mensuration provides solutions to chapter 3 in class 8 maths.
here find out exercises and answers for chapter 3, class 8 maths.
Exercise:3.1
1 Given here are some figures.
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon .
Answer : From the given figures,
(1), (2), (5), (6) and (7) are simple curves (a).
Figures (1), (2), (5), (6) and (7) are simple closed curves (b)
Figures (1) and (2) are polygon (c).
Figure (2) is convex polygon.
Figure (1) is concave polygon.
2. How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle .
Answer: (a) There are 2 diagonals in a convex Quadrilateral.
(b) Number of diagonals =ⁿ⁽ⁿ⁻³⁾/₂ , here n=number of sides.
In hexagon number of diagonals=⁶⁽⁶⁻³⁾/₂=9 diagonals.
(c) A triangle does not have any diagonals in it.
3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non convex Quadrilateral and try)
Answer : The sum of measure of the angles of a convex Quadrilateral is 360° as convex Quadrilateral is made of two triangles.
Here ABCD is a convex Quadrilateral, made of two triangles ∆ABC, ∆ADC.
∴ The sum of all the interior angles of this Quadrilateral will be same as the sum of all the interior angles of these two triangles.
i.e. 180°+180°=360°.
This property also holds true for a Quadrilateral which is not convex. Because any Quadrilateral can be divided into two triangles.
ABCD is a concave Quadrilateral, made of two triangles ∆ABD, ∆DBC.
∴ Sum of all the interior angles of this concave Quadrilateral will also 360°.
4. Examine the table. (Each figure is divided into triangles and the sum of the angles d deduced from that.)
What can you say about the angle sum of a convex polygon with number of sides?
(a) 7 (b) 8 (c) 10 (d) n
Answer : From given table
(a) Sum of interior angles of polygon of 7 sides = (7-2) × 180°=900°
(b) Sum of interior angles of polygon of 8 sides=(8-2) × 180°
=1080°
(c)Sum of interior angles of polygon of 10 sides=(10-2) × 180°=1440°
(d)Sum of interior angles of convex polygon of n sides =(n-2) × 180°.
5. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Answer : A convex polygon is called s regular polygon if all its sides have equal length and measure of all angles are equal
(i) Equilateral triangle
. (ii) Square
(iii) Regular hexagon.
6. Find the angle measure x in the following figures.
Answer : (a) Sum of measures of all interior angles of a Quadrilateral is 360°.
∴50°+130°+120°+x°=360°
300°+x=360°
x=60°
(b)
From the figure,
90°+a=180° (∵ linear pair of angles)
a=180°-90°=90°
Sum of measures of all interior angles of a Quadrilateral is 360°.
∴ 60°+70°+x+90°=360°
220°+x=360°
x=140°
(c)
From the figure,
70°+a=180° , 60+b=180° (∵ linear pair of angles)
a=180°-70°=110° , b=180°-60°=120°
a=110°, b=120°
Sum of measures of all interior angles of a pentagon is 540°
∴120°+110°+30°+x°+x°=540°
2x°+260°=540°
2x=540°-260°=280°
x=²⁸⁰/₂=140°
(d) Sum of all interior angles of a pentagon is 540°
Given figure is regular pentagon.
So x+x+x+x+x=540°
5x=540°
x=⁵⁴⁰/₅=108°
7
(a) Find x + y + z (b) Find x + y + z + w
Answer: (a) From figure,
x+90°=180° ( ∵ linear pair of angles)
x=90°
z+30°=180° ( ∵ linear pair of angles)
⇒z=180-30=150°
y=90°+30 (∵ In a ∆, Exterior angle=sum of two opposite interior angles)
y=120°
x+y+z=90°+120°+150°=360°
(b)
From figure sum of the measures of all interior angles of a Quadrilateral is 360°
∴ a+60°+80°+120°=360°
a=360°-260°=100°
a=100°
x+120°=180° (∵ linear pair of angles)
x=60°
y+80°=180° (∵ linear pair of angles)
y=100°
z+60°=180° (∵ linear pair of angles)
z=120°
w+100°=180° (∵ linear pair of angles)
