1. How many diagonals does each of the following have? (a) A convex quadrilateral (b) A regular hexagon (c) A triangle Answer: (a) 2 (b) 9 (c) 0 2. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? Using the formula: (n - 2)180° 3.What can you say about the angle sum of a convex polygon with number of sides?(a) 7 (b) 8 (c) 10 4. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides 5. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides 8. How many sides does a regular polygon have if the measure of an exterior angle is 24°? Number of sides of a polygon =360/exterior angle 9. How many sides does a regular polygon have if each of its interior angles is 165°? Exterior Angle = 180°-interior angle 10. (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? [ If interior angle is 22° then the exterior angle = 180°-22°=158°On dividing 360° by 158° we can’t get answer in whole number, so such a polygon is not possible.]
11. (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: The polygon with minimum number of sides is a triangle, and each angle of an equilateral triangle measures 60°, so 60° is the minimum value of the possible interior angle for a regular polygon. For an equilateral triangle the exterior angle is 180°-60°=120° and this is the maximum possible value of an exterior angle for a regular polygon.] 12. Can a quadrilateral ABCD be a parallelogram if (i) <D + <B=180? (ii) AB=DC= 8cm, AD= 4cm,and BC = 4.4 cm (iii) <A = 70 and <C = 65? Answer: (i)It can be , but not always as you need to look for other criteria as well. (ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram. (iii) Here opposite angles are not equal, so it is not a parallelogram. 13. 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. 14. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram. 15. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them. 16. . The following figures GUNS and RUNS are parallelograms. Find x and y. (x=6 , y =9 ) 17. In the given figure both RISK and CLUE are parallelograms. Find the value of x.{x=10} 18. State whether True or False. a) All rectangles are squares Answer: All squares are rectangles but all rectangles can’t be squares, so this statement is false. (b) All kites are rhombuses. Answer: All rhombuses are kites but all kites can’t be rhombus (c) All rhombuses are parallelograms Answer: True (d) All rhombuses are kites. Answer: True (e) All squares are rhombuses and also rectangles Answer: True; squares fulfill all criteria of being a rectangle because all angles are right angle and opposite sides are equal. Similarly, they fulfill all criteria of a rhombus, as all sides are equal and their diagonals bisect each other. (f) All parallelograms are trapeziums. Answer: False; All trapeziums are parallelograms, but all parallelograms can’t be trapezoid. (g) All squares are not parallelograms. Answer: False; all squares are parallelograms (h) All squares are trapeziums. Answer: True 20. Identify all the quadrilaterals that have. (a) four sides of equal length (b) four right angles Answer: (a) If all four sides are equal then it can be either a square or a rhombus. (b) All four right angles make it either a rectangle or a square.
21. Explain how a square is. (i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle Answer: (i) Having four sides makes it a quadrilateral (ii) Opposite sides are parallel so it is a parallelogram (iii) Diagonals bisect each other so it is a rhombus (iv) Opposite sides are equal and angles are right angles so it is a rectangle. 22.. Name the quadrilaterals whose diagonals. (i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal Answer: Rhombus; because, in a square or rectangle diagonals don’t intersect at right angles. 23. Explain why a rectangle is a convex quadrilateral. Answer: Both diagonals lie in its interior, so it is a convex quadrilateral. 24. 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. Answer: If we extend BO to D, we get a rectangle ABCD. Now AC and BD are diagonals of the rectangle. In a rectangle diagonals are equal and bisect each other. So, AC = BD , AO = OC ,BO = OD, And AO = OC = BO = OD So, it is clear that O is equidistant from A, B and C.