Karnataka Class 8 Math Solution Chapter Understanding Quadrilaterals by Experience Math Teacher. Karnataka Board Class 8 Math Part 1 PDF Chapter 4 Exercise 4.1, Exercise 4.2, Exercise 4.3, Exercise 4.4 Solution.
Karnataka Class 8 Math Solution Chapter Understanding Quadrilaterals
Exercise 4.1
Classify each of them on the basis of the following.
(a) Simple curve
Ans:
Simple Curve are
(b) Simple closed curve
Ans:
Simple closed curve are
(c) Polygon
Ans:
(d) Convex polygon
Ans:
Convex polygon are
(e) Concave polygon
Ans:
Concave polygon are
2.) How many diagonals does each of the following have?
(a) A convex quadrilateral
Ans:
A convex quadrilateral has two (2) diagonal.
(b) A regular hexagon
Ans:
A regular hexagon has nine (9) diagonal.
(c) A triangle
Ans:
A triangle has no diagonal.
4.) Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
What can you say about the angle sum of a convex polygon with number of sides?
(a) 7
Ans:
When n = 7
We know, Angle sum of a polygon = (n – 2) x 180 deg
= (7 – 2) x 180 deg
= 5 x 180 deg
= 900 deg
(b) 8
Ans:
When n = 8
We know,
Angle sum of a polygon = (n – 2) x 180 deg
= (8 – 2) x 180 deg
= 6 x 180 deg
= 1080 deg
(c) 10
Ans:
When n = 10
We know,
Angle sum of a polygon = (n – 2) x 180 deg
= (10 – 2) x 180 deg
= 8 x 180 deg
= 1440 deg
(d) n
Ans:
When n = n
We know,
Angle sum of a polygon = (n – 2) x 180 deg
5.) What is a regular polygon? State the name of a regular polygon of
A polygon having all sides of equal length and the interior angles of equal measure is known as regular polygon.
(i) 3 sides
Ans:
Regular Polygon having 3 sides is called Triangle.
(ii) 4 sides
Ans:
Regular Polygon having 4 sides is Quadrilateral.
(iii) 6 sides
Ans:
Regular Polygon having 6 sides is Hexagon.
6.) Find the angle measure x in the following figures
Ans:
We know,
Sum of the measures of all interior angles of a quadrilateral is 3600
50 deg + 130 deg + 120 deg + x = 360 deg
300 deg + x = 360 deg
x = 360 deg – 300 deg
x = 60 deg
Ans:
We know,
Sum of the measures of all interior angles of a quadrilateral is 3600
90 deg + 60 deg + 70 deg + x = 360 deg
220 deg + x = 360 deg
220 deg + x = 360 deg
x = 360 deg – 220 deg
x = 140 deg
EXERCISE 4.2
1.) Find x in the following figures.
Ans:
We know,
The sum of the measures of the exterior angles of any polygon is 3600
Sum of the measures of the external angles,
125 deg + 125 deg + x deg = 360 deg
250 deg + x deg = 360 deg
x deg = 110 deg
Ans:
We know,
The sum of the measures of the exterior angles of any polygon is 3600
From fig, y = 90 deg because it is right angle.
Sum of the measures of the external angles,
60 deg + 90 deg + 70 deg + x + 90 deg = 360 deg
310 deg + x = 360 deg
x = 50 deg
2.) Find the measure of each exterior angle of a regular polygon of
(i) 9 sides
Ans:
We know, Total measure of all exterior angles = 360°
Each exterior angle = sum of exterior angle / number of sides
360°/ 9 =40°
Each exterior angle = 40°
(ii) 15 sides
Ans:
We know, Total measure of all exterior angles = 360°
Each exterior angle = sum of exterior angle / number of sides
360°/ 15 =24°
Each exterior angle = 24°
3.) How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Ans:
We know,
Total measure of all the exterior angles of the regular polygon = 360 deg
Let, number of sides of regular polygon = n
Measure of each exterior angle = 24 deg
Number of sides = Sum of exterior angles / each exterior angle
= (360 deg)/ (24 deg)
= 15
Regular polygon has 15 sides.
4.) How many sides does a regular polygon have if each of its interior angles is 165°?
Ans:
Given that,
Measure of each interior angle = 165°
We have to Exterior angle.
