NCERT Solutions for class 8 Maths Chapter 3: Understanding QuadrilateralsExercise 3.11. Given here are some figures. Classify each of them on the basis of the following. (a) Simple curve Answer: 1, 2, 5, 6, 7 Explanation: Simple curve is defined as the curve formed by joining multiple points on a paper without lifting the pencil. A simple curve is classified as open curve and closed curve. An open curve is not bounded from all the sides, while the closed curve is completely enclosed in an area. Figures 1, 2, 5, 6, and 7 are defined as simple curves. (b) Simple closed curve Answer: 1, 2, 5, 6, 7 Explanation: Simple closed curve is defined as the curve formed by joining multiple points on a paper without lifting the pencil. The closed curve has no endpoint. It is completely enclosed in an area. Figures 1, 2, 5, 6, and 7 are defined as simple closed curves. All the curves are forming the enclosed figure. (c) Polygon Answer: 1, 2 Explanation: A polygon is a closed figure formed with three or more line segments. Figure 1 is a polygon with five line segments and figure 2 is a polygon with four line segments. (d) Convex polygon Answer: 2 Explanation: All the vertices of the convex polygon face outside away from the centre. It means that no portion of its diagonal lies in the exterior. Figure 2 is a type of convex polygon as its vertices lies outside away from the centre. (e) Concave polygon Answer: 1 Explanation: Concave polygons generally have irregular shape. It is the opposite of the convex polygon. Some vertices of the concave polygon face inside, i.e., near from the centre. Figure 1 is a type of concave polygon as its vertices face inside, i.e., near from the centre. 2. What is a regular polygon? State the name of a regular polygon of Answer: A regular polygon is a closed figure formed with three or more line segments. All the sides and interior angles of the regular polygon are equal. (i) 3 sides Answer: Equilateral Triangle Explanation: Equilateral Triangle is an enclosed figure formed with three line segments. All the sides and interior angles of the equilateral triangle are equal. Hence, we can define equilateral triangle as a regular polygon. (ii) 4 sides Answer: Square Explanation: A square is an enclosed figure formed with four line segments. All the sides and interior angles of the square are equal. Hence, we can define square as a regular polygon. (iii) 6 sides Answer: Regular hexagon Explanation: A regular hexagon is an enclosed figure formed with six line segments. All the sides and interior angles of the regular hexagon are equal. Hence, we can define regular hexagon as a regular polygon. Exercise 3.21. Find x in the following figures. (a) Answer: 110° Explanation: 125°, 125°, and x° are the exterior angles of the given figure. The sum of measures of exterior angles of any polygon is 360°. 125° + 125° + x° = 360° 250° + x° = 360° x° = 360° - 250° x° = 110° Thus, the value of angle x is 110°. (b) Answer: 50° Explanation: The right angle is equal to 90°. 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°. 2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides Answer: 40° Explanation: A regular polygon is a closed figure formed with three or more line segments. All the sides and interior angles of the regular polygon are equal. 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°. (ii) 15 sides Answer: 24° Explanation: A regular polygon is a closed figure formed with three or more line segments. All the sides and interior angles of the regular polygon are equal. 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°. 3. How many sides does a regular polygon have if the measure of an exterior angle is 24°? Answer: 15 sides Explanation: 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 = 24° Number of sides of a polygon × 24°= 360° Number of sides of a polygon = 360°/24° Number of sides of a polygon = 15 4. How many sides does a regular polygon have if each of its interior angles is 165°? Answer: 24 Explanation: Exterior angle = 180° - Interior angle 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 (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°? Answer: No. A polygon with measure of each exterior angle as 22° is not possible. Explanation: 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 = 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°. (b) Can it be an interior angle of a regular polygon? Why? Answer: No. Angle 360 is also not divisible by an interior angle of 158°. Hence, it is not possible. Explanation: Interior angle = 180° - Exterior angle 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°. (a) What is the minimum interior angle possible for a regular polygon? Why? Answer: 60° Explanation: A polygon is formed from the three or more line segments. It means that the minimum sides of a polygon are three. A regular three sided polygon is known as an equilateral triangle. Each angle of an equilateral triangle is equal to 60 degrees. Hence, the minimum interior angle possible for a regular polygon is 60°. (b) What is the maximum exterior angle possible for a regular polygon? Answer: 120° Explanation: For the maximum exterior angle, the interior angle should be the least. 