Geometry. Slide 1 / 190 Slide 2 / 190. Slide 4 / 190. Slide 3 / 190. Slide 5 / 190. Slide 5 (Answer) / 190. Angles

Transcription

1 Slide 1 / 190 Slide 2 / 190 Geometry ngles Slide 3 / 190 Table of ontents click on the topic to go to that section Slide 4 / 190 Table of ontents for Videos emonstrating onstructions ngles ongruent ngles ngles & ngle ddition Postulate Protractors Special ngle Pairs Proofs Special ngles ngle isectors ocus & ngle onstructions ngle isectors & onstructions PR Released Questions ongruent ngles ngle isectors click on the topic to go to that video Slide 5 / 190 Slide 5 (nswer) / 190 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: onstruct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: ttend to precision. MP7: ook for & make use of structure. dditional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: onstruct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: ttend to precision. MP7: ook for & make use of structure. Math Practice dditional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

2 Slide 6 / 190 Slide 7 / 190 ngles ngles efinition 8: plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. Whenever lines, rays or segments in a plane intersect, they do so at an angle. xº Return to Table of ontents Slide 8 / 190 ngles The measure of angle is the amount that one line, one ray or segment would need to rotate in order to overlap the other. Slide 9 / 190 ngles In this course, angles will be measured with degrees, which have the symbol º. In this case, Ray would have to rotate through an angle of x degrees in order to overlap Ray. or a ray to rotate all the way around from ray, as shown, back to ray would represent a 360º angle. xº xº Slide 10 / 190 Measuring angles in degrees Slide 11 / 190 Measuring angles in degrees The use of 360 degrees to represent a full rotation back to the original position is arbitrary. The use of 360 for a full rotation is thought that it come from ancient abylonia, which used a number system based on 60. ny number could have been used, but 360 degrees for a full rotation has become a standard. 360º Their number system may also be linked to the fact that there are 365 days in a year, which is pretty close to is a much easier number to work with than 365 since it is divided evenly by many numbers. These include 2, 3, 4, 5, 6, 8, 9, 10 and 12.

3 Slide 12 / 190 Right ngles efinition 10: When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. Slide 13 / 190 Right ngles ourth Postulate: That all right angles are equal to one another. Not only are adjacent right angles equal to each other as shown below, all right angles are equal, even if they are not adjacent, for instance, all three of the below right angles are equal to one another. The only way that two lines can intersect as shown and form equal adjacent angles, such as the angles shown here where m = m, is if they are right angles, 90º. xº xº xº xº 90º Slide 14 / 190 Right ngles This definition is unchanged today and should be familiar to you. Perpendicular lines, segments or rays form right angles. Slide 15 / 190 Right ngles There is a special indicator of a right angle. If lines intersect to form adjacent equal angles, then they are perpendicular and the measure of those angles is 90º. It is shown in red in this case to make it easy to recognize. 90º When perpendicular lines meet, they form equal adjacent angles and their measure is 90º. Slide 16 / 190 Obtuse ngles efinition 11: n obtuse angle is an angle greater than a right angle. Slide 17 / 190 cute ngles efinition 12: n acute angle is an angle less than a right angle. 135º 45º

4 Slide 18 / 190 Straight ngle definition that we need that was not used in The lements is that of a "straight angle." That is the angle of a straight line. 2 questions to discuss with a partner: Is this an acute or obtuse angle? xplain why. What is the degree measurement of the angle? Slide 18 (nswer) / 190 Straight ngle definition that we need that was not used in The lements is that of a "straight angle." That is the angle of a straight line. This is a type of obtuse angle. It is greater than 90º. 180º nswer 2 questions to discuss with a partner: Questions on this slide Is this an acute or obtuse angle? xplain why. address MP3 What is the degree measurement of the angle? Slide 19 / 190 Reflex ngle Slide 20 / 190 ngles nother modern definition that was not used in The lements is that of a "reflex angle." That is an angle that is greater than 180º. 235º In the next few slides we'll use our responders to review the names of angles by showing angles from 0º to 360º in 45º increments. ngles can be of any size, not just increments of 45º, but this is just to give an idea for what a full rotation looks like. This is also a type of obtuse angle. Slide 21 / This is an example of a (an) angle. hoose all that apply. acute Slide 21 (nswer) / This is an example of a (an) angle. hoose all that apply. acute obtuse right 0º obtuse right nswer 0º reflex reflex straight straight

5 Slide 22 / This is an example of a (an) angle. hoose all that apply. Slide 22 (nswer) / This is an example of a (an) angle. hoose all that apply. acute obtuse right reflex 45º acute obtuse right reflex nswer 45º straight straight Slide 23 / This is an example of a (an) angle. hoose all that apply. Slide 23 (nswer) / This is an example of a (an) angle. hoose all that apply. acute acute obtuse right obtuse right nswer reflex straight 90º reflex straight 90º Slide 24 / This is an example of a (an) angle. hoose all that apply. Slide 24 (nswer) / This is an example of a (an) angle. hoose all that apply. acute acute obtuse right obtuse right nswer reflex 135º reflex 135º straight straight

6 Slide 25 / This is an example of a (an) angle. hoose all that apply. Slide 25 (nswer) / This is an example of a (an) angle. hoose all that apply. acute acute obtuse right reflex 180º obtuse right reflex nswer and 180º straight straight Slide 26 / 190 Slide 26 (nswer) / This is an example of a (an) angle. hoose all that apply. 6 This is an example of a (an) angle. hoose all that apply. acute obtuse right 235º acute obtuse right nswer and 235º reflex straight reflex straight Slide 27 / This is an example of a (an) angle. hoose all that apply. Slide 27 (nswer) / This is an example of a (an) angle. hoose all that apply. acute acute obtuse right 270º obtuse right nswer 270º and reflex reflex straight straight