Measure of each exterior angle = 180° – 165° = 15°
Let number of sides be n.
Number of sides = Sum of exterior angles / each exterior angle
360° / 15°
n = 24
The regular polygon has 24 sides
5.) (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
Ans:
We know,
Total measure of all exterior angles = 360 deg
Given that, Measure of each exterior angle = 22°
Let number of sides be = n
The number of sides = Sum of exterior angles / each exterior angle
= (360 deg)/ (22 deg)
=16.36
We cannot have regular polygon with each exterior angle = 22°
(b) Can it be an interior angle of a regular polygon? Why?
Ans:
Measure of each interior angle = 22 deg
We find Exterior angle.
Measure of each exterior angle = 180 deg – 22 deg
= 158 deg
Number of sides = Sum of exterior angles / each exterior angle
= (360 deg)/ (158 deg)
= 2.27
We cannot have regular polygon with each interior angle as 220
6.) (a) What is the minimum interior angle possible for a regular polygon? Why?
Ans:
We know,
Sum of all the angles of a triangle = 180 deg
x + x + x = 180 deg
3x = 180 deg
x = (180 deg)/3
x = 60 deg
Minimum interior angle possible for a regular polygon = 60 deg
EXERCISE 4.3
1.) Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = —–
Ans:
We know, the opposite sides of a parallelogram are of equal length.
AD = BC
(ii) Angle DCB =———–
Ans:
We know, in a parallelogram, opposite angles are equal in measure.
Angle DCB = Angle DAB
(iii) OC =
Ans:
We know, we know, in a parallelogram, diagonals bisect each other.
OC = OA
(iv) m angle DAB+m angle CDA = …….
Ans:
We know, in a parallelogram, adjacent angles are supplementary to each other.
m angle DAB+ m angle CDA = 180°
2) Consider the following parallelograms. Find the values of the unknowns x, y,z
Ans:
From Fig,
Since D is opposite to B.
We know, opposite angles of a parallelogram are equal
So, y = 100 deg
We know, the adjacent angles in a parallelogram are supplementary
angle C+ angle B = 180 deg
We know, the adjacent angles in a parallelogram are supplementary
x + 100 deg = 180 deg
Therefore x = 180 deg – 100 deg
= 80 deg
We know, opposite angles of a parallelogram are equal
x = z = 80 deg
Ans:
We know, the adjacent angles in a parallelogram are supplementary.
x + 50 deg = 180 deg
x = 180 deg – 50 deg
= 130 deg
We know, opposite angles of a parallelogram are equal
x = y = 130 deg
x = z = 130 deg (Corresponding angles)
Ans:
We know, Angle sum property of triangles
x + y + 30 deg = 180 deg
From fig, x = 90 deg (Vertically opposite angles)
90 deg + y + 30 deg = 180 deg
y + 120 = 180 deg
y = 180 deg – 120 deg
y = 60 deg
We know, Alternate interior angles are equal
z = y = 60 deg
3.) Can a quadrilateral ABCD be a parallelogram if
(1) Angle D+ angle B= 180 deg
Ans:
We know, sum of angles of a quadrilateral is 360 deg.
angle A+ angle B+ angle D + angle C = 360 deg
angle A+ angle C + 180 deg = 360 deg
angle A+ angle C = 360 deg – 180 deg
We know, Opposite angles in parallelogram, should also be of same measures.
angle A+ angle C = 180 deg
For angle D+ angle B = 180 deg, is a parallelogram.
(ii) AB = DC = 8cm AD = 4cm and BC= 4.4cm
Ans:
We know,
The opposite sides of a parallelogram are of equal length.
Opposite sides AD and BC are of different lengths.
So ABCD is not parallelogram
(iii) Angle A = 70 deg and angle C =65 deg
Ans:
We know,
In a parallelogram opposite angles are equal.
angle A and angle C are opposite angles
angle A = 70 deg and angle C = 65 deg are not equal.
So ABCD is not parallelogram
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.
Ans:
Given that,
The measures of two adjacent angles of a parallelogram are in the ratio 3:2.
Let, The adjacent angles are 3x and 2x respectively.
