.
3. With C as centre, draw an arc of radius 6 cm on each side of AC. These arcs intersect the arcs of step 2 at B and D.
4. Join AB, AD, CD and CB.

#### Question 11:

ABCD is a rhombus and its diagonals intersect at O.
(i) Is ∆BOC ≅ ∆DOC? State the congruence condition used?
(ii) Also state, if ∠BCO = ∠DCO.

(i) Yes

(ii) Yes
By c.p.c.t:
$\angle BCO=\angle DCO$

#### Question 12:

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

#### Question 13:

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

#### Question 14:

The diagonals 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?

#### Question 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.

(i) True
(ii) False
(iii) True
(iv) True
(v) False
(vi) False
The diagonals are not perpendicular to each other.
(vii) True
(viii) False
The diagonals are not perpendicular to each other.
(ix) False
All sides are not equal.
(x) True
(xi) True
(xii) False
All squares are parallelogram.

#### Question 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.
(iv) Its diagonals are equal to its sides.

(i) True
(ii) True
(iii) True
(iv) False
This is because the hypotenuse in any right angle triangle is always greater than its two sides.

#### Question 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 .....

(i) A rectangle is a parallelogram in which each angle is a right angle.

(ii) A square is a rhombus in which each angle is a right angle.

(iii) A square is a rectangle in which the adjacent sides are equal.

#### Question 4:

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

No, since diagonals of a rectangle are equal.

#### Question 5:

In a rectangle ABCD, prove that ∆ACB ≅ ∆CAD.

AB = CD (rectangle property)

AC ( common side )

Hence, by SSS criterion, it is proved that $∆ACB\cong ∆CAD$.

#### Question 6:

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

#### Question 7:

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

#### Question 8:

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

#### Question 9:

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

(i) Draw a side AB, equal to 8 cm.
(ii) With A as the centre, draw an arc of length 10 cm.
(iii) Draw $\angle$ABX = 90$°$, which intersects the arc at C.
(iv)Draw $\angle$BAY = 90$°$.
(v) With C as the centre, draw an arc of length 8 cm.
(vi) Join CD.
Thus, ABCD is the required rectangle.

#### Question 10:

Draw a square whose each side measures 4.8 cm.

#### Question 11:

Identify all the quadrilaterals that have:
(i) Four sides of equal length
(ii) Four right angles

(i) If all four sides are equal, then it can be either a square or a rhombus.

(ii) All four right angles, make it either a rectangle or a square.

#### Question 12:

Explain how a square is
(ii) a parallelogram?
(iii) a rhombus?
(iv) a rectangle?

(i) Since a square has four sides, it is a quadrilateral.

(ii) Since the opposite sides are parallel and equal, it is a parallelogram.

(iii) Since the diagonals bisect each other and all the sides are equal, it is a rhombus.

(iv) Since the opposite sides are equal and all the angles are right angles, it is a rectangle.

#### Question 13:

(i) bisect each other
(ii) are perpendicular bisector of each other
(iii) are equal.

(i) Rhombus, parallelogram, rectangle and square
(ii) Rhombus and square
(iii) Rectangle and square

#### Question 14:

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

#### Question 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?

(i) By measuring each angle - Each angle of a rectangle is 90$°$.

(ii) By measuring the length of the diagonals - Diagonals of a rectangle are equal.

(iii) By measuring the sides of rectangle - Each pair of opposite sides are equal.

#### Question 1:

Given below is a parallelogram ABCD. Complete each statement along with the definition or property used.
(ii) ∠DCB =
(iii) OC =
(iv) ∠DAB + ∠CDA =

The correct figure is

#### Question 2:

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