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lide 1 / 209 lide 2 / 209 New ersey enter for eaching and earning eometry rogressive athematics Initiative his material is made freely available at and is intended for the non-commercial

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lide 1 / 209 lide 2 / 209 New ersey enter for eaching and earning eometry rogressive athematics Initiative his material is made freely available at and is intended for the non-commercial use of students and teachers. hese materials may not be used for any commercial purpose without the written permission of the owners. N maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. ongruent ris lick to go to website: lide 3 / 209 lide 4 / 209 able of ontents lassifying ris Interior ngle heorems xterior ngle heorems Isosceles ri heorem lassifying ris ongruence & ris ongruence ongruence ongruence ongruence ongruence eturn to able of ontents ri ongruence roofs ri oordinate roofs lide 5 / 209 arts of a tri Vertex ide opposite arts of a tri (cont'd) and are adjacent sides ide hypotenuse interior Vertex ide lide 6 / 209 Vertex leg leg leg base leg Vertex (vertices) - points joining the sides of tris In a right tri, the hypotenuse is the side opposite the right. he legs are the 2 sides that form the right. djacent ides - two sides sharing a common vertex In an isosceles tri, the base is the side that is not congruent to the other two sides (legs). If an isosceles tri has 3 congruent sides, it is an equilateral tri. lide 7 / 209 lide 8 / 209 tri is formed by line segments joining three noncollinear points. tri can be classified by its sides and s. lassification by ides efinitions olygon - a closed plane figure composed of line segments quilateral ri - three-sided polygon ides - the line segments that make up a polygon calene Isosceles Vertex (vertices) - the endpoints of the sides cute ri - all s 90 3 congruent sides Obtuse tri - one is between, 90 congruent sides No congruent sides lassification by ngles ight ri - one 90 cute quiangular ri - 3 congruent s quiangular ight Obtuse quilateral ri - 3 congruent sides 3 acute s Isosceles ri - 2 congruent sides calene tri - No congruent sides lide 9 / 209 lassify the tris by sides and s 3 congruent s (also acute) 1 right 1 obtuse lide 10 / 209 xample easure and lassify the tris by sides and s scalene acute equilateral isosceles acute equiangular isosceles obtuse isosceles right isosceles, right lick for for lick lide 11 / 209 isosceles, lick for foracute lick scalene, obtuse lick for for lick lide 12 / 209 xample 1 lassify the tri with the given information: ide lengths: 3 cm, 4 cm, 5 cm quilateral cute Isosceles quiangular calene ight Obtuse easure and lassify the tris by sides and s scalene, obtuse lick for for lick lick for for lick scalene, acute equilateral, acute/equiangular lick for lick for lide 13 / 209 lassify the tri with the given information: ide lengths: 5 cm, 5 cm, 5 cm 3 quilateral cute quilateral cute Isosceles quiangular Isosceles quiangular calene ight calene ight Obtuse Obtuse lide 15 / 209 lide 16 / 209 lassify the tri with the given information: ngle easures: 25, 120, 35 5 quilateral cute quilateral cute Isosceles quiangular Isosceles quiangular calene ight calene ight Obtuse Obtuse lide 17 / 209 lide 18 / 209 lassify the tri with the given information: ide lengths: 3 cm, 4 cm, 5 cm ngle measures: 37, 53, 90 quilateral cute quilateral cute Isosceles quiangular Isosceles quiangular calene ight calene ight Obtuse Obtuse 7 lassify the tri with the given information: ngle easures: 60, 60, 60 6 lassify the tri with the given information: ngle easures: 30, 60, 90 4 lassify the tri with the given information: ide lengths: 3 cm, 2 cm, 3 cm 2 lide 14 / 209 lide 19 / lassify the tri with the given information: ide lengths: 3 cm, 3 cm, 3 cm ngle measures: 60, 60, 60 cute Isosceles calene ight quilateral Isosceles cute quiangular ight Obtuse lide 21 / 209 lide 22 / 209 lassify the tri by sides and s quilateral 11 Isosceles Obtuse cute N quiangular ight Obtuse lide 23 / lide 24 / 209 n isosceles tri is an equilateral tri. 13 n obtuse tri is an isosceles tri. ometimes ometimes lways lways Never Never 12 ight 85 calene Isosceles cute quiangular lassify the tri by sides and s quilateral calene calene quiangular Obtuse quilateral lassify the tri by sides and s 8 lide 20 / 209 lide 25 / 209 tri can have more than one obtuse. 