Understanding Quadrilaterals

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Joleen Roxanne Hampton
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1 12 Understanding Quadrilaterals introduction In previous classes, you have learnt about curves, open and closed curves, polygons, quadrilaterals etc. In this chapter, we shall review, revise and strengthen these. We shall also learn the properties of quadrilaterals. ngle sum property of a quadrilateral Types of quadrilaterals trapezium, kite, parallelogram, rhombus, rectangle, square Properties of parallelogram Properties of rhombus Properties of rectangle and square curves Flat surfaces are known as planes. For eample, a page of your notebook, top of a table, floor of a room etc. are all planes. If we put the sharp tip of a pencil on a sheet of paper and move from one point to the other, without lifting the pencil, then the shapes so obtained are called plane curves. Some plane curves are shown below: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) Open curve. The curves which have different beginning and end points are called open curves. In the above figure (i), (vi), (vii) and (viii) are all open curves. losed curve. The curves which have same beginning and end points are called closed curves. In the above figure (ii), (iii), (iv) and (v) are all closed curves. Simple curve. curve which does not cross itself at any point is called a simple curve. In the above figure (i), (ii), (iii), (vi) and (vii) are all simple curves. Note that curves (ii) and (iii) are simple closed curves.

2 204 Learning Mathematics VIII Interior and eterior of a closed curve simple closed curve divides the region of the plane into three parts: (i) The region of the plane that lies inside the curve is called the interior of the curve. In the adjoining figure, point lies in the interior of the curve. (ii) The region of the plane that lies outside the curve is called the eterior of the curve. In the adjoining figure, point lies in the eterior of the curve. (iii) The collection of points that lie on the curve is called the boundary of the curve. In the adjoining figure, point c lies on the boundary of the curve. Polygons simple closed curve made up entirely of line segments is called a polygon. Look at the following figures: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) In the above figure (i), (ii), (iii) and (iv) are all simple closed curves made up entirely of line segments, so these are all polygons. In the above figure (v), (vi), (vii) and (viii) are not polygons (why?) Sides, vertices and diagonals of a polygon The line segments forming a polygon are called its sides. In the adjoining figure E is a polygon. It has five sides,,, E and E. The meeting point of a pair of sides is called a verte of the polygon. It has five vertices,,, and E. E The line segments formed by joining non-adjacent vertices are called diagonals of the polygon. In the above polygon, the line segments,,, E and E are all diagonals of the polygon.

3 Understanding Quadrilaterals 205 onve and concave polygons onve polygon. polygon in which each interior angle is less than 180 is called a conve polygon. Or Polygons in which no portion of any of their diagonals lies eterior to the polygon are called conve polygons. onve polygons oncave polygon. polygon in which atleast one interior angle is greater than 180 is called a concave polygon. Or Polygons in which a portion of atleast one diagonal lies eterior to the polygon are called concave polygons. diagonal more than 180 oncave polygons Note. In this chapter, by a polygon we would mean a conve polygon only. lassification of polygons We can classify the polygons according to the number of sides or vertices they have: Number of sides or vertices lassification Sample figure Number of sides or vertices lassification Sample figure 3 Triangle 8 Octagon 4 Quadrilateral 5 Pentagon 9 Nonagon 6 Heagon 10 ecagon 7 Heptagon n n-gon

4 206 Learning Mathematics VIII ctivity 7 To verify that sum of interior angles of a quadrilateral is 360 by cutting and pasting. Steps 1. Take a sheet of paper and draw any quadrilateral on it and cut off the four angles i.e.,, and as shown in adjoining figure (i). 2. Mark any point O on the copy. Fig. (i) 3. Paste all the four cut out angles in such a way that the vertices of these angles are at the marked point and they form adjacent angles as shown in figure (ii). These angles forms a complete angle at the point O = 360. Thus, we have verified: The sum of all the interior angles of quadrilaterals is 360. Regular and irregular polygons O Fig. (ii) polygon which has all its sides are of equal length and all its interior angles are of equal measure is called a regular polygon. For eample: Equilateral triangle and square are regular polygons. Equilateral triangle Square polygon which is not regular is called irregular polygon. For eample: Rectangle Rhombus (i) Rectangle is equiangular but all of its sides are not equal, so it is an irregular polygon. (ii) Rhombus has all its sides equal but it is not equiangular, so it is also an irregular polygon. Thus, in a regular polygon: ll sides are equal in length. ll interior angles are of equal measure. ll eterior angles are of equal measure. ll regular polygons are conve.

