3 - Understanding Quadrilaterals

Exercise 3.1

Question 1- Given here are some figures.

(1)	(2)	(3)
(4)	(5)	(6)
(7)	(8)

Classify each of them on the basis of the following.

(a) Simple curve

(b) Simple closed curve

(c) Polygon

(d) Convex polygon

(e) Concave polygon

Answer - (a) 1, 2, 5, 6, 7

(b) 1, 2, 5, 6, 7

(c) 1, 2

(d) 2

(e) 1

Question 2- How many diagonals does each of the following have?

(a) A convex quadrilateral

(b) A regular hexagon

(c) A triangle

Answer -  (a) There are 2 diagonals in a convex quadrilateral.

(b) There are 9 diagonals in a regular hexagon.

(c) A triangle does not have any diagonal in it.

Question 3- What is the sum of the measures of the angels of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Answer - The sum of the measures of the angles of a convex quadrilateral is 360° as a convex quadrilateral is made of two triangles.

Here, ABCD is a convex quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, the sum of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles i.e., 180º + 180º = 360º

Yes, this property also holds true for a quadrilateral which is not convex. This is because any quadrilateral can be divided into two triangles.

Here again, ABCD is a concave quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, sum of all the interior angles of this quadrilateral will also be 180º + 180º = 360º

Question 4- Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

Figure
Side	3	4	5	6
Angle sum	180°	2 × 180° = (4 − 2) × 180°	3 × 180° = (5 − 2) × 180°	4 × 180° = (6 − 2) × 180°

What can you say about the angle sum of a convex polygon with number of sides?

(a) 7

(b) 8

(c) 10

(d) n

Answer - From the table, it can be observed that the angle sum of a convex polygon of n sides is (n −2) × 180º. Hence, the angle sum of the convex polygons having number of sides as above will be as follows.

(a) (7 − 2) × 180º = 900°

(b) (8 − 2) × 180º = 1080°

(c) (10 − 2) × 180º = 1440°

(d) (n − 2) × 180°

Question 5- What is a regular polygon?

State the name of a regular polygon of

(i) 3 sides

(ii) 4 sides

(iii) 6 sides

Answer - A polygon with equal sides and equal angles is called a regular polygon.

(i) Equilateral Triangle

(ii) Square

(iii) Regular Hexagon

Question 6- Find the angle measure x in the following figures.

(a)	(b)
(c) Answer -(a) Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral, 50° + 130° + 120° + x = 360° 300° + x = 360° x = 60° (b) From the figure, it can be concluded that, 90º + a = 180º (Linear pair) a = 180º − 90º = 90º Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral, 60° + 70° + x + 90° = 360° 220° + x = 360° x = 140° (c) From the figure, it can be concluded that, 70 + a = 180° (Linear pair) a = 110° 60° + b = 180° (Linear pair) b = 120° Sum of the measures of all interior angles of a pentagon is 540º. Therefore, in the given pentagon, 120° + 110° + 30° + x + x = 540° 260° + 2x = 540° 2x = 280° x = 140° (d) Sum of the measures of all interior angles of a pentagon is 540º. 5x = 540° x = 108° Question 7-  (a) find x + y + z (b) find x + y + z + w Answer - (a) x + 90° = 180° (Linear pair) x = 90° z + 30° = 180° (Linear pair) z = 150° y = 90° + 30° (Exterior angle theorem) y = 120° x + y + z = 90° + 120° + 150° = 360° (b) Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral, a + 60° + 80° + 120° = 360° a + 260° = 360° a = 100° x + 120° = 180° (Linear pair) x = 60° y + 80° = 180° (Linear pair) y = 100° z + 60° = 180° (Linear pair) z = 120° w + 100° = 180° (Linear pair) w = 80° Sum of the measures of all interior angles = x + y + z + w = 60° + 100° + 120° + 80° = 360° Exercise -3.2 Question 1-  Find x in the following figures. (a) Answer - We know that the sum of all exterior angles of any polygon is 360º. (a) 125° + 125° + x = 360° 250° + x = 360° x = 110° (b) 60° + 90° + 70° + x + 90° = 360° 310° + x = 360° x = 50° Question 2-  Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides Answer - (i) Sum of all exterior angles of the given polygon = 360º Each exterior angle of a regular polygon has the same measure. Thus, measure of each exterior angle of a regular polygon of 9 sides =  (ii) Sum of all exterior angles of the given polygon = 360º Each exterior angle of a regular polygon has the same measure. Thus, measure of each exterior angle of a regular polygon of 15 sides =  Question 3- How many sides does a regular polygon have if the measure of an exterior angle is 24°? Answer - Sum of all exterior angles of the given polygon = 360º Measure of each exterior angle = 24º Thus, number of sides of the regular polygon\ Question 4-  How many sides does a regular polygon have if each of its interior angles is 165°? Answer - Measure of each interior angle = 165° Measure of each exterior angle = 180° − 165° = 15° The sum of all exterior angles of any polygon is 360º. Thus, number of sides of the polygon 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? Answer -  The sum of all exterior angles of all polygons is 360º. Also, in a regular polygon, each exterior angle is of the same measure. Hence, if 360º is a perfect multiple of the given exterior angle, then the given polygon will be possible. (a) Exterior angle = 22° 360º is not a perfect multiple of 22º. Hence, such polygon is not possible. (b) Interior angle = 22° Exterior angle = 180° − 22° = 158° Such a polygon is not possible as 360° is not a perfect multiple of 158°. Question 6- (a) What is the minimum interior angle possible for a regular polygon? (b) What is the maximum exterior angel possible for a regular polygon? Answer - Consider a regular polygon having the lowest possible number of sides (i.e., an equilateral triangle). The exterior angle of this triangle will be the maximum exterior angle possible for any regular polygon. Exterior angle of an equilateral triangle  Hence, maximum possible measure of exterior angle for any polygon is 120º. Also, we know that an exterior angle and an interior angle are always in a linear pair. Hence, minimum interior angle = 180º − 120° = 60º	(  d)