15 tri can have more than one right. rue alse alse rue 14 lide 26 / 209 lide 27 / 209 ach in an equiangular tri measures n equilateral tri is also an isosceles tri rue rue alse alse 16 lide 28 / 209 lide 29 / 209 lide 30 / ri um heorem he measures of the interior s of a tri sum to 180 Interior ngle heorems eturn to able of ontents If you have a tri, then you know the sum of its three interior s is 180 Why is this true? lick here to go to the lab titled, ri um heorem lide 31 / 209 lide 32 / xample: ri um heorem What is the measurement of the missing? ind the measure of the missing heorem 1. he ri um heorem says that the interior s of must sum to = 180 m = and substituting the information from the diagram + 52 = 180 = 128 heck: =180 lide 33 / 209 lide 34 / 209 What is the measurement of the missing? 20 What is the measure of the missing? 57 N x m N = x= lide 35 / In, if m is 84 and m what is the m? (draw a diagram) lide 36 / 209 is 36, o, 53 In, if m is 63 and m what is the m? (draw a diagram) is 12, lide 37 / 209 lide 38 / xample olve for x in the diagram. We can solve more complicated problems using the ri um heorem. (12x+8) rom the ri um heorem (8x-3) 55 5x 55 + (12x+8) + (8x-3) = 180 ubstituting from the diagram 20x + 60 = 180 ombining like terms 20x = 120 Isolating x using inverse operations x=6 What is m m xtension m lick to reveal lide 39 / x olve for x 8x lide 40 / 209 What is the measure of? orollary to ri um heorem he acute s of a right tri are complementary. ince 1. the ri um heorem says the interior s of a tri must sum to 180. o, (the right ) = 90 left between and. olve for x 3x-17 +x+40 int +2x-5 = 180 ecall: two s that add up to 90 are called complementary lick to reveal lide 41 / 209 lide 42 / 209 he measure of one acute of a right tri is five times the measure of the other acute. ind the measure of each acute. 5x x ince this is a right tri, we can use the orollary to the ri um heorem which says the two acute s are complementary. o, x + 5x = 90 (using the ri um 6x = 90 heorem is a little more work) x = 15 One acute is 15 and the other is In a right tri, the two acute s sum to 90 rue alse xample lide 43 / 209 What is the measurement of the missing? 27 olve for x 57 N What are the measures of the hallenge three s? lick to reveal lide 45 / lide 46 / 209 olve for x 29 In the right tri given, what is the measurement of each acute? 2x x 26 lide 44 / 209 What are the measures of the hallenge three s? lick to reveal lide 47 / m 31 o 2 = 1 m 1+m o 3 = m 30 lide 48 / 209 lide 49 / lide 50 / 209 ind the value of x in the diagram X xterior ngle heorems ark your int vertical s! to reveal eturn to able of ontents 20 lide 51 / 209 lide 52 / 209 xterior s are adjacent to the interior s. he adjacent s form a straight line so the sum of the two measures will be 180o xterior s and interior s together form a straight line. he sum of an exterior and an interior is 180 degrees. xterior Interior Interior Interior Interior Interior Interior xterior xterior xterior lide 53 / 209 lide 54 / 209 he adjacent s form a straight line so the sum of the two measures will be 180o Interior xterior Interior Interior Interior xterior Interior Interior he adjacent s form a straight line so the sum of the two measures will be 180o lide 55 / 209 lide 56 / 209 he xterior ngle heorem says : m 1=m +m Interior Interior Interior he measure of the exterior is equal to the sum of the two s that are not adjacent to the exterior. xterior he sum of the interior s of a tri add up to 180 degrees m + m + m = 180o 1 he sum of an interior and an adjacent exterior is 180 degrees 1 roof of the xterior ngle heorem We know the following is true : 1. m + m + m = 180o m + m 1 = 180o 1 2. m + m 1 = 180o his implies that m 1 = m + m and the xterior ngle heorem is proved true lide 57 / 209 lide 58 / 209 xample: Using the xterior ngle heorem xample olve for x using the xterior ngle heorem Xo What is the value of X? o Xo he measure of the exterior is equal to the sum of the two s that are not adjacent to the exterior. 