Sect 8.1 Lines and Angles

Transcription

1 7 Sect 8. Lines and nges Objective a: asic efinitions. efinition Iustration Notation point is a ocation in space. It is indicated by aking a dot. Points are typicay abeed with capita etters next to the dot. Point and point ine is deterined by two different points and extends infinitey in two directions. ine can be abeed by sa case etter or by two different points. If a picture contains two or ore ines, subscripts can be used to denote the ines. E F 2 Line or ine or and EF or ines and 2 ray is deterined by two different points and extends infinitey in one direction. Ray or Ray or ine segent is deterined by two different points and extends in no direction. The two different points are caed endpoints. The distance between the endpoints is caed the ength of the ine segent. ft Line segent or. The ength of or = ft. Note: refers to the ine segent and refers to the ength. Sove the foowing: Ex Given = 6 inches and = 9 inches in the diagra beow, find.

2 76 Since =, then = 6 9 = 7 inches. Objective b & c: Understanding types of anges. efinition Iustration Notation Two rays that share a coon endpoint for an ange. Rays and for ange. The coon endpoint,, is caed the vertex of the ange and the two rays are caed the sides of the ange. n ange is easured using a too caed a protractor. protractor is arked in equa increents caed degrees. There are 80 in an ange whose sides for a straight ine. Vertex 80 Sides nge or Soeties its usefu to use a three points to denote an ange, i.e., ange or. The vertex is aways the idde etter in this notation. is a straight ange. The easure of, denoted, is 80. We can say: = 80 n acute ange is an ange whose easure is between 0 and 90. is an acute ange since < 90. right ange is an ange whose easure is exacty 90. enotes a right ange is a right ange since = 90. n obtuse ange is an ange whose easure is between 90 and 80. is an obtuse ange since > 90 and < 80. straight ange is an ange whose easure is exacty is a straight ange since = 80.

3 77 efinition Iustration Notation refex ange is an ange whose easure is between 80 and 360. E E is a refex ange since E > 80 and E < 360. Two anges are caed adjacent anges if they are side by side and have the sae vertex. and are adjacent anges since they are side by side and have the point as their vertex. Two acute anges are caed opeentary anges if the su of their easures is 90. Each is caed the copeent of the other. and are copeentary anges since + = 90. Two anges easuring ess than 80 are caed Suppeentary anges if the su of their easures is 80. Each is caed the suppeent of the other. X W Y Z XYW and WYZ are suppeentary anges since XYW + WYZ = 80 Note that copeentary and suppeentary anges are not aways adjacent anges. Sove the foowing: Ex. 2 Given that = 36, find the easure of the copeent and the suppeent of. Since copeentary anges tota 90, then the easure of the copeent of is 90 = = 4. Since suppeentary anges tota 80, then the easure of the suppeent of is 80 = = 44.

4 Ex. 3 Given the diagra beow, find the easure of the copeent and the suppeent of Since copeentary anges have to be acute anges (ess than 90 ) and the > 90, it is not possibe for to have a copeent. Hence, has no copeent. Since suppeentary anges add up to 80, then the easure of the suppeent of is 80 = 80 9 = efinition Iustration Notation Two or ore ines are caed intersecting if they cross, eet, or share a coon point. E The two ines, and, intersect at point E. Two intersecting ines for vertica anges. These anges coe in pairs which have equa easures. a d b c The vertica pairs are: a & c; b & d. a = c b = d Ex. 4 Find the vaue of x and y in the foowing diagra: X y 34 Since x and 34 are vertica anges, they are equa. Thus, x = 34 y is a suppeentary ange to x, so y = 80 x = = 46

5 79 Objective d: Parae and Perpendicuar Lines. efinition Iustration Notation Two intersecting ines (or rays or ine segents) that for right anges (90 ) are caed perpendicuar ines. Line is perpendicuar to ine or. Two or ore ines are caed parae ines if they never "cross", "eet", or have no points in coon. 2 Lines and 2 are parae or 2. transversa is a ine that intersects two or ore different ines at different points. If the transversa intersects two parae ines, it produces a tota eight anges with soe specia properties. t Line t is the transversa since it intersects & 2 at two different points. If, then = 4 = = 8 2 = 3 = 6 = 7 Ex. Given that b = 63 in the diagra beow, find a the other anges. ssue that. d c a b h g Since f = h = d = b, then f = h = d = 63. Since c and b are suppeentary anges, c = 80 b = = 7. Hence, g = e = a = c = 7. e f

6 80 efinition Iustration Notation orresponding anges are those anges that share the sae ocation in their respective intersections. If two ines that are cut by a transversa are parae, then the corresponding anges are equa There are four pairs of corresponding anges. They are: & ; 2 & 6; 3 & 7; 4 & 8. If, then = ; 2 = 6; 3 = 7; 4 = 8. ternate interior anges are those anges inside the two ines sharing the opposite ocations, i.e., top - eft and botto - right. If the two ines are parae, then the aternate interior anges are equa There are two pairs of aternate interior anges. They are: 3 & ; 4 & 6. If, then 3 = ; 4 = 6. ternate exterior anges are those anges outside the two ines sharing the opposite ocations, i.e., top - eft and botto - right. If the two ines are parae, then the aternate exterior anges are equa There are two pairs of aternate exterior anges. They are: & 7; 2 & 8. If, then = 7; 2 = 8. Ex. 6 Using the diagra beow, fi in the foowing banks. ssue that. d c a b h g e f

7 8 a) a and e are anges. b) d and f are anges. c) g and e are anges. d) h and b are anges. e) d and c are anges. a) a and e are corresponding anges. b) d and f are aternate exterior anges. c) g and e are vertica anges. d) h and b are aternate interior anges. e) d and c are suppeentary anges. Ex. 7 In the diagra beow, find a the issing anges. ssue Since and 6 are vertica anges, = 6. Since and 43 are aternate exterior anges, = 43. ut and 3 are vertica anges, thus 3 = 43., 2, and 6 for a straight ine and = 43, hence 80 = Soving for 2 yieds 2 = 8. Yet, 2 and 4 are vertica anges which eans 4 = 8. Since corresponds to 8, 8 = = 6. ut 6 and 8 are vertica anges, so 6 = 6. 7 and 8 are suppeentary anges, therefore 7 = 80 8 = 80 6 = and 9 are vertica anges and thus 9 = and 43 are vertica anges which eans 0 = 43. Since 0 and are suppeentary anges, = = 37. ut, and 2 are vertica anges, so 2 = 37. Therefore: = 43 2 = 8 3 = 43 4 = 8 = 6 6 = 6 7 = 24 8 = 6 9 = 24 0 = 43 = 37 2 = 37