NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Introduction

A paper is a very common example of a plane surface. The curve obtained by joining several points consecutively without lifting the pencil from the paper is called a plane curve or simple curve or simply curve.

Some examples of simple curves are as follows

A circle is a very common example of a plane curve.

Some types of plane curves are as follows :

1. Open curve: A curve that does not cut itself is called an open curve.

Read and Learn More NCERT Solutions For Class 8 Maths

For example:

   is an open curve.

2. Closed curve:  A curve that cuts itself is called a closed curve.

For example:

 is a closed curve.

3. Simple closed curve: A closed curve that does not pass through a point more than once is called a simple closed
curve.

For example: 

is a simple closed curve.

Note: A curve can be straight also, i.e., A straight line is also a curve. It is an open

Question 1. Match the following: (Caution A figure may match more than one type)

Solution:

Compare your matchings with those of your friends. Do they agree?

Solution:

1-C

2-B

3-A

4-D

Please compare yourself your matchings with those of friends and find out whether they agree.

Polygons

A polygon is a simple closed-curve formed of only line segments. A triangle is a very common example of a polygon

Question 1. Try to give a few more examples and non-examples for a polygon.

Solution:

The following curves are examples of a polygon:

The following curves are non-examples of a polygon:

Question 2. Draw a rough figure of a polygon and identify its sides and vertices.

Solution:

Sides : AB, BC, CD, DE, EF, FG, GH, HI, IJ, JK, KL and LA.

Vertices: A, B, C, D, E, F, G, H, I, J, K and L.

Convex And Concave Polygons

A polygon is said to be convex if it has no portion of its diagonals in its exterior otherwise it is said to be a concave polygon.

Question 1. Can you find how these types of polygons differ from one another

Solution:

Yes, some of these are convex polygons while the others are concave polygons.

These polygons differ in the sense that in a convex polygon, the line segment joining any two points inside it lies completely inside the polygon whereas this property does not hold well in a concave polygon.

Question 2. Polygons that are convex have no portions of their diagonals in their exteriors. Is it true with concave polygons?

Solution:  No, it is not true with concave poly polygons

Question 3. Study the figures given below

Then try to describe in your own words what we mean by a convex polygon and what we mean by a concave polygon. Give two rough sketches of each kind

Solution:

By a convex polygon, we mean a polygon that has no portion of its diagonals in its exterior. On the contrary, by a concave polygon, we mean a polygon that has some portion of its diagonals in its exterior

1. Two rough sketches of a convex polygon

2. Two rough sketches of a concave polygon

Regular And Irregular Polygons

A polygon that is both ‘equiangular’ (has all angles of equal measure) and ‘equilateral’ (has all sides of equal measure) is called a regular polygon,

For example: A square, an equilateral triangle.

A polygon which is equiangular but not equilateral is called an irregular polygon. For example: a rectangle

Question 1. Is a rectangle a regular polygon? my?

Solution: No, A rectangle is not a regular polygon because it is equiangular but not equilateral

Question 2. Is an equilateral triangle a regular polygon? Why?

Solution: Yes, An equilateral triangle is a regular polygon because it is both equiangular and equilateral.

Question 3. Is there a triangle that is equilateral but not equiangular?

Solution:

No, there is no triangle which is equilateral but not equiangular

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.1

Question 1.  Given Here are some figures

Classify each of them based on the following;

1. Simple curve
2. Simple closed curve
3. Polygon
4. Convex polygon
5. Concave polygon

Solution:

1. Simple curve-1,2,5,6,7
2. Simple closed curve-1,2,5,6,7
3. Polygon – 1,2
4. Convex polygon – 2
5. Concave polygon – 1

Question 2. What is a regular polygon? State the name of a regular polygon of

1. 3 sides
2. 4 sides
3. 6 sides.

Solution:

A polygon, both ‘equilateral’ and ‘equiangular’, is called a regular polygon.

1. 3 sides: The name of the regular polygon of 3 sides is an equilateral triangle.
2. 4 sides: The name of the regular polygon of 4 sides is square.
3. 6 sides: The name of the regular polygon of 6 sides is a regular hexagon.

Sum Of The Measures Of The Exterior Angles Of A Polygon

The sum of the measures of the exterior angles of a polygon is 360°

Draw a polygon on the floor, using a piece of chalk. (In the figure, a pentagon ABCDE is shown)

We want to know the total measure of angles, i.e.

m∠1 + m∠2 + m∠3 + m∠4 + m∠5

Start at A. Walk along \(\overline{\mathrm{AB}} \text {. }\). On reaching B, we need to turn through an angle of m∠1, to walk along \(\overline{\mathrm{BC}} \text {. }\)

When we reach C, we need to turn E D through an angle of m∠2 to walk along CD.

We continue to move in this manner until we return to side AB. We would have made one complete turn.

∴  m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360°

This is true whatever the number of E sides ofthe polygon.

Therefore, the sum of the measures of y the external angles of any polygon is 360°

Take a regular hexagon

Question 1. What is the sum of the measures of its exterior angles r, y, z, p, q, r?

