Lines and Angles. introduction

Transcription

1 9 Lines and nges intrductin In cass VI, you have earnt soe basic concepts and ters of geoetry point, ine, pane, ine segent, ray, ange and types of anges. In this chapter, we sha earn about soe pairs of reated anges copeentary anges, suppeentary anges, adjacent anges, inear pair, su of anges at a point, su of anges at a point on one side of a straight ine and verticay opposite anges. We sha aso earn about pairs of intersecting and parae ines, transversa, anges ade by a transversa and properties of anges associated with a pair of parae ines. Reca, the basic concept of a ine is its straightness and it etends indefinitey in both directions i.e. it has no definite ength and it has no end points. ine segent is a portion of a ine. It has two end points and a definite ength. ray is a part of a ine that etends ony in one direction. It has one end point (caed initia point) and has no definite ength. ine ine segent We know that when two rays or two ine segents or two ines eet, an ange is fored. The coon point is caed the verte of the ange. Look at the figures given beow: ray In fig., two ray and eet at, they for ange. Point is the verte and two rays and are the ars (or sides) of the ange. The ange is denoted by. In fig., the pairs of ine segents, ;, ;, eet at the points, and respectivey. Three anges fored by these pairs of ine segents are, and respectivey. In fig. (iii), the pair of ines and eet at the point. Four anges fored are,, and. The size (agnitude or easure) of an ange is the aount by which one of the ars needs to be rotated about the verte so that it ies on the top of the other ar. To easure an ange, one copete turn is divided into 60 equa parts and each part is caed one degree and is written as. We sha easure anges in degrees. Usuay, the easure of i.e. is written as. Two anges are caed equa if they have sae easure. (iii)

2 68 Learning Matheatics VII Reated nges opeentary anges Two anges are caed copeentary anges if the su of their easures is 90. ach ange is caed the copeent of the other. Look at the figures given beow: 5 In fig., + F = = 90 i.e. su of their easures is 90, therefore, and F are copeentary. In fig., + F = = 00 i.e. su of their easures is not 90, therefore, and F are not copeentary. Suppeentary anges F 65 Two anges are caed suppeentary anges if the su of their easures is 80. ach ange is caed the suppeent of the other. Look at the figures given beow: 0 60 F F 0 In fig., + F = = 80 i.e. su of their easures is 80, therefore, and F are suppeentary. In fig., + F = = 70 i.e. su of their easures is not 80, therefore, and F are not suppeentary. djacent anges Two anges are caed adjacent anges if they have a coon verte, a coon ar and their non-coon ars ie on either side of the coon ar. Look at the figures given beow: 50 F (iii)

3 Lines and nges 69 In fig., and have a coon verte, coon ar and the non-coon ars and ie on either (opposite) sides of the coon ar, therefore, and are adjacent anges. In fig., and have a coon verte, coon ar but the non-coon ars and ie on the sae side of the coon ar, therefore, and are not adjacent anges. In fig. (iii), and do not have a coon verte, therefore, and are not adjacent anges. Note. Two adjacent anges have no coon interior points. Linear pair Two adjacent anges are said to for a inear pair if their non-coon ar are opposite rays i.e. they are in a straight ine. In the adjoining figure, and are two adjacent anges and their non-coon ars and are the opposite rays, therefore, and for a inear pair. s and are opposite rays, and are in a straight ine. = 80. Fro given fig., = + + = 80. Thus, two anges in a inear pair are suppeentary. onversey, if two adjacent anges are suppeentary i.e. if the su of their easures is 80, then the non-coon ars are in a straight ine and hence they for a inear pair. In the adjoining figure, and are two adjacent anges such that + = = 80, therefore, and for a inear pair. Note. If a pair of suppeentary anges are paced adjacent to each other, then they for a inear pair. nges at a point In the adjoining diagra, the four anges together ake one copete turn, so they add upto 60. This is true no atter how any anges are fored at a point. Thus: Su of anges at a point = 60. nges on one side of a straight ine In the adjoining diagra, the three anges together ake a straight ine, so they add upto 80. This is true no atter how any anges ake up the straight ine. Thus: Su of anges at a point on one side of a straight ine =

4 70 Learning Matheatics VII Verticay opposite anges When two straight ines intersect each other, they for four anges at their point of intersection say,, and. and are caed verticay opposite anges to each other and so are and. They are caed verticay opposite anges because they have the sae verte and are opposite to each other. In fact, verticay opposite anges are fored by the non-coon ars. ctivity 6 Verticay opposite anges are equa Materias required Steps sheet of white paper. n a sheet of paper, draw two straight ines and Ruer intersecting at the point. Four anges are fored at (iii) Tracing paper the point, say,, and. (iv) Pin, for one pair of verticay opposite anges and, for another pair of verticay opposite anges.. Make a trace copy (repica) of the fig. on a tracing paper.. Pace the trace copy on the origina figure such that one of the anges atch its copy, then the other anges wi atch the copy.. Fi a pin at the point and rotate the copy through 80. bservation The ines of the copy wi coincide with the origina figure (as shown in fig. ). We note that coincides with and coincides with. Resut = and =. If foows that verticay opposite anges are equa. Verticay opposite anges are equa When two straight ine intersect each other, they for four anges at their point of intersection, say,, and. Look at the figure, and for a inear pair.

