Sec.. ngles and ngle Measure LSSIFITION OF NGLES Secion. ngles and ngle Measure. Righ angles are angles which. Sraigh angles are angles which measure measure 90. 80. Every line forms a sraigh angle. 90 80 3. cue angles are angles which have. Obuse angles are angles which measure a (posiive) measure less han 90. greaer han 90 and less han 80. 5 50 JENT NGLES 5. djacen angles are any wo angles ha share a common side, forming an even larger angle; all hree angles,,, and, have he same verex, and he shared side mus be in he inerior of he larger angle. ll of he poins on he ray (excep he endpoin, ) are in he inerior of he larger angle. In his case, we can say ha and are adjacen angles. Togeher, hey form he larger angle,. In fac, he measure of he larger angle is he sum of he measures of he wo smaller angles: m = m + m. Likewise, he measure of one of he smaller angles is he difference beween he measure of he larges angle and he measure of he oher smaller angle: m = m m. Sec.. ngles and ngle Measure Rober H. Prior, 06 opying is prohibied.
Sec.. ngles and ngle Measure Example : Given he diagram below, find m given he measures of and. a) m = 80 and m = 5 b) m = 85 39 5 and m = 5 6 nswer: Use m = m + m a) m = 80 + 5 = 5 b) m = 85 39 5 + 5 6 85 39 5 djus he seconds, hen he minues, o where hey + 5 6 are boh less han 60: 36 63 77 36 6 7 37 0 7 Example : Given he diagram below, find m given he measures of and. a) m = 0 and m = 85 b) m = 8 39 and m = 53 5 3 nswer: Use m = m m a) m = 0 85 = 55 b) m = m m m = 8 39 53 5 3 Noice ha he minues and seconds change as we borrow: Make ino 60 Make ino 60 Now subrac 8 39 8 38 7 7 98 7 53 5 3 53 5 3 53 5 3 m = 7 53 Sec.. ngles and ngle Measure Rober H. Prior, 06 opying is prohibied.
Sec.. ngles and ngle Measure 3 SUPPLEMENTRY N OMPLEMENTRY NGLES 6. Supplemenary angles are any wo angles wih measures ha add o 80. Supplemenary angles can be adjacen, bu hey don have o be. 5 35 5 Z 35 Y X We say ha and are supplemenary. We also say ha is he supplemen of, and vice-versa. XYZ is he supplemen of. m + m = 80 m + m XYZ = 80 m = 80 m. m XYZ = 80 m. Example 3: and XYZ are supplemenary angles. Given m, find m XYZ. a) m = 0 b) m = 5 8 5 nswer: Use m XYZ = 80 m a) m XYZ = 80 0 = 0 b) m XYZ = 80 5 8 5 Make ino 60 Make ino 60 Now subrac 80 00 00 79 60 00 79 59 60 5 8 5 5 8 5 5 8 5 m XYZ = 5 3 5 ngles ha are boh adjacen and supplemenary form a sraigh angle, a line: m = 80 Sec.. ngles and ngle Measure 3 Rober H. Prior, 06 opying is prohibied.
Sec.. ngles and ngle Measure 7. omplemenary angles are any wo angles wih measures ha add o 90. omplemenary angles can be adjacen, bu hey don have o be. 55 35 We can say ha and are complemenary, or ha is he complemen of. X 55 35 Y Z Likewise, XYZ is he complemen of. Example : and XYZ are complemenary angles. Given m, find m XYZ a) m = 53 b) m = 5 8 5 nswer: Use m XYZ = 90 m a) m XYZ = 90 53 = 37 b) m XYZ = 90 5 8 5 Make ino 60 Make ino 60 Now subrac 90 00 00 89 60 00 89 59 60 5 8 5 5 8 5 5 8 5 m XYZ = 35 3 5 In Secion.3 we discuss riangles and prove ha he sum of he hree angles in a riangle is 80. his poin hough, we will assume ha o be rue for he purposes of his nex, brief discussion: In a righ riangle (a riangle wih one righ angle), he wo acue angles are complemenary. This is easy o demonsrae: m + m + m = 80 90 + m + m = 80 m + m = 90 So, and are complemenary. Sec.. ngles and ngle Measure Rober H. Prior, 06 opying is prohibied.