Solution:

The sum of the measures of its exterior angles x, y, z, p , q, r is 360°.

Question 2. Is x = y = z = p = q = r? Why?

Solution:

Yes, x = y = z = p = q = r because

x + a = y + a = z + a = p + a = q + a = r + a

= 180°

Linear pair property

Question 3. What is the measure of each?

1. Exterior angle
2. Interior angle

Solution:

1. The measure of each exterior angle is \(\frac{360^{\circ}}{6}=60^{\circ}.\) = 60°

The sum of the measures of the exterior angles of a polygon is 360°. Also, all interior angles are of equal Consequently, all exterior angles are of equal measure (by Unear pair property).

2. The measure of each interior angle is 180°-60° = 120°

Note:

1. Each exterior angle of an n–sided regular polygon

⇒ \(\frac{360^{\circ}}{6}=60^{\circ} .\)

2. Number of sides of a regular polygon

= \(\frac{360^{\circ}}{n} .\)

3. Each interior angle of an n-sided regular polygon

= \(\frac{(n-2) 180^{\circ}}{n} \text { Or } \frac{(2 n-4) 90^{\circ}}{n}\)

=  \(\frac{(2 n-4)}{n}\)right angles.

Question 4. Repeal, this activity in the cases of

1. A regular octagon
2. A regular 20-gon.

Solution:

1. A regular octagon

1. The measure of each exterior angle = \(\frac{360^{\circ}}{8}=45^{\circ} .\)
2. The measure of each interior angle is 180° -45° =135°

2. A regular 20-gon

1. The measure of each exterior angle \(\frac{360^{\circ}}{20}=18^{\circ} .\)
2. The measure of each interior angle is 180° -18° =162°

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.2

Question 1. Find x in the following figures

Solution:

1.  x°+ 125° + 125° = 360°.

∵ The sum of the measures of the exterior angles of a polygon is 360°

x + 250° = 360°

x = 360° -250°

x  = 110°

2. x + 70° + 90° + 60° + (180°- 90°)

= 360°

x + 160° + 60° + 90° = 360°

x + 220° + 90° = 360°.

The sum of the measures of the exterior angles of a polygon is 360° and the linear pair property.

x+ 310° = 360°

x = 360° – 310°

x = 50°.

Question 2. Find the measure of each exterior angle of a regular polygon of

1. 9 sides
2. 15sides

Solution:

1. 9 sides:

Here, n = 9

Measure of each exterior angle

= \(\frac{360^{\circ}}{n}=\frac{360^{\circ}}{9}\)

= \(40^{\circ}\)

2. 15 sides:

No, since 22 is not a factor of

Hero, n = 15

Measure of each exterior angle

= \(\frac{360^{\circ}}{n}=\frac{360^{\circ}}{15}\)

= \(24^{\circ}\)

Question 3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:

Let the number of sides be n. Then

n = \(\frac{360^{\circ}}{\text { Each exterior angle }} \text {. }\)

n = \(\frac{360^{\circ}}{24^{\circ}}\)

= 15

Hence, the number of sides is 15.

Question 4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solution: 

Each interior angle = 165°

∴  Each exterior angle

= 180° – 165° = 15°

∴ Linear pair property

Let the number of sides be n. Then

n = \(\frac{360^{\circ}}{\text { Each exterior angle }}\)

n = \(\frac{360^{\circ}}{15^{\circ}}\)

= 24

Hence, the number of sides is 24.

Question 5.

1. Is it possible to have a regular polygon with a measure of each exterior angle as 22°?
2. Can it be an interior angle of a regular polygon? Why?

Solution:

1. No, since 22 is not a factor of m 360°

2. No, because by linear pair property, each exterior angle is 180° -22°= 158°

Which 360°n – 360°15 = 24°. is not. a factor of 360

Question 6.

1. What is the minimum interior angle possible for a regular polygon? Why ?
2. What is the maximum exterior angle possible for a regular polygon

Solution:

1. Minimum number of sides of a regular polygon = 3.

• A regular polygon of 3 sides is an equiangular triangle.
• Each interior angle of an equilateral triangle

n = \(\left(\frac{n-2}{n}\right) 180^{\circ}\)

= \(\left(\frac{3-2}{3}\right) 180^{\circ}\)

= 60°

Hence, the minimum interior angle possible for a regular polygon is 60°.

2. We know that the measure of an interior angle + the measure of a corresponding exterior angle

= 180°

By linear pair property

∵ Minimum interior angle possible for a regular polygon = 60°

∴ Maximum exterior angle possible for a regular polygon = 180°- 60° = 120°

Kinds Of Quadrilaterals

The important types of quadrilaterals are as follows :

• Trapezium
• Kite
• Parallelogram
• Rhombus
• Rectangle
• Square

Trapezium

A quadrilateral, which has only one pair of parallel sides, is called a trapezium.

Question 1. Study the above, [inures and discuss with your friends why some of them are trapeziums while some are not,(Note: The arrow marks indicate parallel lines).

Solution:

Some of them are trapeziums because they have only one pair of parallel sides. Some of them are not trapeziums because they have no pair of parallel sides.