5 Lines and nges 7 + = 80 gain fro figure, and for a inear pair. + = 80 For and, we get + = + =. In the sae way, we can show that =. Hence, verticay opposite anges are equa ape. an two acute anges be copeent to each other? an two obtuse anges be copeent to each other? (iii) an two acute anges be suppeentary? (iv) an two adjacent obtuse anges for a inear pair? Soution. Yes; pairs of anges ike 0 and 60 ; 5 and 65 are copeents of each others. No; as the su of two obtuse anges is aways greater than 80, so they can never be copeent of each other. (iii) No; as the su of two acute anges is aways ess than 80, so they can never be suppeentary anges. (iv) No; as the su of two obtuse anges is aways greater than 80, so they cannot for a inear pair. ape. In the given figure, straight ines and intersect each other at : Is adjacent to? Is adjacent to? (iii) o and for a inear pair? (iv) re and suppeentary? (v) Is verticay opposite to? (vi) What is the verticay opposite ange of 5? Soution. Yes; it is cear fro figure. No; is the coon ar of and but their non-coon ars and ie on the sae side of the coon ar, therefore, and are not adjacent anges. (iii) Yes; because and are adjacent anges and their non-coon ars are in a straight ine. (iv) Yes; as is a straight ine, + = 80, therefore these anges are suppeentary. (v) Yes, it is cear fro figure. (vi) is verticay opposite to 5. ape. In the adjoining figure, straight ines and intersect each other at and. Nae the foowing pairs of anges: obtuse verticay opposite anges adjacent copeentary anges 5

6 7 Learning MatheMatics Vii (iii) equa suppeentary anges (iv) unequa suppeentary anges (v) adjacent anges that do not for a inear pair. Soution. and and (iii) and (iv) Pairs of unequa suppeentary anges are:, ;, ;, ;,,, (v) Pair of adjacent anges that do not for a inear pair are:, ;, ;,. ape. Find the vaue of in each of the foowing diagras: Soution. s the su of anges at a point = 60, = 60 = = 0 s the su of anges at a point on one side of a straight ine is 80, = 80 = = 05 = 5. (ii ) ape 5. Find the vaues of, y and z in each of the foowing diagras: You ust state which geoetrica fact you are using to find the ange y 55 z 0 5 y z (i ) (ii ) Soution. = 55 (verticay opposite anges) 55 + y = 80 (inear pair) y = y = 5 z = y (verticay opposite anges) z = 5.

7 Lines and nges 7 s the su of anges at a point on one side of a straight ine is 80, = 80 = = y = 80 (inear pair) y = 80 0 y = 0 z = 0 (verticay opposite anges) ape 6. If the difference in the easures of two copeentary anges is, then find the easures of the anges. Soution. Let one ange be, then the other ange is ( + ). s the given anges are copeentary anges, + ( + ) = 90 = 90 = 78 = 9 and + = 9 + = 5. Hence, the easures of the required anges are 9 and 5. ape 7. Two copeentary anges are in the ratio :, find these anges. Soution. Since the given anges are in the ratio :, et the anges be and. s the given anges are copeentary anges, + = 90 5 = 90 = 8 = 6 and = 5. Hence, the required anges are 6 and 5. ape 8. If two anges are suppeentary anges and one ange is 0 ess than twice the other, find the anges. Soution. Let one ange be, then the other ange = ( 0). s the given anges are suppeentary anges, + ( 0) = 80 = = 0 = 70 and 0 = 70 0 = 0. Hence, the required anges are 70 and 0. ercise 9.. an two right anges be copeentary? an two right anges be suppeentary? (iii) an two adjacent anges be copeentary? (iv) an two adjacent anges be suppeentary? (v) an two obtuse anges be adjacent? (vi) an an acute ange be adjacent to an obtuse ange? (vii) an two right anges for a inear pair?

8 7 Learning MatheMatics Vii. Find the easure of the copeent of each of the foowing anges: 5 65 (iii) (iv) 5. Find the copeent of each of the foowing anges: (iii). Find the easure of the suppeent of each of the foowing anges: (iii) 55 (iv) 5 5. Find the suppeent of each of the foowing anges: (iii) 6. Identify which of the foowing pairs of anges are copeentary and which are suppeentary: 65, 5 6, 7 (iii) 0, 50 (iv), 68 (v) 5, 5 (vi) 7, Find the ange which is equa to its copeent. Find the ange which is equa to its suppeent. 8. Two copeentary anges are ( + ) and ( 7), find the vaue of. 9. Two suppeentary anges are ( + 8) and ( 58), find. 0. Two suppeentary anges are in the ratio of : 7, find the anges.. ong two suppeentary anges, the easure of the onger ange is ore than the easure of the saer ange. Find their easures.. If an ange is haf of its copeent, find the easure of anges.. n ange is greater than 5. Is its copeentary ange greater than 5 or equa to 5 or ess than 5?. In the adjoining figure, and are suppeentary. If is decreased, what change shoud take pace in so that both anges sti reain suppeentary?

9 Lines and angles In the adjoining figure, is adjacent to? Give reasons. 6. In the adjoining figure, write pairs of anges which are: verticay opposite anges inear pairs 5 7. Find the vaue of in each of the foowing diagras: (iii) 0 8. Find the vaues of, y and z in each of the foowing diagras: z y y 5 z y z 5 (iii) Pairs F Lines intersecting ines Two ines and are intersecting ines if they have a point in coon. In the adjoining figure, ines and intersect each other at the point. The point is caed the point of intersection.