Sec.. ngles and ngle Measure 5 NGLE ISETORS n angle bisecor splis an angle ino wo smaller, congruen angles. Said more formally, an angle bisecor is a ray ha begins a he angle s verex and passes hrough an inerior poin so ha i forms wo smaller, congruen angles. Example 5: Ray bisecs. Given he measure of, find he measure of. Wrie answers in MS form. a) m = 8 b) m = 53 c) m = 65 3 0 d) m = 7 8 3 nswer: Use m = m a) m = 8 = b) m = 53. 53 is an odd number, bu we can make i even by borrowing from i and making m = 5 60. These are boh divisible by : m = 53 = (5 60 ) = 6 30 c) m = 65 3 0. oh 65 and 3 are odd numbers, bu we can make hem even by borrowing degree (or minue) from each o make hem boh even: 65 3 0 = 6 03 0 = 6 0 70 We can now easily divide each of hese by : m = (6 0 70 ) = 3 5 35. d) The number of seconds is odd, 3, and here is nohing ha will make i even. ecause he oher values are already even, we can divide as follows: m = (7 8 3 ) = 36 5.5. Sec.. ngles and ngle Measure 5 Rober H. Prior, 06 opying is prohibied.
6 Sec.. ngles and ngle Measure VERTIL NGLES When wo lines inersec in a plane, several pairs of angles are formed: djacen angles are supplemenary (forming a sraigh line) and 3 non-adjacen angles are verical angles. I can easily be shown ha verical angles are congruen o each oher: 30 30 30 3 3 50 50 3 50 Given: m = 30 3 is supplemenary o so Likewise, is supplemenary m 3 = 80 30 = 50. o, so m is also 50. Therefore, m = m 3, and 3. I follows ha m is also 30. is verical o, and. Example 6: nswer: In he diagram of inersecing lines, given m = 7, find he measures of he oher hree angles. m 3 = m and and are boh supplemenary o, so, a) m 3 = 7 b) m = 80 7 = 53 c) m = 53. PERPENIULR LINES N LINE SEGMENTS Two lines in a plane ha inersec o form four righ angles are said o be perpendicular. If wo lines or line segmens, and, are perpendicular, hen we may wrie. Sec.. ngles and ngle Measure 6 Rober H. Prior, 06 opying is prohibied.
Sec.. ngles and ngle Measure 7 PRLLEL LINES N TRNSVERSLS Two lines in a plane, and, are parallel if hey never inersec. We someimes use arrows going in he same direcion o indicae ha wo lines (or line segmens) are parallel o each oher. line ha inersecs wo (or more) parallel lines is called a ransversal. The ransversal and he parallel lines form a oal of eigh angles. There are many pairs of congruen angles and many pairs of supplemenary angles. 5 6 7 8 3 ransversal If he ransversal is no perpendicular o he parallel lines, hen four acue angles and four obuse angles formed. This leads us o he following: (i) (ii) all of he acue angles are congruen o each oher and all of he obuse angles are congruen o each oher; each acue angle is supplemenary o each obuse angle. To alk abou hese eigh angles easily, we refer o hem in his way (fill in he blanks): (i) lef side of ransversal: (ii) righ side of ransversal: (iii) direcly above a parallel line: 5 6 7 8 3 ransversal (iv) direcly below a parallel line: (v) inerior angles: (vi) exerior angles: Sec.. ngles and ngle Measure 7 Rober H. Prior, 06 opying is prohibied.
8 Sec.. ngles and ngle Measure In his seing, a pair of corresponding angles are angles ha are on he same side of he ransversal (lef or righ) and are eiher boh above or boh below he parallel lines. corresponding angles For example, and 5 are a pair of corresponding angles because hey are boh o he above-lef angles. 5 orresponding angles are congruen. lernae inerior angles are on opposie sides of he ransversal (one on he lef, one on he righ) and are beween he parallel lines: below and above. For example, and 5 are alernae inerior angles. alernae inerior angles line 5 lernae inerior angles are congruen. lernae exerior angles are on opposie sides of he ransversal (one on he lef, one on he righ) and are ouside of he parallel lines: above and below. alernae exerior angles For example, and 8 are alernae inerior angles. lernae exerior angles are congruen. 8 s we already know, all of he verical angles are congruen; for example and 6 7. lso, all of he adjacen angles form lines (sraigh angles), so each pair of adjacen angles is supplemenary. Sec.. ngles and ngle Measure 8 Rober H. Prior, 06 opying is prohibied.