w=80°
Sum of the measures of all angles=x+y+z+w
=60°+100°+120°+80°=360°
Exercise 3.2
1 Find the x in the following figures.
Answer : we know that sum of exterior angles of a polygon is 360°
∴ From figure (a) 125°+125°+x°=360°
x+250°=360°
x=360°-250°
x=110°
(b) From figure (b)
If two adjacent angles on straight line. One angle is 90°, then another angle is also 90°
So from above figure 60°+90°+70°+x+90°=360°
310°+x=360°
x=360°-310°
x=50°
2 Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) sides.
Answer : (i) Sum of exterior angles of polygon=360°.
Each exterior angle of regular polygon has the same measure.
∴ Measure of each exterior angle of regular polygon of 9 sides =
(ii) Sum of exterior angles of polygon=360°.
Each exterior angle of regular polygon has the same measure.
∴ Measure of each exterior angle of regular polygon of 15 sides =
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Answer: Given measure of an exterior angle of regular polygon =24°
Exterior angle of regular polygon=
∴ 24°=360°÷Number of sides.
Number of sides=360°÷24°=15 sides.
Number of sides of the regular polygon=15 sides.
4. How many sides does a regular polygon have if each of its interior angles is 165°?
Answer : Given that measure of each interior angle of regular polygon =165°
Then each exterior angle=180°-165°=15°
{ ∵ In regular polygon
Sum of interior angle +exterior angle=180°}
Exterior angle of regular polygon=360°/n.
{Here n=number of sides of polygon}
15°=360°/n
n=³⁶⁰/₁₅=24
∴ Number of sides of polygon=24 sides.
5 (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?
Answer : The sum of exterior angles of all polygons is 360°.
And In a regular polygon each exterior angle is of the same measure. If 360° is a perfect multiple of the given exterior angle, then the given polygon will be possible.
(a) Given exterior angle=22°
360° is not a perfect multiple of 22°.
So such polygon is not possible.
(b) Given interior angle=22°
Exterior angle=180°-22°=158°
360° is not a perfect multiple of 158°.
Such a polygon is not possible.
6 (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
Answer : Let a polygon having the lowest possible number of sides.
Let 3 sided regular polygon i.e. equilateral triangle.
Exterior angle of equilateral triangle=³⁶⁰/₃ =120°
Interior angle of equilateral triangle=180°-120°=60°
∴ Minimum possible measure of interior angle for regular polygon=60°
∴ Maximum possible measure of exterior angle for regular polygon=120°
class 8 maths chapter 3 exercise 3.3 solution
1 Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD=… (ii) ∠DCB=….
(iii) OC=…. (iv) m ∠DAB+m ∠CDA=….
Answer : (i) In a parallelogram, opposite sides are equal.
So AD=BC
(ii) In a parallelogram, opposite angles are equal in measure.
∠DCB=∠DAB.
(iii) In parallelogram diagonals bisects each other.
So OC=OA.
(iv) In a parallelogram sum of adjacent angles are equal to 180°.
∴ m ∠DAB + m ∠CDA=180°
2 Consider the following parallelograms. Find the values of the unknowns x, y, z.
Answer : In parallelogram opposite angles are equal in measure and sum of adjacent angles are equal to 180°.
(i)
y=100° (∵ opposite angles are equal)
x+100°=180° (∵ sum of adjacent angles=180°)
⇒x=180°-100°=80°
∴x=z=80° (∵ opposite angles are equal)
(ii)
From figure
y+50°=180° (∵ sum of adjacent angles are equal to 180°)
⇒y=180°-50°=130°
y=x=130° (∵ opposite angles are equal)
x=z=130° (∵ corresponding angles are equal)
(iii)
From figure
x=90° (∵ vertically opposite angles are equal)
x+y+30°=180° (∵sum of angles in a triangle=180°)
x+y=180°-30°
x+y=150°
90°+y=150° (∵ x=90°)
y=150°-90°=60°
z=y=60° (∵ Alternate interior angles)
(iv)
From figure.