5 Understanding Quadrilaterals 207 ngle sum property of a polygon In class VII, you have learnt that sum of all interior angles of a triangle is 180. an you tell that what is the sum of all interior angles of a quadrilateral, a pentagon, a heagon and so on? To find this, let us try this method: Take a quadrilateral. ivide it into two triangles and by drawing a diagonal. We get si angles 1, 2, 3, 4, 5 and 6. y angle sum property of a triangle In, we have = 180 In, we have = 180 Now in quadrilateral (i) (ii) = or = (iii) From equations (i), (ii) and (iii), we have = = 360. Thus, the sum of measures of all the interior angles of a quadrilateral is 360. We can etend this idea to other polygons. Triangle Quadrilateral Pentagon Heagon Figure i i ii i iii i iv ii ii iii Number of sides Number of triangles Sum of interior angles = 360 or (4 2) 180 = = 540 or (5 2) 180 = = 720 or (6 2) 180 = 720 Hence, from the above table we can conclude that: Sum of measures of all interior angles of a n-sided polygon = (n 2) 180. Each interior angle of a n-sided regular polygon = ( n 2 ) # 180c. n Remark If a polygon has n sides, then the number of diagonals of the polygon = nn ( 3). 2 Eample 1. Some figures are given below: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

6 208 Learning Mathematics VIII lassify each of them on the basis of (a) simple curve (b) simple closed curves (c) polygon (d) conve polygon (e) concave polygon. Solution. (a) figures (i), (ii), (iii), (iv) and (vii) are simple curves (b) figures (i), (iv) are simple closed curves (c) figures (i) and (iv) are polygons (d) figure (i) is conve polygon (e) figure (iv) is concave polygon. Eample 2. Find the sum of measures of all the interior angles of a polygon with (i) 7 sides (ii) 9 sides. Solution. We know that: Sum of measures of all interior angles of a n-sided polygon = (n 2) 180. (i) Sum of measures of all interior angles of a 7-sided polygon = (7 2) 180 = = 900 (ii) Sum of measures of all interior angles of a 9-sided polygon = (9 2) 180 = = Eample 3. How many sides does a regular polygon have if each of its interior angles is 120? Solution. We know that: Each interior angle of a n-sided regular polygon = ( n 2) # 180c. n Given each interior angle = 120, ( n 2) # 180c = 120 n (n 2) 3 = 2n 3n 6 = 2n 3n 2n = 6 n = 6. Number of sides of regular polygon = 6. Eample 4. Find the angle measure in the following figures: (i) (iii) (iii) Solution. (i) s the sum of all interior angles of a quadrilateral is 360, = =

7 Understanding Quadrilaterals 209 = = 130 (ii) s E is a straight line = 180 = = 40...(i) s the sum of all interior angles of a quadrilateral is 360, = = 360 (from (i)) = 360 = = 100 (iii) s the given figure is a regular heagon, each interior angle of a regular heagon = (6 2) # 180c 6 = 4 # 180c 6 E = 120. = 4 30 = 120 Eample 5. If the angles of a quadrilateral are in the ratio 5 : 8 : 11 : 12, find the angles. Solution. Since the angles of quadrilateral are in the ratio 5 : 8 : 11 : 12, let these angles be 5, 8, 11 and 12. s the sum of measures of all interior angles of a quadrilateral is = = c = = c The angles of the quadrilaterals are 5 10 = 50, 8 10 = 80, = 110 and = 120 Hence, the angles of the quadrilateral are 50, 80, 110, 120. Eample 6. From the adjoining diagram, find + y + z. Solution. s an eterior angle of a triangle = sum of two opposite interior angles z = z = 130 is a straight line,...(i) = 180 = (ii) gain F is a straight line, 60 + y = 180 y = (iii) From (i), (ii) and (iii), we have E z 60 y F + y + z = = 360.