Note: One of these figures is not a trapezium although it has only one pair of parallel  sides because a trapezium is essentially a quadrilateral whereas this figure is a pentagon

Question 2. Take identical cut-outs of congruent triangles of sides 3 cm, 4 cm, and 5 cm. Arrange them as shown

You get a trapezium. (Check it!) Which are the parallel sides here? Should the non-parallel sides be equal? You can get two more trapeziums using the same set of triangles. Find them out and discuss their shapes

Solution:

DE = CB (= 3 cm)

DC = EB (= 5 cm)

∴ Quadrilateral DCBE parallelogram

∴ DC || EB => DC|| AB

∴ ABCD is a trapezium. Its parallel sides are AB and DC.

Its non-parallel sides DA and CB are D 5 cm c not equal.

Thus, the non-parallel sides need not to be equal.

We get two more trapeziums using the same set of triangles. These are shown below

Question 3. Take four sol squares from your and your friend’s instrument boxes. Use different numbers of them to place side-by-side and obtain different trapeziums.

If the non-parallel sides of a trapezium arc are of equal length, we call it an isosceles trapezium. Did you get an isosceles trapezium in any of your investigations?

Solution:

Taking the different numbers of set-squares and pinning (hem side-by-side, we obtain the following trapeziums

No, we did not get an isosceles trapezium in any of our investigations.

Kite

A quadrilateral, which has exactly two pairs of equal consecutive sides, is called a kite.

Question 4. Take a thick white sheet. Fold the paper once. Draw two line segments of different lengths as shown in the figure

Cut along the line segments and open up. You have the shape of a kite. Has the kite any line symmetry?

Fold both the diagonals of the kite. Use the set square to check if they cut at right angles. Are the diagonals equal in length?

Verify by (paper-folding or measurement) if the diagonals bisect each other, i j By folding an angle of the kite on its opposite, check for angles of equal measure.

Observe the diagonal folds; do they indicate any diagonal being an angle bisector? Share your findings with others and list them.

Solution:

The kite has only one line of symmetry shown by a dotted line segment (a diagonal) AC and ∠B = ∠D

The diagonal AC is an angle bisector as it bisects ∠A and ∠C both.

Yes, the diagonals are cut at right angles.

No, the diagonals are not equal in length.

Yes, one of the diagonals bisects the other.

Question 5. Show that ΔABC mid ΔADC congruent, What do we infer from this?

Solution:

In ΔABC and  ADC

AB = AD

One pair of consecutive sides

BC = DC

Another pair of consecutive sides common

AC = AC

Δ ABC = Δ ADC

SSS Congruence Axiom

∴ ∠AC = ∠DAC

∠BCA = ∠DCA

i-e., diagonal AC bisects ∠BAD and ∠BCD both.

Parallelogram

A quadrilateral whose opposite sides are parallel is called a parallelogram

Question 1.  Study these figures and try to describe in your own words what we mean by a parallelogram. Share your observations with your friends

Solution:

By a parallelogram, we mean a quadrilateral whose opposite sides are parallel.

We observe that in a parallelogram,

Opposite sides are parallel and equal

Opposite angles are equal

Adjacent angles are supplementary

Diagonals bisect each other but the not equal

Elements Of A Parallelogram

The elements of a parallelogram are as follows :

1. Two pairs of opposite sides
2. Four pairs of adjacent sides
3. Two pairs of opposite equal angles
4. Four pairs of adjacent angles.

Question 2.  Are \(\overline{B C} \text { and } \overline{C D}\) adjacent sides too? Try to find two more pairs of adjacent sides.

Solution:

Yes, \(\overline{B C} \text { and } \overline{C D}\) are also adjacent sides. Two more pairs of adjacent sides are CD and DA; DA and AB.

Question 3. Identify other pairs of adjacent angles of the parallelogram

Solution:

The other pairs of adjacent angles of the parallelogram are∠C, ∠D, and ∠D, ∠A.

Question 4.

1.  Place \(\overline{A^{\prime} B^{\prime}}\) over \( \overline{D C}\). Do they coincide?

Solution:

Yes, they coincide.

2. What can you say about the lengths \( \overline{A B}\) and \( \overline{D C}\)

Solution:  These lengths are equal

3. Examine the lengths \( \overline{A D}\) and \( \overline{B C}\) What do you find?

Solution: These lengths are equal

Question 5.  Take two identical set squares with angles 30°- 60°- 90°andplace them adjacently to form a parallelogram as shown in the figure. Does this help you to verify the above property?

Solution: 

Yes, this helps us to verify the above property. We can further strengthen this idea through a logical argument.

Consider a parallelogram ABCD. Draw any one diagonal, say \(\overline{\mathrm{AC}}\). Looking at the angles,

∠1 =∠2 and ∠3 = ∠4

Alternate interior angles

Since in triangles ABC and ADC,

∠1 = ∠2, ∠3 =∠4 (alternate interior angles), and \(\overline{\mathrm{AC}}\) is common;

So, by ASA congruency condition,

Δ ABC≅ Δ CDA

This gives AB = DC and BC = AD

Question 6. Does this tell you anything about the measures of angles A and C ? Examine the same for angles B and D. State your findings.