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 3.31. Given a parallelogram ABCD. Complete each statement along with the definition or property used. (i) AD =...... Answer: AD = BC Opposite sides of the parallelogram are equal. Explanation: The properties of the parallelogram are as follows:
(ii) ∠ DCB =...... Answer: ∠ DCB = ∠DAB Explanation: Opposite angles of the parallelogram are equal. The properties of the parallelogram are as follows:
(iii) OC =...... Answer: OC = OA Diagonals of the parallelogram bisect each other. Explanation: The properties of the parallelogram are as follows:
(iv) m ∠ DAB + m ∠ CDA = ...... Answer: m ∠DAB + m ∠CDA = 180° Explanation: The sum of the opposite interior angles is equal to 180°. 2. Consider the following parallelograms. Find the values of the unknowns x, y, z. (i) Answer: x = 80° y = 100° z = 80° Explanation: ABCD is a parallelogram. ∠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° (ii) Answer: x = 130° y = 130° z = 130° Explanation: y and 50° are the opposite interior angles. The sum of the opposite interior angles is equal to 180°. 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° (iii) Answer: x = 90° y = 60° z = 60° Explanation: x and right angle (90°) are the vertically opposite angles. The vertically opposite angles of a parallelogram are equal. 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° (iv) Answer: x = 100° y = 80° z = 80° Explanation: x and 80° are the opposite interior angles. The sum of the opposite interior angles is equal to 180°. 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° (v) Answer: x = 28° y = 112° z = 28° Explanation: y and 112° are the opposite angles. Opposite angles of the parallelogram are equal. 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° 3. Can a quadrilateral ABCD be a parallelogram if (i) ∠ D + ∠ B = 180°? Answer: Can be; but need not to be Explanation: A parallelogram is shown below: It has the following properties:
∠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. (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? Answer: No, AD is not equal to BC. Explanation: A parallelogram is shown below: It has the following properties:
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. (iii) ∠ A = 70° and ∠ C = 65°? Answer: No, ∠A is not equal to ∠C Explanation: A parallelogram is shown below: It has the following properties:
∠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. 4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure. Answer: The figure is shown below: PQRS is a quadrilateral, but not a parallelogram. Q and S are the two opposite angles of equal measure. 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: 108°, 72° Explanation: The sum of the adjacent angles of parallelogram is equal to 180°. 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° 6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram. Answer: If each is a right angle Explanation: The sum of the adjacent angles of parallelogram is equal to 180°. 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. 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: x = 110° y = 40° z = 30° Explanation: 70° and ∠HOP are the linear pairs. ∠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. 8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm) (i) Answer: x = 6 cm y = 9 cm Explanation: The opposite angles of a parallelogram are equal. 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 (ii) Answer: x = 3 y = 13 Explanation: The diagonals of the parallelogram bisect each other. 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 9. In the above figure both RISK and CLUE are parallelograms. Find the value of x. Answer: x = 50° Explanation: Let the centre point be O. ∠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° 10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.26) Answer: NM is parallel to KL. It is because the sum of opposite angles of the parallelogram is equal to 180°. Explanation: ∠M and ∠L are the opposite angles of the trapezium. If the sum of the opposite angles is equal to 180°, the given two sides are parallel. ∠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. 