140o = xo + xo x x = = x o, the exterior x = 55 lide 59 / 209 w = w = 125o What does w + x equal? = o Xo We also know what y is 125o? What does x + y have to equal? 180o lide 60 / 209 m 4 = 131 wo 34 he xterior ngle heorem says that the exterior, marked x, is equal to the two nonadjacent interior s. 140 = 2x xample: What are w and x? x = x = x = 55o y m 3 = 53, fill in all the s. 3 50o 4 131o 53o 131o 53o 2 127o 127o 1 78o 78o 180o 180o 49o 49o lide 61 / 209 olve for the exterior, x. 60 Y m 1 = 25 and m 4 = 83 ind m 3 =? 34 x lide 63 / Y ind the value of y in the figure below lide 65 / 209 Y lide 66 / 209 Using the xterior ngles heorem, find the value 38 What is the value of Y? of x Y 37 2 ind the value of x using the xterior ngles heorem? 34 4 lide 64 / Y 33 lide 62 / 209 lide 67 / 209 ind the value of x. 40, what is the value of w? (x + 2) o (3x - 5) 20 bisects w 5o 11 lide 69 / lide 68 / 209 lide 70 / ind the measure of 1. xample ind the missing s in the diagram. 40o eacher Note 1 60o lide 71 / 209 lide 72 / ind the measure of ind the measure of o o 40o o 1 lide 73 / 209 lide 74 / ind the measure of ind the measure of o o 60o 5 60o lide 75 / 209 lide 76 / 209 arts of an Isosceles ri n isosceles tri has at least two congruent sides (an equilateral tri is an isosceles tri w/three congruent sides) If an isosceles tri has exactly two congruent sides, the: - two congruent sides are called legs, - the noncongruent side is called the base, - the two s adjacent to the base are the base s, Isosceles ri heorem vertex leg eturn to able of ontents leg base s he vertex is the opposite the base O it is the included by the legs base lide 77 / 209 lide 78 / 209 xamples: 3. ase ngles heorem () If two sides of a tri are congruent, the s opposite them are congruent. If ind the values of x & y in the isosceles tri below. x = 44; ase ngles are ongruent y, then orollary to (3) 44 x ind the values of x & y in the isosceles tri below. If a tri is equilateral, then it is equiangular. x = y; ase ngles are ongruent 52 y = 180; ri um h. y + 88 = 180 y = 92 x y x + y + 52 = 180; ri um h. x + x + 52 = 180; ubstitution 2x + 52 = 180 2x = 128 x = 64 1 lide 79 / lide 80 / 209 olve for the measurements of the s x and y 47 olve for x and y. 72 y 35 x y lide 81 / 209 lide 82 / he vertex of an isosceles tri is 38. What is the measure of each base? What are the measurements of the base s? lide 83 / 209 lide 84 / onverse of the ase ngles heorem What is the measurement of? If two s of a tri are congruent, then the sides opposite them are congruent. If 50, then orollary to onverse of the (4) If a tri is equiangular, then it is equilateral x lide 85 / lassify the tri by sides and s equilateral equiangular equilateral equiangular isosceles acute isosceles acute scalene obtuse scalene obtuse right 40o 7 right lide 87 / 209 lide 88 / 209 lassify the tri by sides and s 54 lassify the tri by sides and s equilateral equiangular equilateral equiangular isosceles acute isosceles scalene obtuse scalene obtuse right 3 113o 3 acute right lide 89 / 209 lide 90 / 209 xample ind the value of x and y 4. wo adjacent s whose non-shared sides form a straight line are a linear pair. y 1. irst, consider the top tri. he 3 marks indicate this is an equilateral tri 2. rom the orollary to the (3), we know that an equilateral tri is also equiangular lassify the tri by sides and s 51 lide 86 / he supplement to 60 is 120 ( = 180 ) x 6. Using the ase ngles heorem (3) and the ri um theorem (1), we can determine x y 3. ince the ri um heorem (1) says the interior s must sum to 180, y = 60. x x + x = 180 2x = 180 2x = 60 x = x x x lide 91 / 209 What is the value of y? What is the value of x? lide 92 / 209 y 50 lide 93 / x lide 94 / 209 olve for x in the diagram. 