Solution:

Findings: Angles A and C are equal.

Similarly, angles B and D are equal.

Also, pairs of consecutive angles (adjacent angles) ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A are supplementary.

Angles Of A Parallelogram

Note point

Property :

The opposite angles of a parallelogram are of equal measure

Question 1.  Take two identical 30°- 60°- 90° set squares and form a parallelogram as before. Does the figure obtained help us to confirm the above property

Solution:

Yes, the figure obtained helps us to confirm the above property. We can further justify this idea through logical arguments.

If \( \overline{A C}\) and \( \overline{B D}\) are the diagonals of the parallelogram, we find that

∠ 1 = ∠2 and ∠3 = ∠4

Alternate interior angles

Studying Δ ABC and Δ ADC separately, helps us to see that, by ASA-congruency condition,

Δ ABC ≅  Δ CDA

1 ∠1 = ∠2, ∠3 = ∠4, \( \overline{A C}\) and \( \overline{B D}\). This shows that ∠B and ∠D have the same measure. In the same way, we can get m ∠A = m∠C.

Question 2. Identify two more pairs of supplementary angles from the figure

Solution:

Two more pairs of supplementary angles from the figure are ∠B, ∠C, and ∠C, ∠D

Note Point

Property: The adjacent angles in a parallelogram are supplementary.

Question 3.  After showing m ∠R = m ∠N = 70°, can you find m ∠I and m∠G by any other method?

Solution:

RG || IN and RI is a transversal

∴ ∠R + ∠ I = 180°

Consecutive interior angles

70° + ∠ I = 180°

∠l  = 180° – 70° = 110°

Again,  RI || GN and RG is a transversal

∴∠R + ∠ G = 180°

Consecutive interior angles

70° +  ∠G = 180°

∠G = 180° -70° = 110°

∠I = ∠G = 110°.

Question 4. Is the mid-point, the same as O?

Solution: Yes

Question 5. Does this show that diagonal \(\overline{D B}\) bisects the diagonal \(\overline{A C}\)at point O ? Discuss it with your friends

Solution: Yes

Question 6. Repeat this activity to find where the I mid-point of \(\overline{D B}\) could lie.

Solution:

The midpoint of \(\overline{D B}\) will also lie at

Diagonals Of A Parallelogram

Note Point

Property:

The diagonals of a parallelogram bisect each other at the point. of their intersection.

Proof: Consider the parallelogram given below:

In Δ AOB and Δ COD

∠1 = ∠2

∠3 = ∠ 4

Alternate interior ∠ s

AB = DC

Opp. sides of parallelogram

∴  AOB ≅ COD

ASA congruence criterion

This gives AO = CO and BO = DO

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.3

Question 1. Given a parallelogram ABCD. x + 100° = 180° Complete each statement along with the definition or property used

AD  = _______________

∠DCB =_______________

OC =_______________

m∠DAB + m∠CDA = _______________

Solution:

1. AD = BC

∵ Opposite sides of a supplementary parallelogram are equal

2. ∠DCB =∠DAB

∵ Opposite angles of a parallelogram are equal

3. OC = OA

∵ Diagonals of a parallelogram bisect each other

m∠DAB + m∠CDA = 180°

∵ Adjacent angles of a parallelogram are supplementary.

Question 2. Consider the following parallelo¬ grams. Find the values ofthe unknowns x, y, z.

Solution:

1. y = 100° 

∵ Opposite angles of a parallelogram are equal

x + 100° = 180°

Adjacent angles in a parallelogram are supplementary

x = 180° – 100°

x = 80°

z = x = 80°

∵ Opposite angles of a parallelogram are of equal measure

Thus, x = 80°, y = 100° and z = 80°

2. x + 50° = 180°

Adjacent angles in a parallelogram are supplementary

x= 180°- 50° = 130°

y = x = 130°

∵ Opposite angles of a parallelogram are of equal measure

180° – z = 50°

∵ Linear pair property and opposite angles of a parallelogram are of equal measure

z = 180° – 50° = 130°

Thus, x = 130°, y = 130° and z = 130°

3. x = 90°

x + y 4- 30° = 180°

Vertically opposite angles are equal

X + y + 30° = 180°

By angle sum property of a triangle

90° + y + 30° = 180°

120° + y = 180°

y = 180°- 120° = 60°

∵ AD|| BC and AC is a transversal

∴ y = z

Alternate interior angles

But y = 60°

z = 60°

Thus, .x = 90°, y = 60° and z = 60°

4. y = 80°

∵ Opposite angles of a parallelogram are of equal measure

x + 80° = 180°

Adjacent angles in a parallelogram are supplementary

x = 180° – 80°

x = 100°

180° – z + 80° = 180°

∵ Linear pair property and adjacent angles in a parallelogram are supplementary. Solution:

z = 80°

Thus, x = 100°, y = 80° and z = 80°

5. y = 112°

∵ Opposite angles of a parallelogram are equal

x+ y + 40° = 180°

∵ By angle sum property of a triangle

⇒  x + 112° + 40° = 180°

⇒  x + 152° = 180°

x = 180° – 152°

x = 28°

Z = X

Alternate interior angles

But x = 28°

∴ z = 28°

Thus, x= 28°, y = 112° and z = 28°

Question 3. Can a quadrilateral ABCD be a parallelogram if

1. ∠D + ∠B = 180°?
2. AB = DC = 8 cm, AD = 4 cm, and BC = 4.4 cm?
3. ∠A = 70° and ZC = 65° ?