We know,
The sum of the measures of adjacent angles is 180° for a parallelogram.
angle A+ angle B = 180 deg
3x + 2x = 180 deg
5x = 180 deg
x = (180 deg)/5
x = 36 deg
angle A = angle C = 3x because they are opposite angles.
= 108 deg
angle B= angle D = 2x because they are opposite angles.
= 72 deg
The measures of the angles of the parallelogram are 108 deg, 72 deg, 108 deg and 72 deg.
7.) The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Ans:
We know, sum of angles of linear pair is 180 deg
angle HOP + 70 deg = 180 deg
angle HOP = 180 deg – 70 deg
angle HOP = 110 deg
We know, opposite angles are equal
angle O = angle E
x = 110 deg
From fig, Alternate interior angles are equal
y = 40 deg
z + 40 deg = 70 deg (They are Corresponding angles)
z = 70 deg – 40 deg
z = 30 deg
x = 110 deg
y = 40 deg
z = 30 deg
8.) The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
We know,In a parallelogram, the opposite sides have same length.
SG = NU
3x = 18
x = 18/3
x = 6
And,
SN = GU
26 = 3y – 1
3y = 26 + 1
y = 27/3
y = 9
The measures of x and y are 6 cm and 9 cm respectively.
We know, the diagonals of a parallelogram bisect each other.
y + 7 = 20
y = 20 – 7
y = 13
x + y = 16
x + 13 = 16
x = 3
9.) In the above figure both RISK and CLUE are parallelograms. Find the value of x.
Ans:
In parallelogram RISK
RKS+ISK = 180°
1200 + angle ISK = 180 deg
angle ISK = 180 deg – 120 deg
angle ISK = 60 deg
We know, in parallelogram opposite angles are equal
angle I = angle K
= 120 deg
In parallelogram CLUE
We know, in parallelogram opposite angles are equal
angle L = angle E = 70 deg
We know,
The sum of the measures of all the interior angles of a triangle is 180 deg
x + 60 deg + 70 deg = 180 deg
x + 130 deg = 180 deg
x = 180 deg – 130 deg
x = 50 deg.
EXERCISE 4.4
1.) State whether True or False.
(a) All rectangles are squares
Ans: False.
(b) All rhombuses are parallelograms
Ans: True.
(c) All squares are rhombuses and also rectangles
Ans: True.
(d) All squares are not parallelograms.
Ans: False.
(e) All kites are rhombuses.
Ans: False.
(f) All rhombuses are kites.
Ans: True.
(g) All parallelograms are trapeziums.
Ans: True.
(h) All squares are trapeziums.
Ans: True.
2.) Identify all the quadrilaterals that have.
(a) four sides of equal length
Ans: The Quadrilateral having four sides of equal length are Rhombus and Square.
(b) Four right angles
Ans: The Quadrilateral having four right angles are Square and Rectangle.
3.) Explain how a square is.
(i) a quadrilateral
Ans:
Quadrilateral has four sides. Square has also four sides.
(ii) a parallelogram
Ans:
In parallelogram, pairs of opposite sides equal. In square also pairs of opposite sides equal.
(iii) a rhombus
Ans:
In rhombus, four sides are of same length and the diagonals are perpendicular bisectors of each other.
In Square also four sides are of same length and the diagonals are perpendicular bisectors of each other.
(iv) a rectangle
Ans:
In rectangle, each angle is right angle.
In Square alsoeach angle is right angle.
4.) Name the quadrilaterals whose diagonals.
(i) bisect each other
Ans: The quadrilaterals whose diagonalsbisect each other are Parallelogram, Rhombus, Rectangle and Square
(ii) are perpendicular bisectors of each other
Ans:The quadrilaterals whose diagonalsare perpendicular bisectors of each other are Rhombus and
Square.
(iii) are equal
Ans:The quadrilaterals whose diagonalsare equal are Rectangle and Square.
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).
Ans:
ABCD is a rectangle.
We know, In rectangle opposite sides are equal and parallel to each other also each interior angleis of 90 deg.
AD ||BC, AB ||DC
AD = BC AB = DC
In a rectangle, diagonals are of equal length and also these bisect each other.
Hence, AO = OC = BO = OD
Since, two right triangles make a rectangle where O is equidistant point from A, B, C and D
O is equidistant from A, B, C and D.
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