From figure.
z=80° (∵ Corresponding angles)
y=80° (∵ opposite angles are equal)
x+y=180° (∵ sum of adjacent angles are equal to 180°)
⇒x+80°=180°
x=180°-80°=100°.
(v)
From figure.
y=112° (∵ opposite angles are equal)
x+y+40°=180° (∵ sum of interior angles of triangle is =180°)
x+112°+40°=180°
x+152°=180°
x=180°-152°=28°
z=x=28° (Alternate interior angles).
3 Can a quadrilateral ABCD be a parallelogram if
(i) ∠D+∠B=180° ?
(ii) AB=DC=8 cm, AD=4 cm and BC=4.4 cm ?
(iii) ∠A=70° and ∠C=65° ?
Answer : (i) For ∠D+∠B=180°, Quadrilateral ABCD may or may not be a parallelogram. If in a quadrilateral sum of adjacent angles =180° and adjacent angles are equal then that quadrilateral is said to be parallelogram.
Along with given condition the above two conditions satisfied then we say ABCD is a parallelogram.
(ii) with Given measurement ABCD can’t form a parallelogram because opposite sides AD, BC are different lengths.
(iii)In a parallelogram opposite angles.
Given that ∠A≠ ∠C i.e. opposite angles are not equal.
∴ ABCD is not a parallelogram.
4 Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Answer: Quadrilateral (kite) ABCD has two of its interior angles ∠A and ∠C of same measure. But quadrilateral ABCD not a parallelogram because remaining pair of opposite angles not equal. i.e. ∠B≠ ∠D.
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.
Answer : Let the two adjacent angles ∠A and ∠B of parallelogram ABCD are in the ratio 3:2.
Let ∠A=3x and ∠B=2x.
We know that in parallelogram sum of the measures of adjacent angles is 180°.
∠A+∠B=180°
3x+2x=180°
5x=180°
x=¹⁸⁰/₅
x=36°
∠A=∠C=3x=3(36°)= 108° (∵ A, C are opposite angles are equal)
∠B=∠D=2x=2(36°)=72° (∵ opposite angles are equal).
∴ The measures of the angles of the parallelogram are 108°, 72°, 108° and 72°.
6 Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Answer: In parallelogram sum of adjacent angles are equal to 180°.
Let ABCD is required parallelogram.
∠A+∠B=180°
According to problem.
Adjacent angles of given parallelogram have equal measure.
∠A=∠B.
∠A+∠A=180°
2 ∠A=180°
∠A=90°
∠A=∠B=90°
∠A=∠C=90° and ∠B=∠D=90° (∵ opposite angles are equal)
∴Each angle of parallelogram measure 90°.
7 The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Answer: From figure y=40° (∵ Alternate interior angles).
z+40°=70° (∵ Corresponding angles)
z=70-40=30°
x+(z+40°)=180° (∵ sum of adjacent angles=180°).
x+30+40=180
x=180-70
x=110°.
8 The following figures GUNS and RUNS are parallelograms. Find x and y. (Length are in cm)
Answer: In parallelogram opposite sides are equal.
From figure (i) GU=SN
3y-1=26
3y=26+1 =27
y=²⁷/₃
y=9.
SG=NU
3x=18
x=6
∴ x=6 cm and y=9 cm.
(ii) We know that the diagonals of parallelogram bisect each other.
From figure (ii)
16=x+y and 20=y+7
16=x+y and 20-7=y ⇒ y=13
Sub y=13 in 16 =x+y
16=x+13
16-13=x
⇒x=3 cm
∴ x=3 and y=13cm
9
In the above figure both RISK and the CLUE are parallelograms. Find the value of x.
Answer : Given in figure RISK and CLUE are parallelograms.
Adjacent angles of parallelograms are supplementary.
∠RKS +∠ISK =180°
120°+∠ISK=180°
∠ISK=180°-120°
∠ISK=60°.
In figur upper ∆, ∠S=60° ………(a)
In parallelogram CLUE,
∠CEU=∠CLU=70°(∵ opposite angles are equal)
In figure upper ∆, ∠E=70°….(b)
In figure upper ∆.