Solution:

1. Can be, but needs not be.

2. No, in a parallelogram, opposite sides are equal; but here, AD ≠ BC.

3. No, in a parallelogram, opposite angles are of equal measure; but here ∠A ≠∠ C.

Question 4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure
Solution:

ABCD is a quadrilateral that is not a x parallelogram. Here,  ∠A = ∠ C. It is a kite. °

Question 5. The measures of two adjacent angles of a parallelogram are in the ratio 3: 2. Find the measure of each of the angles of the parallelogram.

Solution:

Let the two adjacent angles be 3x° and 2x

Then,

3x° + 2x° = 180°

∵ The sum of the two adjacent angles of a parallelogram is 180°

5x° = 180°

x° = \(\frac{180^{\circ}}{5}\)

x° = 36°

3x° = 3 × 36° = 108°

2x° = 2 × 36° = 72°

Since the opposite angles of a parallelogram are of equal measure, therefore the measures of the angles of the parallelogram are 72°, 108°; 72°, and 108°.

Question 6. Two adjacent angles of a parallelogram have equal measure. Find the immure of each of the angles of the parallelogram.

Solution:

Let the two adjacent, angles of a parallelogram be .v° each.

Then, x° + x° = 180°

The sum of the two adjacent angles of a parallelogram is 180°

2x° = 180°

x° =  \(\frac{180^{\circ}}{2}\)

x° = 90°

Since the opposite angles of a parallelogram are of equal measure, therefore, the measure of each of the angles of the parallelogram is 90°, i.e., each angle of the parallelogram is a right angle

Question 7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y, and z. State the properties you use to find them

Solution:

x = ∠HOP = 180° – 70° = 110°

The opposite angles of a parallelogram are of equal measure

∵ HOPE is a ||gm

∴ HE || OP

And HP is a transversal

∴ y = 40°

An alternate interior angles

40° + z + x = 180°

The adjacent angles in a parallelo¬ gram are supplementary

40° + z + 110° = 180°

z + 150° = 180°

z = 180° – 150°

z = 30°.

Thus, x =110°, y = 40°  and z = 30°

Question 8. The following figures GUNS mid RUNS are parallelograms. Find x and y. (Lengths are in cm)

Solution:

1. For GUNS:

Since the opposite sides of a parallelogram are of equal length, therefore, 3x = 18

x = \(\frac{18}{3}\) = 6

And, – 1 = 26

3y = 26 + 1

3y = 27

x = \(\frac{27}{9}\)

Hence, x = 6; y = 9.

2. For RUNS:

Since the diagonals of a parallelogram bisect each other,

∴ x + y = 16______________ (1)

y + 7 = 20 ______________ (2)

From (2), y = 20 – 7 = 13

Putting y = 13 in (1), we get

x + 13 = 16 ⇒  x = 16 – 13

= 3.

Hence, x = 3; y = 13

Question 9. In the below figure, both RISK and CLUE are parallelograms. Find the value of x

Solution:  ∠RISK is a parallelogram

∴ ∠RIS = ∠RKS = 120°

The opposite angles of a parallelogram are of equal measure

Also, ∠RIS + ∠ISK = 180°.

The adjacent angles in a parallelogram are supplementary

120° + ∠ISK = 180°

∠ISK = 180° – 120°

∠ISK = 60°_____________ (1)

CLUE is a parallelogram

∠CES = ∠CLU = 70°_____________ (2)

The opposite angles of a parallelogram are of equal measure

In triangle EST,

x° + ∠TSE + ∠TES = 180°

By angle sum property of a triangle

⇒ x° + ∠ISK + ∠CES = 180°

⇒ x° + 60° + 70° = 180°

From (1) and (2)

x° + 130° = 180°

x° = 180° – 130°

x° = 50°.

Question 10. Explain how this figure is a trapezium. Which of its two sides is parallel

Solution:

∠KLM + ∠NML = 80°+ 100°

= 180°

∴ KL || NM

∵ The sum of consecutive interior angles is 180°

∴ KLMN is a trapezium.

Its two sides \(\overline{\mathrm{KL}} \text { and } \overline{\mathrm{NM}}\) are parallel.