11. Find m ∠ C in Fig 3.27 if AB|| DC. Answer: ∠C = 60° Explanation: The given figure is a trapezium. The two sides DC and AB are parallel. The sum of opposite angles of the trapezium is equal to 180°. Thus, ∠B + ∠C = 180° 120° + ∠C = 180° ∠C = 180° - 120° ∠C = 60° The measure of angle C is 60°. 12. Find the measure of ∠P and ∠S if SP|| RQ in Fig 3.28. (If you find m∠R, is there more than one method to find m∠P?) Answer: ∠Q = 50° ∠S = 90° Explanation: ∠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 3.41. State whether True or False. (a) All rectangles are squares Answer: False Explanation: All the sides of the square are equal. In the case of rectangle, opposite sides are equal. A square can be called as a rectangle of equal sides. But, a rectangle cannot be always called as a square. Hence, all rectangles are not squares. (b) All rhombuses are parallelograms Answer: True Explanation: Rhombus, rectangle, and square are the types of parallelograms. (c) All squares are rhombuses and also rectangles Answer: True Explanation: All the sides of the square are equal. In the case of rectangle, opposite sides are equal. A square can be called as a rectangle of equal sides. 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. (d) All squares are not parallelograms. Answer: False Explanation: Rhombus, rectangle, and square are the types of parallelograms. (e) All kites are rhombuses. Answer: False Explanation: All the sides of rhombus are equal. A rhombus can be a kite, but a kite cannot always be a rhombus. It is because all the sides of the kite are not equal. Rhombus acts as a special case of kite. Hence, all kites are not rhombuses. (f) All rhombuses are kites. Answer: True Explanation: All the sides of rhombus are equal. A rhombus can be a kite, but a kite cannot always be a rhombus. A rhombus can be a kite of equal sides. Hence, all rhombuses are kites. (g) All parallelograms are trapeziums. Answer: True Explanation: Only one pair of opposite sides of a trapezium is equal. In the case of a parallelogram, both the pairs of opposite sides of a parallelogram are equal. A parallelogram can be a trapezium, but a trapezium cannot be a parallelogram. Hence, all parallelograms can be trapeziums. (h) All squares are trapeziums. Answer: True Explanation: All the sides of a square are equal, while only one pair of opposite sides of a trapezium is equal. A square can be a trapezium, but a trapezium cannot be a square. Hence, all squares can be trapeziums. 2. Identify all the quadrilaterals that have. (a) Four sides of equal length Answer: Square or rhombus Explanation: A square is a quadrilateral with all four sides of equal length. A rhombus is also a quadrilateral with all four sides of equal length. (b) Four right angles Answer: Square or rectangle Explanation: A square is a quadrilateral with all four sides of equal length and each side at right angle with the other. A rectangle is also a quadrilateral with all four sides of equal length and each side at right angle with the other side. 3. Explain how a square is. (i) A quadrilateral Answer: A quadrilateral is a four sided enclosed figure. A square is a quadrilateral with all four sides of equal length. Hence, a square is called as a quadrilateral. (ii) A parallelogram Answer: The opposite sides of a parallelogram are equal. A square is also a parallelogram with opposite parallel sides. (iii) A rhombus Answer: All the sides of a rhombus are equal. Similarly, all the sides of a square are also equal. Hence, a square is a rhombus. (iv) A rectangle Answer: The opposite sides of a rectangle are equal. Similarly, the opposite sides of a square are also equal. 4. Name the quadrilaterals whose diagonals. (i) Bisect each other Answer: Parallelogram; square, rhombus, rectangle Explanation: The diagonals of parallelogram, square, rhombus and the rectangle bisect each other. (ii) Are perpendicular bisectors of each other? Answer: rhombus, square Explanation: The diagonals of rhombus and square are the perpendicular bisector of each other. (iii) Are equal Answer: Square, rectangle Explanation: The diagonals of square and rectangle equal. 5. Explain why a rectangle is a convex quadrilateral. Answer: Both of its diagonals lie in its interior. Explanation: All the vertices of the convex quadrilateral face outside away from the centre. It means that no portion of its diagonal lies in the exterior. The diagonals of the rectangle also lie in its interior. Hence, we can define a rectangle as a convex quadrilateral. 6. ABC is a right-angled triangle and O is the midpoint 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: 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. |