Sum of angles in the ∆=180°
∴ ∠E+∠S+x=180°
70°+60°+x=180° {∵ from (a) and (b)}
x=180°-130°
x=50°
10 Explain how this figure is a trapezium. Which of its two sides are parallel?
Answer : If a transversal line intersecting two lines at distinct points and sum of measures of interior angles of same side is 180°.
Then the two lines are parallel to each other.
From figure, ∠NML+∠KLM=180°
∴ line segment NM∥KL
Given Quadrilateral KLMN has a pair of parallel lines,
∴ Quadrilateral KLMN is a trapezium.
11 Find m∠C in fig, if
line segment AB∥ line segment DC.
Answer : Given line segment AB∥ line segment DC.
∠B+∠C=180° (AB∥DC, Angles on the same side of transversal)
120°+∠C=180°
∠C=180°-120°=60°
12 Find the measure of ∠P and ∠S if line segment SP ∥ line segment RQ in the figure.
Answer : From figure,
∠P+∠Q=180° (∵Angles of same side of transversal,and SP∥RQ)
∠P+130°=180°
∠P=180°-130°
∠P=50°
∠R+∠S=180° (∵Angles of same side of transversal,and SP∥RQ)
90°+∠R=180°
∠R=180°-90°
∠R=90°.
There is one more method to find the m ∠P.
m ∠R and m ∠Q are given.
After finding m ∠S, the angle sum property of a Quadrilateral can be applied to find m∠P.
Exercise 3.4
1. State whether True or False.
(a) All rectangles are squares
(b) All rhombuses are parallelograms
(c) All squares are rhombuses and also rectangles
(d) All squares are not parallelograms.
(e) All kites are rhombuses.
(f) All rhombuses are kites
(g) All parallelograms are trapeziums.
(h) All squares are trapeziums.
Answer : (a) False, All rectangles are not squares.
(b) True, All rhombuses are parallelograms. Because opposite sides of rhombus are equal and parallel to each other.
(c)True, All squares are rhombuses as all sides of a square are equal lengths.All squares are also rectangles as each internal angle measure 90°
(d)False, All squares are parallelograms as opposite sides are equal and parallel.
(e) False All kites are not rhombuses. Because a kite does not have all sides of the same length.
(f) True, All rhombuses are kites.
(g)True, All parallelograms are trapeziums. Since parallelograms have a pair of parallel sides.
(h) True. All squares are trapeziums. Since all squares have a pair of parallel sides.
2 Identify all the quadrilaterals that have.
(a) four sides of equal length (b) four right angles .
Answer: (a) A rhombus and square are the Quadrilaterals that have 4 sides of equal length.
(b) A Square and rectangle are the Quadrilaterals that have 4 right angles.
3 Explain how a square is.
(i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle
Answer : (i) A square is a Quadrilateral. Since it has 4 sides.
(ii) A square is a parallelogram since it’s opposite sides are parallel to each other.
(iii) A square is a rhombus since it’s four sides are of the same length.
(iv) A square is a rectangle since each interior angle measures 90°.
4 Name the quadrilaterals whose diagonals.
(i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal.
Answer : (i) The diagonals of a parallelogram, rhombus, rectangle and squre bisect each other.
(ii) The diagonals of a rhombus and square perpendicularly bisects each other.
(iii) The diagonals of square and rectangle are equal.
5. Explain why a rectangle is a convex quadrilateral.
Answer : In a rectangle, There are two diagonal. Both lying in the interior of the rectangle.
∴ Rectangle is a convex Quadrilateral.
6. 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. (The dotted lines are drawn additionally to help you).
Answer :
Draw line segments AD and DC, such that AD∥BC, AB∥DC
AD=BC, AB=DC
ABCD is a rectangle. In this opposite sides are equal and parallel to each other and all the interior angles are equal to 90°.
In a rectangle, diagonals are of equal length and also bisects each other.
Hence, AO=OC=BO=OD
∴ O is equidistant from A,B and C.