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Exercise - 3.2

Understanding Quadrilaterals

**Question-1 :-** Find x in the following figures.

(a) By linear pair angle, 125° + m = 180° m = 180° - 125° m = 55° By linear pair angle, 125° + n = 180° n = 180° - 125° n = 55° Exterior angle x = sum of opposite interior angles x = 55° + 55° x = 110°

(b) Sum of angles of a pentagon = (n-2) x 180° = (5-2) x 180° = 3 x 180° = 540° By using of linear pair angles, a + 90° = 180° a = 180° - 90° a = 90° b + 60° = 180° b = 180° - 60° b = 120° c + 90° = 180° c = 180° - 90° c = 90° d + 70° = 180° d = 180° - 70° d = 110° e + x = 180° e = 180° - x Now, a + b + c + d + e = 540° 90° + 120° + 90° + 110° + 180° - x = 540° 590° - x = 540° x = 590° - 540° x = 50°

**Question-2 :-** Find the measure of each exterior angle of a regular polygon of

(i) 9 sides (ii) 15 sides

(i) Sum of angles of a regular polygon = (n-2) x 180° = (9 - 2) x 180° = 7 x 180° = 1260° Each interior angle = sum of interior angles ÷ no. of sides = 1260° ÷ 9 = 140° So, each exterior angle = 180° - 140° = 40° (ii)Sum of exterior angles of a regular polygon = 360° Each interior angle = sum of interior angles ÷ no. of sides = 360° ÷ 15 = 24°

**Question-3 :-** How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Let no. of sides be n. Sum of exterior angles of a regular polygon = 360° No. of sides = sum of exterior angles ÷ each inetrior angle = 360° ÷ 24° = 15 Hence, the regular polygon has 15 sides.

**Question-4 :-** How many sides does a regular polygon have if each of its interior angles is 165°?

Let no. of sides be n. Exterior angle = 180° - 165° = 25° Sum of exterior angles of a regular polygon = 360° No. of sides = sum of exterior angles ÷ each inetrior angle = 360° ÷ 15° = 24 Hence, the regular polygon has 24 sides.

**Question-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?

(a) No, because 22° is not divisior of 360°. (b) No, because each exterior angle is 180° - 22° = 158°, which is not divisior of 360°.

**Question-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?

(a) The Equilateral Trianglebeing a regular polygon of 3 sides has the last measure of an interior angles of 60°. Sum of all angles of triangle = 180° x + x + x = 180° 3x = 180° x = 180°/3 x = 60° (b) By first (a), we can observe that the greatest interior angle = 180° - 60° = 120°.

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