7 Slide 28 / This is an example of a (an) angle. hoose all that apply. Slide 28 (nswer) / This is an example of a (an) angle. hoose all that apply. acute acute obtuse right 315º obtuse right nswer 315º and reflex straight reflex straight Slide 29 / 190 Slide 29 (nswer) / This is an example of a (an) angle. hoose all that apply. 9 This is an example of a (an) angle. hoose all that apply. acute acute obtuse right reflex 360º obtuse right reflex nswer 360º and straight straight Slide 30 / 190 Slide 31 / 190 Naming ngles Interior of ngles n angle has three parts, it has two sides and one vertex, where the sides meet. ny angle with a measure of less than 180º has an interior and exterior, as shown below. In this example, the sides are the rays and and the vertex is. side vertex θ side xterior θ Interior

8 Slide 32 / 190 Naming ngles Slide 33 / 190 Naming ngles n angle can be named in three different ways: The angle shown can be called,, or. y its vertex ( in the below example) y a point on one leg, its vertex and a point on the other leg (either or in the below example) leg Or by a number or a symbol placed inside the angle (e.g. Greek letter, θ, in the figure) vertex θ leg When there is no chance of confusion, the angle may also be identified by its vertex. The sides of are rays and 32 The measure of is 32 degrees, which can be rewritten as m = 32º. Slide 34 / 190 Naming ngles Slide 34 (nswer) / 190 Naming ngles Using the vertex to name an angle doesn't work in some cases. Why would it be unclear to use the vertex to name the angle in the image below? Using the vertex to name an angle doesn't work in some There cases. is Why more would than it be 1 unclear angle to use the vertex with to name as the its angle vertex. in the image below? nswer There are 3 angles. ow many angles do you count in the image? θ α ow many angles Questions do on this slide you count in the image? address MP3 θ α Slide 35 / 190 Naming ngles Slide 35 (nswer) / 190 Naming ngles What other ways could you name, and in the case below? (using the side - vertex - side method) What other ways could you, name, and and in the case below? (using the side - vertex - side method) θ α nswer θ, α, θ + α Questions on this slide address MP2 θ α ow could you name those 3 angles using the letters placed inside the angles? ow could you name those 3 angles using the letters placed inside the angles?

9 Slide 36 / 190 Intersecting ines orm ngles Slide 37 / 190 Intersecting ines orm ngles When an angle is formed by either two rays or segments with a shared vertex, one included angle is formed. Shown as θ in the below diagram to the left. When two lines intersect, 4 angles are formed, they are numbered in the diagram below to the right. These numbers used have no special significance, but just show the 4 angles. When rays or segments intersect but do not have a common vertex, they also create 4 angles. θ θ Slide 38 / Two lines meet at more than one point. lways Sometimes Slide 38 (nswer) / Two lines meet at more than one point. lways Sometimes Never Never nswer Slide 39 / 190 Slide 39 (nswer) / n angle that measures 90 degrees is a right angle. 11 n angle that measures 90 degrees is a right angle. lways lways Sometimes Never Sometimes Never nswer

10 Slide 40 / n angle that is less than 90 degrees is obtuse. Slide 40 (nswer) / n angle that is less than 90 degrees is obtuse. lways lways Sometimes Never Sometimes Never nswer Slide 41 / n angle that is greater than 180 degrees is referred to as a reflex angle. Slide 41 (nswer) / n angle that is greater than 180 degrees is referred to as a reflex angle. lways lways Sometimes Never Sometimes Never nswer Slide 42 / 190 Slide 43 / 190 ongruence We learned earlier that if two line segments have the same length, they are congruent. ongruent ngles lso, all line segments with the same length are congruent. b a Return to Table of ontents re these two segments congruent? xplain why your answer is true.

11 Slide 43 (nswer) / 190 ongruence We learned earlier that if two line segments have the same length, they are congruent. Slide 44 / 190 ongruence ow about two angles which are formed by two rays with common vertices. re all of those congruent? Math Practice lso, all line segments with the same length are congruent. re these two segments congruent? xplain why your answer is true. a Questions on this slide address MP6 & MP3 b What would have to be the same for each of them to be congruent? Slide 44 (nswer) / 190 ongruence ow about two angles which are formed by two rays with common vertices. re all of those congruent? What would have to be the same for each of them to be congruent? Questions on this slide address MP6 & MP3 Math Practice Slide 45 / 190 ongruence If two angles have the same measure, they are congruent since they can be rotated and moved to overlap at every point. Slide 46 / 190 ongruence owever, if their included angles do not have equal measure, they cannot be made to overlap at every point. or angles to be congruent, they need to have equal measures. Slide 46 (nswer) / 190 owever, if their included angles do not have equal measure, they cannot be made to overlap at every point. Math Practice ongruence or angles to be congruent, they need to have equal measures. Questions on this slide address MP6 & MP3 re these two angles congruent? xplain why your answer is true. re these two angles congruent? xplain why your answer is true.

12 Slide 47 / 190 ongruence owever, if their included angles do not have the same measure, they cannot be made to overlap at every point. or angles to be congruent, they need to have the same measure. Slide 48 / 190 ongruent ngles One way to indicate that two angles have the same measure is to label them with the same variable. or instance, labeling both of these angles with xº indicates that they have the same measure. ere you can see clearly when we rotate the two angles from the previous slide, they do not have the same angle measure. xº xº Slide 49 / 190 ongruent ngles nother way to show angles are congruent is to mark the angle with a line. If there are 2 equal sets of angles, the second set could be marked with two lines. 14 Is congruent to? Yes No Slide 50 / 190 Slide 50 (nswer) / 190 Slide 51 / Is congruent to? 15 ongruent angles have the same measure. Yes No nswer YS lways Sometimes Never

