## NCERT Solutions for class 8 Maths Chapter 3: Understanding Quadrilaterals## Exercise 3.1
Classify each of them on the basis of the following.
Figures 1, 2, 5, 6, and 7 are defined as simple curves.
Figures 1, 2, 5, 6, and 7 are defined as simple closed curves. All the curves are forming the enclosed figure.
Figure 1 is a polygon with five line segments and figure 2 is a polygon with four line segments.
Figure 2 is a type of convex polygon as its vertices lies outside away from the centre.
Figure 1 is a type of concave polygon as its vertices face inside, i.e., near from the centre.
## Exercise 3.2
125° + 125° + x° = 360° 250° + x° = 360° x° = 360° - 250° x° = 110° Thus, the value of angle x is 110°.
60°, 70°, 90°, 90°, and x° are the exterior angles of the given figure. The sum of measures of exterior angles of any polygon is 360°. 60° + 70° + 90° + 90° + x° = 360° 310° + x° = 360° x° = 360° - 310° x° = 50° Thus, the value of angle x is 50°.
There are nine angles in a regular polygon with 9 sides. The sum of measures of exterior angles of any polygon is 360°. Let the exterior angle be x. 9 × x = 360° 9x = 360° x = 360°/9 x = 40° Thus, the measure of each exterior angle of a polygon of 9 sides is equal to 40°.
There are fifteen angles in a regular polygon with 15 sides. The sum of measures of exterior angles of any polygon is 360°. Let the exterior angle be x. 15 × x = 360° 15x = 360° x = 360°/15 x = 24° Thus, the measure of each exterior angle of a polygon of 15 sides is equal to 24°.
Number of sides of a polygon × each angle of a polygon = 360° Each angle = 24° Number of sides of a polygon × 24°= 360° Number of sides of a polygon = 360°/24° Number of sides of a polygon = 15
Exterior angle = 180° - 165° Exterior angle = 15° The sum of measures of exterior angles of any polygon is 360°. Number of sides of a polygon × each angle of a polygon = 360° Each angle = 15° Number of sides of a polygon × 15°= 360° Number of sides of a polygon = 360°/15° Number of sides of a polygon = 24
Number of sides of a polygon × each angle of a polygon = 360° Each angle = 22° Number of sides of a polygon × 22°= 360° Number of sides of a polygon = 360°/22° Angle 360 is not divisible by 22. Hence, it is not possible to have a regular polygon with measure of each exterior angle as 22°.
Interior angle = 180° - 22° Interior angle = 158° Angle 360 is not divisible by 158°. Hence, it is not possible to have a regular polygon with measure of each interior angle as 158°.
Each angle of an equilateral triangle is equal to 60 degrees. Hence, the minimum interior angle possible for a regular polygon is 60°.
It is because, Exterior angle = 180° - Interior angle The minimum interior angle possible for a regular polygon is 60°. Exterior angle = 180° - 60° Exterior angle = 120° Hence, the maximum exterior angle possible for a regular polygon is 120°. ## Exercise 3.3
Opposite sides of the parallelogram are equal.
**Equal opposite sides**- Diagonals bisect each other
- Equal opposite angles
- Supplementary adjacent angles
The properties of the parallelogram are as follows: - Equal opposite sides
- Diagonals bisect each other
**Equal opposite angles**- Supplementary adjacent angles
Diagonals of the parallelogram bisect each other.
- Equal opposite sides
**Diagonals bisect each other**- Equal opposite angles
- Supplementary adjacent angles
y = 100° z = 80°
∠B and ∠C are the opposite interior angles. The sum of the opposite interior angles is equal to 180°. ∠B + ∠C = 180° 100° + ∠C = 180° ∠C = 180° - 110° ∠C = 80° x = 80° ∠B and ∠D are the opposite angles. Opposite angles of the parallelogram are equal. Thus, ∠B = ∠D = 100° y = 100° ∠C and ∠A are the opposite angles. Opposite angles of the parallelogram are equal. Thus, ∠C= ∠C = 80° Z = 80°
y = 130° z = 130°
y + 50° = 180° y = 180° - 50° y = 130° y and x are the opposite angles. Opposite angles of the parallelogram are equal. Thus, y = x = 130° z and 50° forms a liner pair. z + 50° = 180° z = 180° - 50° z = 130°
y = 60° z = 60°
x = 90° The sum of the internal angles of triangle is equal to 180°. x + y + 30° = 180° 90° + y + 30° = 180° y + 120° = 180° y = 180° - 120° y = 60° z and y are the diagonally opposite angles. The diagonally opposite angles of the parallelogram are equal. Thus, z = y = 60° y = 60°
y = 80° z = 80°
x + 80° = 180° x = 180° - 80° x = 100° y and 80° are the opposite angles. Opposite angles of the parallelogram are equal. Thus, y = 80° z and 100° forms a linear pair. z + 100° = 180° z = 180° - 100° z = 80°
y = 112° z = 28°
Thus, y = 112° The sum of angles of a triangle is equal to 180°. 40° + y + x = 180° 40° + 112° + x = 180° 152° + x = 180° x = 180° - 152° x = 28° z and are the diagonally opposite angles. The diagonally opposite angles of the parallelogram are equal. Thus, z = x = 28°
It has the following properties: - Equal opposite sides
- Diagonals bisect each other
- Equal opposite angles
- Supplementary adjacent angles
∠D and ∠B are the opposite angles. The opposite angles of a parallelogram are equal. Hence, ∠D + ∠B = 180° can be true if each of the angle measures 90°. But, the condition is needed not to be true for a parallelogram. The value can differ.