Question 11. Find m∠C in the figure, if \(\overline{A B} \| \overline{D C}\)

Solution:

∴ \(\overline{\mathrm{AB}} \| \overline{\mathrm{DC}}\)

m∠C + m∠B = 180°

The sum of consecutive interior angles is 180

m∠C + 120° = 180°

m∠C = 180° – 120° = 60

Question 12. Find the measure of ∠P and ∠S, if \(\overline{S P} \| \overline{R Q}\) in the figure. (If you find m∠R, is there more than one method to find m∠P

Solution: 

⇒ \(\overline{\mathrm{SP}} \| \overline{\mathrm{RQ}}\) and \(\overline{\mathrm{SR}}\)isatransversal

∴m ∠P + m ∠Q = 180°

∴ The sum of consecutive interior angles is 180°

130° = 180°

m ∠P = 180° – 130°

m∠P = 50°

Again,

∵ \(\overline{\mathrm{SP}} \| \overline{\mathrm{RQ}}\) and \(\overline{\mathrm{SR}}\) and SR is a transversal.

m∠R + m ∠S = 180°

The sum of consecutive interior angles is 180°

90° + m∠S = 180°

m∠S = = 180° – 90°

= 90°

Yes; there is one more method of finding m ∠P if m ∠R is given and that is by using the angle sum property of a quadrilateral.

We have,

m∠P+m∠Q+m∠R+m∠S = 360°

m∠P+ 130° + 90° + 90° = 360°

∠S = 90° as above

m∠P + 310° = 360°

m∠P = 360°-310°

= 50°

Some Special Parallelograms

Rhombus

A quadrilateral whose all four sides are of equal length is called a rhombus.

Note Point

Property: The diagonals of a rhombus are perpendicular bisectors of one other.

A Rectangle

A rectangle is a parallelogram with equal angles.

Question 1. What Is The Full Meaning Of This Definition? Discuss With Your Friends.
Solution:

A rectangle is a parallelogram in which every angle is a right angle.

Note point:

Property: The diagonals of a rectangle are of equal length.

A Square

Property: A square is a rectangle whose all four sides have equal length.

The diagonals of a square are perpendicular bisectors of each other

Question 2. Take a square sheet, say PQRS. Fold along both the diagonals. Are their mid-points the same?

Check if the angle at 0 is 90° by using a set square. see that This verifies the property stated above.

Solution:

Yes, their mid-points are the same.

Yes, the angle at 0 is 90°.

We can justify this also by arguing logically

ABCD is a square whose diagonals meet at 0.

(Since the square is a parallelogram)

By the SSS congruency condition, we now see that

Δ AOD =Δ  COD

OA = OC, OD = OD, AD = CD

m ∠AOD = m∠COD

These angles being a linear pair, each is a right angle

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.4

Question 1. State whether True or False :

1. All rectangles are squares
2. All rhombuses are parallelograms
3. All squares are rhombuses and also rectangles
4. All squares are not parallelograms
5. All kites are rhombuses
6. All rhombuses are kites
7. All parallelograms are trapeziums
8. All squares are trapeziums.

Solution:

(2), (3), (6), (7), (8) are true

(1), (4), (5) are false

Question 2. Identify all the quadrilaterals that have.

1. Four sides of equal length
2. Four right angles

Solution:

The quadrilaterals that have four sides of equal length are square and rhombus.

The quadrilaterals that have right angles are square and rectangle

Question 3. Explain how a square is

1. A quadrilateral
2. A parallelogram
3. A rhombus
4. A rectangle.

Solution:

1.  A quadrilateral: A square is 4 sides, so it is a quadrilateral.
2. A parallelogram: A square has its opposite sides parallel; so it is a parallelogram.
3. A rhombus: A square is a parallelogram with all 4 sides equal, so it is a rhombus.
4. A rectangle: A square is a parallelogram with each angle a right angle, so it is a rectangle

Question 4. Name the quadrilaterals whose diagonals:

1. Bisect each other
2. Are perpendicular bisectors of each other
3. Are equal.

Solution:

1. Bisect each other: The names of the quadrilaterals whose diagonals bisect each other are parallelogram; rhombus, square, and rectangle.
2. Are perpendicular bisectors of each other: The names of the quadrilaterals whose diagonals are perpendicular bisectors of each other are rhombus and square.
3. Are equal: The names of the quadrilaterals whose diagonals are equal are square and rectangle

Question 5. Explain why a rectangle is a convex quadrilateral.
Solution:

A rectangle is a convex quadrilateral because both of its diagonals he wholly in its interior.

Question 6. ABC is a right-angled triangle and AC = BD 0 is the mid-point of the side opposite to the right angle. Explain why is equidistant from A, B, and C. (The dotted lines are drawn addi¬ tionally to help you).

Solution:

Construction: Produce 130 to D such that BO = OD. Join AD and CD.

Proof: AO = OC

∵ O is the mid-point of AC by construction

BO=OD

∴ Diagonals of quadrilateral ABCD bisect each other.

∴  Quadrilateral ABCD isparallelogram.

Now, ∠ ABC = 90°

ABCD is a rectangle.

Since the diagonals of a rectangle bisect each other, therefore,

O is the mid-point of AC and BD both.