13 Slide 51 (nswer) / ongruent angles have the same measure. lways Sometimes Never nswer Slide 52 / and are. ongruent Not ongruent annot be determined 16 and are. Slide 52 (nswer) / and are. Slide 53 / 190 ongruent Not ongruent nswer annot be determined ongruent Not ongruent annot be determined Slide 53 (nswer) / 190 Slide 54 / and are. 18 and are congruent. ongruent Not ongruent nswer annot be determined True alse annot be determined

14 Slide 54 (nswer) / 190 Slide 55 / and are congruent. True alse nswer annot be determined 19 and are congruent. True alse Slide 55 (nswer) / 190 Slide 56 / and are congruent. True alse nswer S ngles & ngle ddition Postulate Return to Table of ontents Slide 57 / 190 djacent ngles Slide 58 / 190 ngle ddition Postulate djacent angles share a vertex and a side. The two angles are side by side, or adjacent. In this case, ngle is adjacent to ngle. The angle addition postulate says that the measures of two adjacent angles add together to form the measure of the angle formed by their exterior rays. In this case, m = m + m

15 Slide 58 (nswer) / 190 ngle ddition Postulate Slide 59 / 190 ngle ddition Postulate The angle addition postulate xplain & emphasize the importance says that the measures and of differences between the two adjacent angles notations add and the symbols when together to form the naming & giving the measurements measure of the angle of angles formed by their exterior e.g. means "angle " rays. whereas m means "the measurement of angle " Math Practice MP6 In this case, m = m + m urther, it says that if any point lies in the interior of an angle, then the ray connecting that point to the vertex creates two adjacent angles that sum to the original angle. If lies in the interior of ngle then m + m = m Which yields the same result we had before. m = m + m Slide 60 / 190 ngle ddition Postulate xample Slide 60 (nswer) / 190 ngle ddition Postulate xample m PQR = 58º m PQS = 32 m SQR = 26 Q P S R Question on this slide addresses m PQS = 32 MP1 nswer m SQR = 26 dditional Q's that could be used: What information 32 do you have? (MP1) What do you need to find? 26 (MP1) ow is this problem related to the ngle ddition Postulate? (MP7) Q P S R What's the measure of PQR? What's the measure of PQR? Slide 61 / 190 ngle ddition Postulate xample Slide 61 (nswer) / 190 ngle ddition Postulate xample is in the interior of N. If m N = (7x +11), m N = (15x + 24), and m N = (9x + 204). Solve for x. N (15x+24) (7x+11) is in the interior m N of N. + m N = m N (7x +11) + (15x+24) = (9x+204) If m N = (7x +11), 22x + 35 = 9x+204 m N = (15x + 24), 13x + 35 = x = 169 and m N = (9x + 204). x = 13 Solve for x. nswer (15x+24) This example addresses MP1 [This object & is MP2 a (7x+11) pull N

16 Slide 62 / Given m = 22 and m = 46. Slide 62 (nswer) / Given m = 22 and m = 46. ind m. ind m. 46 nswer = + 46 = = Slide 63 / Given m OM = 64 and m ON = 53. ind m NM. Slide 63 (nswer) / Given m OM = 64 and m ON = 53. ind m NM O 64 N M nswer OM = ON + O NM 64 = 53 + NM NM = N M Slide 64 / Given m = 95 and m = 48. ind m. Slide 64 (nswer) / Given m = 95 and m = 48. ind m. 48 nswer

17 Slide 65 / Given m K = 145 and m K = 61. ind m. Slide 65 (nswer) / Given m K = 145 and m K = 61. ind m. nswer 84 K 61 K Slide 66 / Given m TRS = 61 and m SRQ = 153. ind m QRT. Slide 66 (nswer) / Given m TRS = 61 and m SRQ = 153. ind m QRT. R nswer R 92 S Q S Q T T Slide 67 / is in the interior of TUV. If m TUV = (10x + 72)⁰, m TU = (14x + 18)⁰and m UV = (9x + 2)⁰ Solve for x. Slide 67 (nswer) / is in the interior of TUV. If m TUV = (10x + 72)⁰, m TU = (14x + 18)⁰and 10x + 72 = 14x x + 2 m UV = (9x + 2)⁰ 10x + 72 = 23x + 20 Solve for x. 13x = 52 nswer x = 4

18 26 is in the interior of. If m = (11x + 66)⁰, m = (5x + 3)⁰and m = (13x + 7)⁰ Solve for x. Slide 68 / is in the interior of. If m = (11x + 66)⁰, Slide 68 (nswer) / 190 m = (5x + 3)⁰and 11x + 66 = 5x x +7 m = (13x + 7)⁰ 11x + 66 = 18x + 10 Solve for x. 7x = 56 nswer x = 8 27 is in the interior of QP. m QP = (3x + 44)⁰ m QP = (8x + 3)⁰ m Q= (5x + 1)⁰ Solve for x. Slide 69 / 190 Slide 69 (nswer) / is in the interior of QP. m QP = (3x + 44)⁰ m QP = (8x + 3)⁰ 3x + 44 = 8x x + 1 m Q= (5x + 1)⁰ Solve for x. nswer 3x + 44 = 13x x = 40 x = 4 Slide 70 / 190 Question 2/25 The figure shows lines r, n, and p intersecting to form angles number 1, 2, 3, 4, 5, and 6. ll three lines lie in the same plane. 28 ased on the figure, which of the individual statements would provide enough information to conclude that r is perpendicular to line p? Select all that apply. m 2 = 90 m 6 = 90 m 3 = m 6 rom OY PR sample test m 1 + m 6 = 90 m 3 + m 4 = 90 m 4 + m 5 = 90 r n p not to scale Slide 70 (nswer) / 190 Question 2/25 The figure shows lines r, n, and p intersecting to form angles number 1, 2, 3, 4, 5, and 6. ll three lines lie in the same plane. 28 ased on the figure, which of the individual statements would provide enough information to conclude that r is perpendicular to line p? Select all that apply. m 2 = 90 m 6 = 90 m 3 = m 6 rom OY PR sample test nswer m 1 + m 6 = 90 m 3 + m 4 = 90 m 4 + m 5 = 90 & r n p not to scale