It has the following properties: - Equal opposite sides
- Diagonals bisect each other
- Equal opposite angles
- Supplementary adjacent angles
AB and DC are the opposite sides. The opposite sides of a parallelogram are equal. AD and BC are also the opposite sides, which should be equal. But, AD and BC given in the statement are not equal. Hence, the above statement signifies that the given quadrilateral cannot be a parallelogram.
It has the following properties: - Equal opposite sides
- Diagonals bisect each other
- Equal opposite angles
- Supplementary adjacent angles
∠A and ∠C are the opposite angles. The opposite angles of a parallelogram are equal. But, the angles given in the statement does not signify the condition. Hence, the above statement signifies that the given quadrilateral cannot be a parallelogram.
PQRS is a quadrilateral, but not a parallelogram. Q and S are the two opposite angles of equal measure.
Let the angles be x. 3x + 2x = 180° 5x = 180° x = 180/5 x = 36° The two adjacent angles of the parallelogram are: 3x = 3 × 36 = 108° 2x = 2 × 36 = 72°
Let the angle be x. x + x = 180° 2x = 180° x = 180/2 x = 90° Sum = 180° can be true if each of the angle measures 90°, i.e., a right angle.
y = 40° z = 30°
∠HOP + 70° = 180° ∠HOP = 180° - 70° ∠HOP = 110° x and ∠HOP are the opposite angles. The opposite angles of a parallelogram are equal. x = ∠HOP = 110° x = 110° In the triangle EHP, the sum of angles is 180°. ∠EHP + ∠HEP + ∠EPH = 180° 40° + x + ∠EPH = 180° 40° + 110° + ∠EPH = 180° 150° + ∠EPH = 180° ∠EPH = 180° - 150° ∠EPH = 30° ∠EHO and ∠EPO are the opposite angles. The opposite angles of a parallelogram are equal. ∠EHO = ∠EPO z + 40° = y + ∠EPH z + 40° = y + 30° Or, y = 40° z = 30° Diagonal of a parallelogram is an angle bisector.
y = 9 cm
GU = SN 3y - 1 = 36 3y = 26 + 1 3y = 27 y = 27/3 y = 9 cm SG = NU 3x = 18 x = 18/3 x = 6 cm
y = 13
Let the centre point of the parallelogram be O. y + 7 = 20 … (1) x + y = 16 … (2) The value of x and y can be calculated from the above two equations. y + 7 = 20 y = 20 - 7 y = 13 x + y = 16 Substituting the value of y, we get: x + 13 = 16 x = 16 - 13 x = 3
∠K and ∠SIR are the opposite angles. The opposite angles of a parallelogram are equal. ∠K = ∠SIR ∠K = ∠SIR = 120° ∠SIR and ∠SIL forms the linear pair. ∠SIR + ∠SIL = 180° 120° + ∠SIL = 180° ∠SIL = 180° - 120° ∠SIL = 60° Sum of ∠L and ∠ECL is equal to 180°. ∠L + ∠ECL = 180° 70° + ∠ECL = 180° ∠ECL = 180° - 70° ∠ECL = 110° ∠ECR and ∠ECL forms the linear pair. ∠ECR + ∠ECL = 180° ∠ECR + 110°= 180° ∠ECR = 180° - 110° ∠ECR = 70° In the triangle IOC, the sum of angles is equal to 180°. ∠ECR + ∠SIL + ∠IOC = 180° 70° + 60° + ∠IOC = 180° 130° + ∠IOC = 180° ∠IOC = 180° - 130° ∠IOC = 50° x and ∠IOC are the vertically opposite angles. The vertically opposite angles are equal. x = ∠IOC = 50° x = 50°
∠M + ∠L = 180° 100° + 80° = 180° 180° = 180° LH = RHS Hence, the two sides NM and KL are parallel. It shows that the given figure is a trapezium.
Thus, ∠B + ∠C = 180° 120° + ∠C = 180° ∠C = 180° - 120° ∠C = 60° The measure of angle C is 60°.
∠S = 90°
∠R is a right angle. The measure of right angle is equal to 90°. SP is parallel to RQ. If the two sides are parallel, the sum of opposite angles is equal to 180°. It means, ∠P + ∠Q = 180° 130 + ∠Q = 180° ∠Q = 180° - 130° ∠Q = 50° ∠S + ∠R = 180° ∠S + 90° = 180° ∠S = 180° - 90° ∠S = 90° ## Exercise 3.4
Hence, all rectangles are not squares.
All the sides of rhombus are equal. A rhombus can be called as a rectangle of equal sides or equal opposite sides. Hence, all squares are rhombuses and also rectangles.
Hence, all kites are not rhombuses.
A rhombus can be a kite of equal sides. Hence, all rhombuses are kites.
A parallelogram can be a trapezium, but a trapezium cannot be a parallelogram. Hence, all parallelograms can be trapeziums.
Hence, all squares can be trapeziums.
Hence, a square is called as a quadrilateral.
The diagonals of the rectangle also lie in its interior. Hence, we can define a rectangle as a convex quadrilateral.
ABCD is a rectangle. Where, AB is parallel to DC AD is parallel to BC Diagonals AC and BD are equal and intersect each other Hence, in such a case, O is the mid-point of the diagonals. |