8 210 Learning Mathematics VIII ctivity 8 Sum of the measures of the eterior angles of a polygon. Steps 1. raw a polygon (a pentagon E as shown in adjoining figure) on the floor, using a piece of chalk. 2. Start walking from. Walk along, on reaching, you need to turn through an angle of m 1 to walk along. On reaching at, you need to turn through an angle of m 2 to walk along. 3. ontinue moving in the same manner, until you return to the side. 4. You will find that you have made one complete turn. Therefore, m 1 + m 2 + m 3 + m 4 + m 5 = 360. Hence, the sum of the measures of the eterior angles of a pentagon is 360. This is true for every polygon. Thus, the sum of measures of all the eterior angles of a polygon = 360. E Eample 7. Find the measure of angle in the following figures: 150 E F F V P T S U 150 R 80 Q Solution. (i) (ii) (iii) (i) Since E is a straight line, E + 90 = 180 ( = 90, given) E = = 90 s the sum of measures of all the eterior angles of a polygon = = = 360 = = 120 (ii) F is a straight line, F + 90 = 180 (Given = 90 ) F = = 90 s the sum of measures of all eterior angles of a polygon = 360, = = 360 = = 130

9 Understanding QUadriLateraLs 211 (iii) QRU is a straight line, URS = 180 URS = 30 STV is a straight line, VTP = 180 VTP = 20 s the sum of measures of all eterior angles of a polygon = 360, = = 360 = 150. Eample 8. Find the number of sides of a regular polygon whose each eterior angle has a measure of 45. Solution. s the sum of measures of all eterior angles of a polygon = 360 and given each eterior angle = c The number of eterior angles = = c Hence, the polygon has 8 sides. Eample 9. Is it possible to have a regular polygon with measure of each eterior angle as 22? Solution. s sum of measures of all eterior angles of a polygon = 360 and given each eterior angle = 22, 360c the number of sides = = , which is not a natural number. 22 c Hence, it is not possible to have a regular polygon with measure of each eterior angle as 22. Eample 10. What is the minimum interior angle possible for a regular polygon? Why? Solution. We know that as number of sides of a regular polygon increases, each interior angle also increases. So, the minimum interior angle is possible in a polygon of least number of sides i.e. in an equilateral triangle. 180c Each angle of an equilateral triangle = = So, the minimum interior angle = 60. Eample 11. What is the maimum eterior angle possible for a regular polygon? Solution. s the sum of measures of interior angle and corresponding eterior angle is 180 (constant), so for a regular polygon each eterior angle will be maimum when each interior angle is minimum. From eample 10, we know that the minimum interior angle = 60. Maimum eterior angle of a regular polygon = = 120. eercise Some figures are given below. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) classify each of them on the basis of the following: (a) Simple curve (b) Simple closed curve (c) Polygon (d) onve polygon (e) concave polygon

10 212 Learning MatheMatics Viii 2. How many diagonals does each of the following have? (a) conve quadrilateral (b) regular heagon (c) triangle. 3. Find the sum of measures of all interior angles of a polygon with number of sides: (i) 6 (ii) 8 (iii) 10 (iv) Find the number of sides of a regular polygon if each of its interior angle is (i) 60 (ii) 90 (iii) 108 (iv) 135 (v) If the angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6, find the angles. 6. If the angles of a pentagon are in the ratio 7 : 8 : 11 : 13 : 15, find the angles. 7. In a quadrilateral,. If : = 2 : 3 and : = 7 : 8, find the measure of each angle. 8. From the adjoining figure, find (i) (ii) (iii) (3 + 10) (3 + 4) (5 + 8) Find the angle measure in the following figures: (i) (ii) (iii) (iv) (v) (vi) 10. (i) In the adjoining figure, find + y + z. 90 z 30 y

1. Revision Description Reflect and Review Teasers Answers Recall of basics of triangles, polygons etc. Review Following are few examples of polygons:

1. Revision Description Reflect and Review Teasers Answers Recall of basics of triangles, polygons etc. Review Following are few examples of polygons: 1. Revision Recall of basics of triangles, polygons etc. The minimum number of line segments required to form a polygon is 3. 1) Name the polygon formed with 4 line segments of equal length. 1) Square