3 2/3 14 3x ongruence & ris eturn to able of ontents lide 95 / 209 lide 96 / 209 ongruence wo figures are congruent if they have the exact size and shape (hey are similar if they have the same shape, but a different size) xample he two tris are congruent 1) a congruence statement 2) identify all congruent corresponding parts ongruent figures have a correspondence between their s and sides where pairs of corresponding s are congruent and pairs of corresponding sides are congruent. O N, write: O N lide 97 / 209 lide 98 / 209 orresponding art lide 99 / 209 lide 100 / roblem What is the corresponding part to (If you need, draw a diagram) orresponding ides ~ = eacher Notes orresponding ngles lide 101 / 209 What is the corresponding part to lide 102 / What is the corresponding part to ~ = ~ = 59 orresponding ide ngle Write a congruence statement for the two tris ~ V = X ~ = ~ V = V ~ = 62 XZ X ZX ZX X V XWZ ZWX WXZ ZXW W Z X Y What else can be marked congruent? Z lide 105 / 209 lide 106 / hird ngles heorem (2 ind the value of x. x+ 40 Y ) U X 1) rom the hird ngle heorem (5), we know m = m Y V W xample If two s of a tri are congruent to two s of another tri, then the third s are congruent. ~ XYZ = omplete the congruence statement 61 lide 104 / 209 lide 103 / 209 3) ubstitute to find x an you give a reason for why this might be true? If the sum of the interior s is 180 o and both sets of s are the same, then the third s will have the same measure. xample: m = m V = 40o & m = m U = 80o lick to reveal degrees, then m = m = 60 o. 2) he m is easy to find with the ri um heorem (1), lide 107 / 209 What is the measurement of 64 olve for x 117 I 32 I lide 108 / 209 (2x+14) lide 109 / lide 110 / roperties of ongruent ris ind the value of x ) ymmetric roperties of ongruent ris (3 x eflexive roperty of ongruent ris very tri is congruent to itself ransitive roperty of ongruent ris lide 111 / 209 lide 112 / 209 ongruence rom the ongruence and ris section, you learned that two tris are congruent if the 3 corresponding pairs of sides and the 3 corresponding pairs of s are congruent. owever, we do not always need all 6 pieces of information to prove 2 tris congruent. eturn to able of ontents lide 113 / 209 ostulate: lide 114 / 209 xample ide-ide-ide () ongruence If three sides of one tri are congruent to three sides of another tri, then the two tris are congruent. olution: he congruence marks on the sides show that: lick here to go to the lab titled, ri ongruence lide 115 / 209 lide 116 / 209 xample 66 he congruence statement is ~= rue ~ = 68 ~=? U lide 119 / 209 lide 120 / 209 Included : the made by two lines with a common vertex 41 ongruence Included side: the side between two s eturn to able of ontents 3 4 alse lide 118 / 209 U rue You need to be very careful that you get the corresponding congruent intparts in the correct order is not congruent to lide 117 / alse lide 121 / 209 lide 122 / 209 ostulate: xample ide-ngle-ide () ongruence N If two sides and the included of one tri are congruent to two sides and the included of a second tri, then the two tris are congruent. 1 2 O Is there any information you can fill in? o, listing the corresponding congruent parts: lick here to go to the lab titled, ri ongruence lide 123 / 209 lide 124 / 209 n i ts 200 What is the included of the given sides of the tri?, sides and 5u 69 Why Not? ove the side with the length of 2 units and create a tri. 2 units int: raw the tri! an a different tri be made than the first one made? 5u n i ts 200 lide 125 / 209 Yes Is ~= XYZ by? Yes No 71 X No 5 V Why? Y 10 ist the congruent parts of the tris below. Is ~ = V? 70 lide 126 / Z lide 127 / 209 Using, what information do you need to show ~= ~ = ~ = Not congruent lide 129 / 209 lide 130 / 209 What type of congruence exists between the two tris? 75 What type of congruence exists between the two tris? Not congruent