19 Slide 71 / 190 Slide 72 / 190 Protractors Protractors ngles are measured in degrees, using a protractor. very angle has a measure from 0 to 180 degrees. ngles can be drawn in any size. Return to Table of ontents Slide 73 / 190 Protractors Slide 74 / 190 Protractors is a 23 degree angle The measure of is 23 degrees is a 118 angle. The measure of is 118. Slide 75 / 190 Protractors rom our prior results we know that m = 118 and m = 23. So, the ngle ddition Postulate tells us that m must be what? Slide 75 (nswer) / 190 Protractors Question on this slide addresses MP6 & MP2 dditional Q's that can be used: What information do you have? (MP1) What do you need to find? (MP1) an you do this mentally? (MP5) an you guess & check? (MP 1 rom our prior results we know [This object that is a pull m = 118 and & MP5) m = 23. Math Practice So, the ngle ddition Postulate tells us that m must be what?

20 Slide 76 / 190 Protractors 29 What is the m? Slide 77 / G Without those prior results, we could just read the values of 118 and 23 from the protractor to get the included angle to be 95. Slide 77 (nswer) / 190 Slide 78 / What is the m? 30 What is the m G nswer G G Slide 78 (nswer) / 190 Slide 79 / What is the m G 31 What is the m? nswer G G

21 31 What is the m? 141 Slide 79 (nswer) / What is the m G? 54 Slide 80 / nswer G G Slide 80 (nswer) / 190 Slide 81 / What is the m G? 33 What is the m? nswer G G Slide 81 (nswer) / 190 Slide 82 / What is the m? 34 m PK = nswer G K M N P O

22 Slide 82 (nswer) / 190 Slide 83 / m PK = 35 m PM = K nswer M 32 N K M N P O P O Slide 83 (nswer) / 190 Slide 84 / m PM = 36 m PO = K nswer M 90 N K M N P O P O Slide 84 (nswer) / 190 Slide 85 / m PO = 37 m P = K nswer M 180 N K M N P O P O

23 Slide 85 (nswer) / 190 Slide 86 / m P = 38 m PN = K nswer M 63 N K M N P O P O Slide 86 (nswer) / 190 Slide 87 / m PN = 39 m NM = M M K nswer 135 N K N P O P O Slide 87 (nswer) / 190 Slide 88 / m NM = 40 m M = K nswer M N K M N P O P O

24 Slide 88 (nswer) / 190 Slide 89 / m M = 41 m K = K nswer M N K M N P O P O Slide 89 (nswer) / 190 Slide 90 / m K = 42 m NK = M M K nswer N K N P O P O Slide 90 (nswer) / 190 Slide 91 / m NK = K nswer M N Special ngle Pairs P O Return to Table of ontents

25 Slide 92 / 190 omplementary ngles omplementary angles are angles whose sum measures 90º. Slide 93 / 190 omplementary ngles djacent complementary angles form a right angle. One such angle is said to complement the other. They may be adjacent, but don't need to be. 25 o ngle and ngle are complementary since they comprise ngle, which is a right angle. 65 o 25 o omplementary adjacent 65 o omplementary nonadjacent Slide 94 / What is the complement of an angle whose measure is 72? Slide 94 (nswer) / What is the complement of an angle whose measure is 72? nswer 18 Slide 95 / What is the complement of an angle whose measure is 28? Slide 95 (nswer) / What is the complement of an angle whose measure is 28? nswer 62

26 Slide 96 / 190 xample Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles? et x = the smaller angle and the larger angle = 2x. nswer Slide 96 (nswer) / 190 xample Since the angles are complementary we know their sum must equal 90 Two angles are complementary. The larger angle degrees. is twice the size of the smaller angle. 90 What = 2x is + the x measure of both angles? 90 = 3x 30 = x = smaller angle et x larger = the smaller angle angle = 2(30) and the larger angle = 2x. larger angle = 60. Slide 97 / n angle is 34 more than its complement. What is its measure? Slide 97 (nswer) / n angle is 34 more than its complement. What is its measure? nswer angle = complement + 34 angle = (90 - x) + 34 x = 90 - x +34 2x = 124 x = 62 Slide 98 / n angle is 14 less than its complement. What is the angle's measure? Slide 98 (nswer) / n angle is 14 less than its complement. What is the angle's measure? nswer angle = complement - 14 angle = (90 - x) - 14 x = 90 - x x = x = 76 x = 38

27 Slide 99 / 190 Supplementary ngles Supplementary angles are angles whose sum measures 180º. Supplementary angles may be adjacent, but don't need to be. One angle is said to supplement the other. Slide 100 / 190 Supplementary ngles ny two angles that add to a straight angle are supplementary. Or, two adjacent angles whose exterior sides are opposite rays, are supplementary. 25 o 155o 25 o Supplementary adjacent a.k.a. inear Pair 155 o Supplementary nonadjacent If ngle is a straight angle, its measure is 180. Then ngle and ngle are supplementary since their measures add to 180. Slide 101 / 190 omplementary vs. Supplementary ngles There are 2 ways that one can remember the difference between complementary & supplementary angles: - Way 1 - Order: Think of the order of the 1st letters in each word & the number that they represent. comes before S in the alphabet & 90 comes before 180 on a number line, so omplementary means that they add up to 90º & Supplementary means that they add up to 180º - Way 2 - Visual: dd a line to each letter to start forming the number associated with it. S y adding a line to the "", you form a 9, for 90º y adding a line to the "S", you form an 8, for 180º Slide 101 (nswer) / 190 omplementary vs. Supplementary ngles There are 2 ways that one can remember the difference between complementary & supplementary angles: The connections shown - Way 1 - Order: Think of the order represent of the 1st MP2 letters & in MP4 each word & the number that they represent. comes before S in the alphabet & 90 comes before 180 on a number line, so omplementary means that they add up to 90º & Supplementary means that they add up to 180º Math Practice - Way 2 - Visual: dd a line to each letter [This to object start is a pull forming the number associated with it. S y adding a line to the "", you form a 9, for 90º y adding a line to the "S", you form an 8, for 180º Slide 102 / What is the supplement of an angle whose measure is 72? Slide 102 (nswer) / What is the supplement of an angle whose measure is 72? nswer 108