But AC = BD

The diagonals of a rectangle are equal

∴  \(\mathrm{OA}=\mathrm{OC}=\frac{1}{2} \mathrm{AC}=\frac{1}{2} \mathrm{BD}=\mathrm{OB}\)

OA = OB = OC

Question 7. 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

Solution:

Different ways:

1. If each angle at the corner is 90° in
2. If the diagonals are equal in length.

Question 8. A square was defined as a rectangle with all sides equal. Can we define it as a rhombus with equal angles? Explore this idea

Solution:

Yes, a square can be defined as a rhombus with equal angles. If a rhombus has equal angles

Then the measure o each angle of the rhombus will be \(\frac{360^{\circ}}{4^{\circ}}\) = 90°

Thus, then it becomes a quadrilateral with all sides equal and each angle of measure 90° and hence a square.

Question 9. Can a trapezium have all angles equal? Can it have all sides equal? Explain.

Solution:

If a trapezium has all angles equal, then either it becomes a rectangle or a square.

If a trapezium has all sides equal, then either it becomes a rhombus or a square

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Multiple Choice Questions

Question 1. What, is the number of sides of a triangle

1. 1
2. 2
3. 3
4. 4

Solution: 3. 3

Question 2. What is the number of vertices of a triangle?

1. 1
2. 2
3. 3
4. 4

Solution: 3. 3

Question 3. What is the number of sides of a quadrilateral?

1. 1
2. 2
3. 3
4. 4

Solution: 4. 4

Question 4. What is the number of vertices of a quadrilateral?

1. 1
2. 2
3. 3
4. 4

Solution: 4. 4

Question 5. How many diagonals does a quadrilateral have?

1. 1
2. 2
3. 3
4. 4

Solution: 2. 2

Question 6. How many diagonals does a triangle have?

1. 0
2. 1
3. 2
4. 4

Solution: 1. 0

Question 7. How many diagonals does a regular hexagon have?

1. 2
2. 0
3. 4
4. 9

Solution: 4. 9

Question 8. What is the name of a regular polygon of 3 sides?

1. Equilateral triangle
2. Square
3. Regular hexagon
4. Regular octagon.

Solution: 1. Equilateral triangle

Question 9. What is the name of a regular polygon of 6 sides?

1. Square
2. Equilateral triangle
3. Regular hexagon
4. Regular octagon.

Solution: 3. Regular hexagon

Question10. What is the name of a regular polygon of 4 sides?

1. Regular hexagon
2. Regular octagon
3. Square
4. Equilateral triangle.

Solution: 3. Square

Question 11. The sum of the measures of the exterior angles of any polygon is

1. 90°
2. 180°
3. 360°
4. 720°

Solution: 3. 360°

Question 12. The number of sides of a regular polygon, whose exterior angle has a measure of 45°. is

1. 4°
2. 6°
3. 8°
4. 10°

Solution: 3. 8°

Number of sides =\(\frac{360^{\circ}}{45^{\circ}}\)

= 8°

Question 13. The measure of each exterior angle of a regular polygon of 9 sides is

1. 30°
2. 40°
3. 60°
4. 45°

Solution: 2. 40°

Required measure =\(\frac{360^{\circ}}{9^{\circ}}\)

= 40°

Question 14. The measure of each exterior angle of a regular polygon of 15 sides is

1. 30°
2. 45°
3. 60°
4. 24°

Solution: 4. 24°

Required measure =\(\frac{360^{\circ}}{24^{\circ}}\)

= 24°

Question 15. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

1. 6°
2. 9°
3. 15°
4. 12°

Solution: 3. 15°

Number of sides =\(\frac{360^{\circ}}{15^{\circ}}\)

Question 16. How many sides does a regular polygon have if each of its interior angles is 165°

1. 12°
2. 24°
3. 9°
4. 6°

Solution: 2. 24°

Each exterior angle = 180° -165°= 15°

Number of sides = \(\frac{360^{\circ}}{15^{\circ}}\)

= 24°

Question 17. Which of (ho following statements is false?

1. All the angles of a rectangle are equal
2. No angle of a rectangle can be obtuse
3. The diagonals of a rectangle bisect each other
4. The opposite sides of a rectangle are not equal.

Solution: 4. The opposite sides of a rectangle are not equal.

Question 18. Which of the following statements is false?

1. A square is a rectangle whose adjacent sides are equal
2. A square is a rhombus whose one angle is a right angle
3. The diagonals of a square bisect each other at right angles
4. The diagonals of a square do not divide the whole square into four equal parts.

Solution: 4. The diagonals of a square do not divide the whole square into four equal parts.

Question 19. Which of the following statements is false?

1. All the rectangles are parallelograms
2. All the squares are rectangles
3. All the parallelograms are rectangles
4. All the rhombuses are parallelograms.

Solution: 3. All the parallelograms are rectangles

Question 20. Which of the following statements is true?

1. All the rectangles are squares
2. All the parallelograms are rhombuses
3. All the squares are rhombuses
4. Each parallelogram is a trapezium.

Solution: 3. All the squares are rhombuses

Question 21. Which ofthe following statements is true?

1. All the rhombuses are squares
2. Each square is a parallelogram
3. Each parallelogram is a square
4. Each trapezium is a parallelogram.