Not congruent 74 What type of congruence exists between the two tris? lide 131 / 209 What type of congruence exists between the two tris? 77 What type of congruence exists between the two tris? Not congruent Not congruent 76 lide 132 / 209 ~= ~ = lide 128 / 209 lide 133 / 209 What type of congruence exists between the two tris? Not congruent ongruence eturn to able of ontents lide 135 / 209 lide 136 / 209 ostulate: xample ngle-ide-ngle () ongruence If two s and the included side of one tri are congruent to two s and the included side of another tri, then the two tris are congruent U Vertical s are congruent V lick here to go to the lab titled, ri ongruence lide 137 / 209 lide 138 / What is the included side for X and W? YX irst: what data is given to you? YW XW X econd: if it is not already marked, check and mark the diagram with that information, hird: check your congruence postulates - what piece of information are you missing (side/) and where does it need to be for your chosen congruence? 78 lide 134 / 209 Y W lide 139 / 209 What is the included side for X and Y 81 XW What piece of information do we need to have congruence between the two tris? N YX X YW W Y lide 141 / 209 lide 142 / 209 What piece of information do we need to have congruence between the two tris? 83 vertical s included s lide 143 / 209 congruent lide 144 / 209 What type of congruence exists between the two tris? Not congruent U 84? Why is 82 O 80 lide 140 / 209 When you have overlapping figures that share sides and/or s, marking the diagram with the given information & pulling the tris apart (when needed) makes it much easier to understand the problem. lide 145 / 209 What type of congruence exists between the two tris? 86 What type of congruence exists between the two tris? Not congruent lide 148 / Not congruent Not congruent ints: tothe revealtris apart! ull to reveal ark the congruent parts! re there any common sides/s (look for to reveal letters that repeat)? int lick vertical eveal tlick thetointersection of two lines you always have s. lide 150 / 209 What type of congruence exists between the two tris? 90 What type of congruence exists between the two tris? Not ongruent Not ongruent N lide 149 / What type of congruence exists between the two tris? What type of congruence exists between the two tris? int ark the diagram with the given information. e careful you don't always use all to reveal information lide 147 / 209 N ints: tothe revealtris apart! ull to reveal ark the congruent parts! re there any common sides/s (look for to reveal letters that repeat)? Not congruent lide 146 / 209 int: to reveal ark the given information into your diagram. Identifying vertical s plays an important part. lide 151 / 209 lide 152 / 209 heorem (7): ngle-ngle-ide () ongruence If two s and the nonincluded side of one tri are congruent to two s and the corresponding nonincluded side of another tri, then the two tris are congruent. ongruence U eturn to able of ontents lide 153 / lide 155 / 209 lide 156 / 209 tris? Not ongruent tris? 92 o, by, congruence statement? ince follows from, is a theorem rather than a postulate he ri um heorem (1) allows us to find the measurement of the third in each tri ( )= ) ark your diagram: lide 154 / 209 xample Why is a heorem? iven two tris: V Not ongruent lide 157 / tris? W Not ongruent tris? Not ongruent lide 159 / 209 lide 160 / 209 tris? Not ongruent tris? Not ongruent lide 161 / 209 lide 162 / 209 tris? 98 tris? Not ongruent Not ongruent 93 lide 158 / 209 lide 163 / 209 lide 164 / 209 heorem (8): ypotenuse-eg () ongruence If the hypotenuse and a leg of one right tri are equal to the corresponding hypotenuse and leg of another right tri, then the two tris are congruent. ongruence eturn to able of ontents O N If you have a right tri, make sure you check if applies lide 165 / 209 lide 166 / 209 xample Why does ongruence work? ecall another theorem for right tris: ythagorean heorem: c2 = a2 + b2 c re the two

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