28 Slide 103 / What is the supplement of an angle whose measure is 28? Slide 103 (nswer) / What is the supplement of an angle whose measure is 28? nswer 152 Slide 104 / The measure of an angle is 98 more than its supplement. What is the measure of the angle? Slide 104 (nswer) / The measure of an angle is 98 more than its supplement. What is the measure of the angle? nswer angle = (180 - x) + 98 x = x x = 278 x = 139 Slide 105 / n measure of angle is 74 less than its supplement. What is the measure of the angle? Slide 105 (nswer) / n measure of angle is 74 less than its supplement. What is the measure of the angle? nswer angle = supplement - 74 x = (180 - x) x = x = 106 x = 53

29 Slide 106 / The measure of an angle is 26 more than its supplement. What is the measure of the angle? Slide 106 (nswer) / The measure of an angle is 26 more than its supplement. What is the measure of the angle? nswer angle = supplement + 26 x = (180 - x) x = x = 206 x = 103 Slide 107 / 190 Vertical ngles Slide 108 / 190 Vertical ngles Vertical ngles are two angles whose sides form two pairs of opposite rays Whenever two lines intersect, two pairs of vertical angles are formed. & are vertical angles, and & are vertical angles. & are vertical angles & are vertical angles. Slide 109 / 190 ngle Pair Relationships We can prove some important properties about these three special cases: angles which are complementary, supplementary or vertical. Two column proofs use one column to make a statement and the column next to it to provide the reason. elow is a 2-column proof format used to find the value of x in the diagram to the right. We're going to use proofs a lot, so we're going to use the format of this example to both prove three theorems. (See next slide.) (5x + 3)⁰ (13x + 7)⁰ (11x + 66)⁰ Slide 110 / 190 ngle Pair Relationships Proof Statements 1) m = (5x + 3) m = (13x + 7) 1) Given m = (11x + 66) Reasons 2) m + m = m 2) ngle ddition Postulate 3) 5x x + 7 = 11x ) Substitution Property of quality 4) ombine ike 4) 18x + 10 = 11x + 66 Terms/Simplify 5) 7x + 10 = 66 5) Subtraction Property of quality 6) 7x = 56 6) Subtraction Property of quality 7) x = 8 7) ivision Property of quality (5x + 3)⁰ (13x + 7)⁰ (11x + 66)⁰

30 Slide 111 / 190 Slide 112 / 190 Two olumn Proofs Proofs Special ngles Proofs all start out with a goal: what it is we are trying to prove. They are not open-ended explorations, but are directed towards a specific end. We know the last statement of every proof when we start, it is what we are trying to prove. We don't know the reason in advance. Return to Table of ontents Slide 113 / 190 ongruent omplements Theorem Theorem: ngles which are complementary to the same angle are equal. Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Slide 114 / 190 ongruent omplements Theorem Theorem: ngles which are complementary to the same angle are equal. Statement 1 ngles 1 and 2 are complementary ngles 1 and 3 are complementary Reason 1 Given What do we know about the sum of the measures of complementary angles? Slide 114 (nswer) / 190 ongruent omplements Theorem Theorem: ngles which are complementary to the same angle are equal. MP7 mphasize that the 1st step to any proof is stating the Statement 1 "Givens". Then, one Reason uses the 1 ngles 1 and 2 properties are complementary of the 1st Given statement ngles 1 and 3 to are ask complementary questions and continue to solve the proof. Math Practice What do we know Question about the on sum this of the slide measures of complementary address angles? MP6. Slide 115 / 190 ongruent omplements Theorem Given: Statement 2 m 1 + m 2 = 90 m 1 + m 3 = 90 ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Reason 2 efinition of complementary angles If both of the equations above equal 90 degrees, how are they related to each other? xplain how you know?

31 Slide 115 (nswer) / 190 ongruent omplements Theorem Given: Statement 2 m 1 + m 2 = 90 m 1 + m 3 = 90 ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Questions on this slide address MP2, MP3 & MP6. Math Practice Reason 2 efinition of complementary angles Statement 3 m 1 + m 2 = m 1 + m 3 Slide 116 / 190 ongruent omplements Theorem Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Reason 3 Substitution property of equality If both of the equations above equal 90 degrees, how are they related to each other? xplain how you know? Is there anything the same on both sides of the equation? If so, what can we do to simplify the equation? Why is this possible? Slide 116 (nswer) / 190 ongruent omplements Theorem Slide 117 / 190 ongruent omplements Theorem Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Math Practice Prove: m 2 = m 3 Questions on this slide address MP2, MP3 & MP6. Statement 3 m 1 + m 2 = m 1 + m 3 Reason 3 Substitution property of equality Prove: m 2 = m 3 Statement 4 m 2 = m 3 Reason 4 Subtraction property of equality Is there anything the same on both sides of the equation? If so, what can we do to simplify the equation? Why is this possible? Which is what we set out to prove Slide 118 / 190 ongruent omplements Theorem Slide 119 / 190 ongruent Supplements Theorem Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Theorem: ngles which are supplementary to the same angle are equal. Prove: m 2 = m 3 Statement ngles 1 and 2 are complementary ngles 1 and 3 are complementary Reason Given Given: ngles 1 and 2 are supplementary ngles 1 and 3 are supplementary Prove: m 2 = m 3 m 1 + m 2 = 90 m 1 + m 3 = 90 m 1 + m 2 = m 1 + m 3 m 2 = m 3 efinition of complementary angles Substitution Property of quality Subtraction Property of quality This is so much like the last proof, that we'll do this by just examining the total proof.