Solution: 2. Each square is a parallelogram

Question 22. One angle of a parallelogram is a right angle. The name ofthe quadrilateral is

1. Square
2. Rectangle
3. Rhombus
4. Kite

Solution: 2. Rectangle

Question 23. Two adjacent sides of a rod angle are equal. The name of the quadrilateral is kite

1. Square
2. Kite
3. Rhombus
4. None of these.

Solution: 1. Square

Question 24. Which of the following statements is false?

1. All the four sides of a parallelogram are equal.
2. The opposite angles of a parallelogram are equal
3. The diagonals of a parallelogram bisect each other
4. All the four sides of a rhombus are equal

Solution: 1. All the four sides of a parallelogram are equal.

Question 25. Which of the following statements is false?

1. All the four angles of a rhombus are equal
2. The diagonals of a rhombus bisect each other at right angles
3. A rectangle is a parallelogram
4. All squares are rectangles.

Solution: 1. All the four angles of a rhombus are equal

Question 26. If one angle of a parallelogram is of 65°, then the measure of the adjacent angle is

1. 65°
2. 115°
3. 25°
4. 90°.

Solution: 2. 115°

The measure of the adjacent angle

= 180° – 65° = 115°

Question 27. If ∠A of a parallelogram ABCD is of60°, then the measure of the opposite angle ∠C is

1. 60°
2. 120°
3. 30°
4. None of these.

Solution: 1. 60°

∠C =∠A = 60°.

Question 28. If all four sides of a parallelogram are equal and the adjacent angles are 120° and 60°, then the name of the quadrilateral is

1. Rectangle
2. Square
3. Rhombus
4. Kite

Solution: 3. Rhombus

Question 29. If the length of a side of a rhombus is 6 cm, (when the perimeter of the rhombus 

1. 6 cm
2. 12 cm
3. 24 cm
4. 3 cm.

Solution: 3. 24 cm

Perimeter = 4 side = 4× 6 = 24 cm

Question 30. In a kite, what is false?

1. The diagonals are perpendicular to each other
2. One ofthe diagonals bisects the other
3. Both the pairs of opposing angles are equal
4. All the four sides are equal.

Solution: 4. All the four sides are equal.

Question 31. ABCD is a rectangle. Its diagonals meet at O.

OA = 2x-1, OD = 3x-2. Find x 

1. 1
2. 2
3. 3
4. -1

Solution: 1. 1

3x – 2 = 3x – 1

3x -2x = -1+ 2

x = 1

Question 32. Find the perimeter of the rectangle ABCD

1. 6 cm
2. 12 cm
3. 3 cm
4. 24 cm.

Solution: 2. 12 cm

Perimeter = 2 (4 + 2) cm = 12 cm

Question 33. The four angles of a quadrilateral are in the ratio 1: 2 : 3: 4. The measure of its smallest angle is

1. 120°
2. 36°
3. 18°
4. 10°.

Solution: 2. 36°

Sum of the ratios =1 + 2+3 + 4=10

∴ Smallest angle =\(\frac{1}{10} \times 360^{\circ}\)

Question 34. In a parallelogram ABCD, ∠A : ∠B = 1: 2. Then, ∠A =

1. 30°
2. 60°
3. 45°
4. 90°.

Solution: 2. 60°

∠A + ∠B = 180°

∠A : ∠B = 1:2

Sum of the ratios =1 + 2 = 3

∴ \(\angle \mathrm{A}=\frac{1}{3} \times 180^{\circ}=60^{\circ}\)

Question 35. Two adjacent angles of a parallelogram are of equal measure. The measure of each angle of the parallelogram is

1. 45°
2. 30°
3. 60°
4. 90°.

Solution: 4. 90°.

x° + x° = 180°

2x° = 180°.

x° = \(\frac{180^{\circ}}{2^{\circ}}\)

x° = 90°

Question 36. ABCD is a parallelogram as shown. Find x and y

1. 1,7
2. 2,6
3. 3,5
4. 4,4

Solution: 3. 3,5

x + y = 8

y + 5 = 10

y=5

x + 5 = 8

x = 3.

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals True Or False

1. The number of diagonals in a decagon is 35 – True

2. The measure of each interior angle of a regular pentagon is 108° – True

3. The minimum number of sides a polygon can have is 4 – False

4. A regular polygon is both ‘equiangular’ and ‘equilateral’ – True

NCERT Solutions For Class 8 Maths Chapter 3 Understanding Quadrilaterals Fill In The Blanks

1. The name of a three-sided regular polygon is → An equilateral triangle

2. The sum of the adjacent angles of a parallelogram is  →  180°

3. The angle between the diagonals of a rhombus is → 90°

4. The maximum number of obtuse angles that a quadrilateral can have is → 3

5. Apala has a beautiful garden in her flat. The shape ofthe garden is triangular. The sum of all the exterior angles taken in order of the triangular garden is → 360°

6. The sides of a rectangle are 24 cm and 10 cm. Find the length of one of the diagonals → 26 cm

7. ABCD is a parallelogram. Find ∠A – ∠C →0

8. Two adjacent angles of a parallelogram are (3x + 3)° and (lx + 7)°. Find x → 17

9. In the following figure, find the value of x → 70°

10. In the following figure, find the measure of ∠ABC – 130°