32 Given: Statement ngles 1 and 2 are supplementary ngles 1 and 3 are supplementary m 1 + m 2 = 180 m 1 + m 3 = 180 m 1 + m 2 = m 1 + m 3 m 2 = m 3 Slide 120 / 190 ongruent Supplements Theorem ngles 1 and 2 are supplementary ngles 1 and 3 are supplementary Prove: m 2 = m 3 Reason Given efinition of supplementary angles Substitution property of equality Subtraction property of equality Slide 121 / 190 Vertical ngles Theorem Vertical angles have equal measure Given: line and line are straight lines that intersect at Point and form angles 1, 2, 3 and 4 Prove: m 1 = m 3 and m 2 = m 4 Slide 122 / 190 Vertical ngles Theorem The first statement will focus on what we are given which makes this situation unique. In this case, it's just the Givens. Slide 122 (nswer) / 190 Vertical ngles Theorem MP7 mphasize that the 1st step to any proof is stating the "Givens". Then, one uses the properties of the 1st statement The first statement will focus on what we are given which to makes ask this questions situation unique. and continue to solve the proof. In this case, it's just the Givens. Math Practice Slide 123 / 190 Vertical ngles Theorem Statement 1 line and line are straight lines that intersect at Point and form angles 1, 2, 3 and 4 Reason 1 Given Then, we know we want to know something about the relationship between the pairs of vertical angles: 1 & 3 as well as 2 & 4 What do you know about these four angles that the givens can help us with? Slide 123 (nswer) / 190 Vertical ngles Theorem Math Practice Question on this slide Statement 1 addresses MP1. Reason 1 line and line are Given straight lines that intersect at Point and form angles 1, 2, 3 and 4 Then, we know we want to know something about the relationship between the pairs of vertical angles: 1 & 3 as well as 2 & 4 What do you know about these four angles that the givens can help us with?

33 Slide 124 / We know that angles. Slide 124 (nswer) / We know that angles. 1 & 4 are supplementary 1 & 2 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary ll of the above 1 & 4 are supplementary 1 & 2 are supplementary 2 & 3 are supplementary nswer 3 & 4 are supplementary ll of the above Slide 125 / 190 Vertical ngles Theorem Slide 125 (nswer) / 190 Vertical ngles Theorem Statement 2 1 & 2 are supplementary 1 & 4 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary Reason 2 ngles that form a linear pair are supplementary Math Practice Statement 2 1 & 2 are supplementary 1 & 4 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary Question on this slide addresses MP7. Reason 2 ngles that form a linear pair are supplementary What do you know about two angles which are supplementary to the same angle, like 2 & 4 which are both supplements of 1? Slide 126 / 190 Vertical ngles Theorem Statement 2 1 & 2 are supplementary 1 & 4 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary Reason 2 ngles that form a linear pair are supplementary What do you know about two angles which are supplementary to the same angle, like 2 & 4 which are both supplements of 1? Slide 126 (nswer) / 190 Vertical ngles Theorem Math Practice Statement 2 1 & 2 are supplementary 1 & 4 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary omment at the bottom of this slide addresses MP7. Reason 2 ngles that form a linear pair are supplementary et's look at the fact that 2 & 4 are both supplementary to 1 and that 1 & 3 are both supplementary to 4, since that relates to the vertical angles we're interested in. et's look at the fact that 2 & 4 are both supplementary to 1 and that 1 & 3 are both supplementary to 4, since that relates to the vertical angles we're interested in.

34 Slide 127 / 190 Vertical ngles Theorem Statement Slide 128 / 190 Vertical ngles Theorem Given: and are straight lines that intersect at Point and form angles 1, 2, 3 and 4 Prove: m 1 = m 3 and m 2 = m 4 Reason Statement 3 m 1 = m 3 m 2 = m 4 Reason 3 Two angles supplementary to the same angle are equal ut those are the pairs of vertical angles which we set out to prove are equal. So, our proof is complete: vertical angles are equal Slide 129 / 190 Vertical ngles Theorem line and line are straight lines that intersect at Point and Given form angles 1, 2, 3 and 4 1 & 2 are supplementary 1 & 4 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary m 1 = m 3 and m 2 = m 4 ngles that form a linear pair are supplementary Two angles supplementary to the same angle are equal Slide 130 / 190 Vertical ngles Given: m = 55, solve for x, y and z. We have proven that vertical angles are congruent. This becomes a theorem we can use in future proofs. lso, we can solve problems with it. y o x o 55 zo Math Practice Slide 130 (nswer) / 190 Vertical ngles This example addresses MP2 Given: m = 55, solve for x, y and z. dditional Q's that could be used: What information are you given? (MP1 What do you need to find? (MP1) What connections do you see? (MP4) x o 55 an you do this mentally? y o zo (MP5) ow is this question related to supplementary [This object is angles? a pull (MP7) Given: m = 55 Slide 131 / 190 Vertical ngles We know that x + 55 = 180, since they are supplementary. nd that y = 55, since they are vertical angles. nd that x = z for the same reason. 125 o 55 o 125 o 55 o

35 Slide 132 / 190 xample Slide 133 / What is the measure of angle 1? ind m 1, m 2 & m 3. xplain your answer m 1 = 180 m 1 = 144 inear pair angles are supplementary none of the above m 2 = 36 ; Vertical angles are congruent (original angle & m 2) m 3 = 144 ; Vertical angles are congruent (m 1 & m 3) Slide 133 (nswer) / What is the measure of angle 1? Slide 134 / What is the measure of angle 2? nswer none of the above none of the above Slide 134 (nswer) / What is the measure of angle 2? Slide 135 / What is the measure of angle 3? nswer none of the above none of the above

36 Slide 135 (nswer) / What is the measure of angle 3? Slide 136 / What is the measure of angle 4? nswer none of the above none of the above Slide 136 (nswer) / What is the measure of angle 4? nswer none of the above ) measure of angle 4 = 68 o Slide 137 / What is the measure of angle 5? none of the above Slide 137 (nswer) / What is the measure of angle 5? 58 What is the m 6? Slide 138 / nswer none of the above none of the above

37 58 What is the m 6? Slide 138 (nswer) / 190 Slide 139 / 190 xample nswer none of the above ind the value of x. The angles shown are vertical, so they are congruent (14x + 7) (13x + 16) Slide 139 (nswer) / 190 xample Slide 140 / 190 xample ind the value of x. nswer The angles shown are vertical, so they are congruent. 13x + 16 = 14x x -13x 16 = x = x ind the value of x. (2x + 8) (3x + 17) The angles shown are supplementary (13x + 16) This example addresses (14x + 7) MP2 & MP4 Slide 140 (nswer) / 190 xample 59 ind the value of x. Slide 141 / 190 ind the value of x. nswer (2x + 8) The angles shown are supplementary 2x x + 17 = 180 5x + 25 = x = 155 (3x + 17) 5 5 x = o (2x - 5) o This example addresses MP2 & MP4

38 59 ind the value of x. Slide 141 (nswer) / ind the value of x. Slide 142 / nswer 85 o (2x - 5) o o (6x + 3) o 60 ind the value of x. Slide 142 (nswer) / ind the value of x. Slide 143 / nswer o (6x + 3) o (9x - 4) o 122 o 61 ind the value of x. Slide 143 (nswer) / ind the value of x. Slide 144 / nswer (9x - 4) o 122 o (7x + 54) o 42o

39 Slide 144 (nswer) / 190 Slide 145 / ind the value of x nswer ngle isectors (7x + 54) o [This object 42o is a pull Return to Table of ontents Slide 146 / 190 Slide 147 / 190 n angle bisector is a ray or line which starts at the vertex and cuts an angle into two equal halves ngle isector X ray X bisects inding the missing measurement. xample: is bisected by ray. ind the measures of the missing angles. isect means to cut it into two equal parts. The 'bisector' is the thing doing the cutting. The angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle. 52 Slide 147 (nswer) / 190 Slide 148 / 190 inding the missing measurement. xample: is bisected by ray. ind the measures of the missing angles. 63 G is bisected by. The m G = 56. ind the measures of the missing angles. nswer m = 52 m = 2(52) = 104 This example addresses MP2 56 o G 52

40 Slide 148 (nswer) / G is bisected by. The m G = 56. ind the measures of the missing angles. Slide 149 / MO bisects MN. ind the value of x. nswer m = 56/2 = 28 o m G = 28 o 56 o M (x + 10) o O (3x - 20) o G N Slide 149 (nswer) / 190 Slide 150 / MO bisects MN. ind the value of x. M nswer m MO = m OMN x + 10 = 3x x -x 10 = 2x - 20 (x + 10) o = O 2x (3x - 20) o = x N 65 Ray NP bisects MNO Given that m MNP = 57, what is m MNO? int: click to reveal What does bisect mean? raw & label a picture. Slide 150 (nswer) / Ray NP bisects MNO Given that m MNP = 57, what is m MNO? Slide 151 / Ray RT bisects QRS Given that m QRT = 78, what is m QRS? nswer m MNO = 2(57) = 114 int: click to reveal What does bisect mean? raw & label a picture.

41 Slide 151 (nswer) / Ray RT bisects QRS Given that m QRT = 78, what is m QRS? Slide 152 / Ray VY bisects UVW. Given that m UVW = 165o, what is m UVY? nswer m QRS = 2(78) = 156 Slide 152 (nswer) / Ray VY bisects UVW. Given that m UVW = 165o, Slide 153 / Ray bisects. ind the value of x. what is m UVY? nswer m UVY = 165/2 = 82.5 (7x + 3) o (11x - 25) o Slide 153 (nswer) / 190 Slide 154 / Ray bisects. ind the value of x. 69 Ray bisects G. ind the value of x. (7x + 3) o nswer (11x - 25) o 7x + 3 = 11x = 4x = 4x 7 = x (9x - 17) o (3x + 49) o G

42 Slide 154 (nswer) / Ray bisects G. ind the value of x. Slide 155 / Ray bisects IK. ind the value of x. (9x - 17) o nswer 9x - 17 = 3x x - 17 = 49 6x = 66 x = 11 (3x + 49) o I (7x + 1) o (12x - 19) o G K Slide 155 (nswer) / 190 Slide 156 / Ray bisects IK. ind the value of x. I nswer (7x + 1) o (12x - 19) o 7x + 1 = 12x = 5x = 5x 4 = x ocus & ngle onstructions K Return to Table of ontents Slide 156 (nswer) / 190 ocus & ngle onstructions Math Practice This entire lesson w/ constructions addresses MP5 Slide 157 / 190 onstructing ongruent ngles Given: G onstruct: such that G Our approach will be based on the idea that the measure of an angle is how much we would have rotate one ray it overlap the other. The larger the measure of the angle, the farther apart they are as you move away from the vertex. Return to Table of ontents G

43 Slide 158 / 190 onstructing ongruent ngles So, if we go out a fixed distance from the vertex on both rays and draw points there, the distance those points are apart from one another defines the measure of the angle. The bigger the distance, the bigger the measure of the angle. If we construct an angle whose rays are the same distance apart at the same distance from the vertex, it will be congruent to the first angle. Slide 159 / 190 onstructing ongruent ngles 1. raw a reference line with your straight edge. Place a reference point () to indicate where your new ray will start on the line. G G Slide 160 / 190 Slide 161 / 190 onstructing ongruent ngles onstructing ongruent ngles 2. Place the compass point on the vertex G and stretch it to any length so long as your arc will intersect both rays. 3. raw an arc that intersects both rays of G. (This defines a common distance from the vertex on both rays since the arc is part of a circle and all its points are equidistant from the center of the circle.) 4. Without changing the span of the compass, place the compass tip on your reference point and swing an arc that goes through the line and above it. (This defines that same distance from the vertex on both our reference ray and the ray we will draw as we used for the original angle.) G G Slide 162 / 190 Slide 163 / 190 onstructing ongruent ngles onstructing ongruent ngles 5. Now place your compass where the arc intersects one ray of the original angle and set it so it can draw an arc where it crosses the other ray. (This defines how far apart the rays are at that distance from the vertex.) 6. Without changing the span of the compass place the point of the compass where the first arc crosses the first ray and draw an arc that intersects the arc above the ray. (This will make the separation between the rays the same at the same distance from the new vertex as was the case for the original angle.) G G

44 Slide 164 / 190 onstructing ongruent ngles 6. Now, use your straight edge to draw the second ray of the new angle which is congruent with the first angle. Slide 165 / 190 onstructing ongruent ngles It should be clear that these two angles are congruent. Ray G would have to be rotated the same amount to overlap Ray G as would Ray to overlap Ray. Notice that where we place the points is not relevant, just the shape of the angle indicates congruence. G G Slide 166 / 190 onstructing ongruent ngles We can confirm that by putting one atop the other. Slide 167 / 190 Try this! onstruct a congruent angle on the given line segment. 1) P Q R G Slide 167 (nswer) / 190 Slide 168 / 190 Try this! Try this! onstruct a congruent angle The on file the for given the line "Try segment. This!" problems 1) is located on the NT website: geometry/points-lines-andplanes/ P under "andouts". Teacher Notes Q onstruct a congruent angle on the given line segment. 2) R K

45 Slide 169 / 190 Slide 170 / 190 Video emonstrating onstructing ongruent ngles using ynamic Geometric Software lick here to see video ngle isectors & onstructions Return to Table of ontents Slide 170 (nswer) / 190 Slide 171 / 190 onstructing ngle isectors s we learned earlier, an angle bisector divides an angle into two adjacent angles of equal measure. ngle isectors & onstructions Math Practice This entire lesson w/ constructions addresses MP5 To create an angle bisector we will use an approach similar to that used to construct a congruent angle, since, in this case, we will be constructing two congruent angles. U Return to Table of ontents Slide 172 / 190 onstructing ngle isectors 1. With the compass point on the vertex, draw an arc that intersects both rays. (This will establish a fixed distance from the vertex on both rays. V W Slide 173 / 190 onstructing ngle isectors 2. Without changing the compass setting, place the compass point on the intersection of each arc and ray and draw a new arc such that the two new arcs intersect in the interior of the angle. (This fixes the distance from each original ray to the new ray to be the same, so that the two new angles will be congruent.) U U V W V W

46 Slide 174 / 190 onstructing ngle isectors 3. With a straightedge, draw a ray from the vertex through the intersection of the arcs and label that point. ecause we know that the distance of each original ray to the new ray is the same, at the same distance from the vertex, we know the measures of the new angles is the same and that m UVX = m XVW U isect the angle 3) Slide 175 / 190 Try This! X V W Slide 175 (nswer) / 190 Try This! Slide 176 / 190 Try This! isect the angle isect the angle 3) Teacher Notes The file for the "Try This!" problems is located on the NT website: geometry/points-lines-andplanes/ under "andouts". 4) Slide 177 / 190 onstructing ngle isectors w/ string, rod, pencil & straightedge Slide 178 / 190 onstructing ngle isectors w/ string, rod, pencil & straightedge verything we do with a compass can also be done with a rod and string. In both cases, the idea is to mark a center (either the point of the compass or the rod) and then draw an part of a circle by keeping a fixed radius (with the span of the compass or the length of the string. 1. With the rod on the vertex, draw an arc across each side. U V W

47 Slide 179 / 190 onstructing ngle isectors w/ string, rod, pencil & straightedge 2. Place the rod on the arc intersections of the sides & draw 2 arcs, one from each side showing an intersection point. Slide 180 / 190 onstructing ngle isectors w/ string, rod, pencil & straightedge 3. With a straightedge, connect the vertex to the arc intersections. abel your point. U U m UVX = m XVW X V W V W Slide 181 / 190 Slide 182 / 190 Try This! Try This! isect the angle with string, rod, pencil & straightedge. 5) isect the angle with string, rod, pencil & straightedge. 6) Slide 183 / 190 onstructing ngle isectors by olding 1. On patty paper, create any angle of your choice. Make it appear large on your patty paper. abel the points, &. Slide 184 / 190 onstructing ngle isectors by olding 2. old your patty paper so that ray lines up with ray. rease the fold.

48 Slide 185 / 190 onstructing ngle isectors by olding 3. Unfold your patty paper. raw a ray along the fold, starting at point. raw and label a point on your ray. isect the angle with folding. 7) Slide 186 / 190 Try This! isect the angle with folding. 8) Slide 187 / 190 Try This! Slide 188 / 190 Videos emonstrating onstructing ngle isectors using ynamic Geometric Software lick here to see video using a compass and segment tool lick here to see video using the menu options Slide 189 / 190 PR Sample Test Questions The remaining slides in this presentation contain questions from the PR Sample Test. fter finishing unit 2, you should be able to answer this question. Good uck! Slide 190 / 190 Question 2/25 The figure shows lines r, n, and p intersecting to form angles number 1, 2, 3, 4, 5, and 6. ll three lines lie in the same plane. 71 ased on the figure, which of the individual statements would provide enough information to conclude that r is perpendicular to line p? Select all that apply. r n p not to scale Return to Table of ontents m 2 = 90 m 6 = 90 m 3 = m 6 m 1 + m 6 = 90 m 3 + m 4 = 90 m 4 + m 5 = 90