Geometry. Slide 1 / 190. Slide 2 / 190. Slide 3 / 190. Angles. Table of Contents

Transcription

1 Slide 1 / 190 Slide 2 / 190 Geometry ngles Table of ontents click on the topic to go to that section Slide 3 / 190 ngles ongruent ngles ngles & ngle ddition Postulate Protractors Special ngle Pairs Proofs Special ngles ngle isectors Locus & ngle onstructions ngle isectors & onstructions PR Released Questions

2 Table of ontents for Videos emonstrating onstructions Slide 4 / 190 click on the topic to go to that video ongruent ngles ngle isectors Throughout this unit, the Standards for Mathematical Practice are used. Slide 5 / 190 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: Look 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. Slide 6 / 190 ngles Return to Table of ontents

3 ngles Slide 7 / 190 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 ngles Slide 8 / 190 The measure of angle is the amount that one line, one ray or segment would need to rotate in order to overlap the other. In this case, Ray would have to rotate through an angle of x degrees in order to overlap Ray. xº ngles Slide 9 / 190 In this course, angles will be measured with degrees, which have the symbol º. For a ray to rotate all the way around from ray, as shown, back to ray would represent a 360º angle. xº

4 Measuring angles in degrees Slide 10 / 190 The use of 360 degrees to represent a full rotation back to the original position is arbitrary. ny number could have been used, but 360 degrees for a full rotation has become a standard. 360º Measuring angles in degrees Slide 11 / 190 The use of 360 for a full rotation is thought that it come from ancient abylonia, which used a number system based on 60. 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. Right ngles Slide 12 / 190 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. 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

5 Right ngles Slide 13 / 190 Fourth 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. xº xº 90º Right ngles Slide 14 / 190 This definition is unchanged today and should be familiar to you. Perpendicular lines, segments or rays form right angles. If lines intersect to form adjacent equal angles, then they are perpendicular and the measure of those angles is 90º. 90º When perpendicular lines meet, they form equal adjacent angles and their measure is 90º. Right ngles Slide 15 / 190 There is a special indicator of a right angle. It is shown in red in this case to make it easy to recognize.

6 Obtuse ngles Slide 16 / 190 efinition 11: n obtuse angle is an angle greater than a right angle. 135º cute ngles Slide 17 / 190 efinition 12: n acute angle is an angle less than a right angle. 45º Straight ngle Slide 18 / 190 definition that we need that was not used in The Elements 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? Explain why. What is the degree measurement of the angle?

7 Reflex ngle Slide 19 / 190 nother modern definition that was not used in The Elements is that of a "reflex angle." That is an angle that is greater than 180º. 235º This is also a type of obtuse angle. ngles Slide 20 / 190 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. 1 This is an example of a (an) angle. hoose all that apply. Slide 21 / 190 acute obtuse right 0º reflex E straight

8 2 This is an example of a (an) angle. hoose all that apply. Slide 22 / 190 acute obtuse right reflex 45º E straight 3 This is an example of a (an) angle. hoose all that apply. Slide 23 / 190 acute obtuse right reflex E straight 90º 4 This is an example of a (an) angle. hoose all that apply. Slide 24 / 190 acute obtuse right reflex E straight 135º

9 5 This is an example of a (an) angle. hoose all that apply. Slide 25 / 190 acute obtuse right reflex 180º E straight 6 This is an example of a (an) angle. hoose all that apply. Slide 26 / 190 acute obtuse right 235º reflex E straight 7 This is an example of a (an) angle. hoose all that apply. Slide 27 / 190 acute obtuse right reflex 270º E straight

10 8 This is an example of a (an) angle. hoose all that apply. Slide 28 / 190 acute obtuse right 315º reflex E straight 9 This is an example of a (an) angle. hoose all that apply. Slide 29 / 190 acute obtuse right reflex 360º E straight Naming ngles Slide 30 / 190 n angle has three parts, it has two sides and one vertex, where the sides meet. In this example, the sides are the rays and and the vertex is. side vertex θ side

11 Interior of ngles Slide 31 / 190 ny angle with a measure of less than 180º has an interior and exterior, as shown below. Exterior Interior θ Naming ngles Slide 32 / 190 n angle can be named in three different ways: 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 vertex θ leg Or by a number or a symbol placed inside the angle (e.g. Greek letter, θ, in the figure) Naming ngles Slide 33 / 190 The angle shown can be called,, or. 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.

12 Naming ngles Slide 34 / 190 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? How many angles do you count in the image? θ α Naming ngles Slide 35 / 190 What other ways could you name, and in the case below? (using the side - vertex - side method) θ α How could you name those 3 angles using the letters placed inside the angles? Intersecting Lines Form ngles Slide 36 / 190 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. θ

13 Intersecting Lines Form ngles Slide 37 / 190 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 Two lines meet at more than one point. Slide 38 / 190 lways Sometimes Never 11 n angle that measures 90 degrees is a right angle. Slide 39 / 190 lways Sometimes Never

14 12 n angle that is less than 90 degrees is obtuse. Slide 40 / 190 lways Sometimes Never 13 n angle that is greater than 180 degrees is referred to as a reflex angle. Slide 41 / 190 lways Sometimes Never Slide 42 / 190 ongruent ngles Return to Table of ontents

15 ongruence Slide 43 / 190 We learned earlier that if two line segments have the same length, they are congruent. a lso, all line segments with the same length are congruent. b re these two segments congruent? Explain why your answer is true. ongruence Slide 44 / 190 How 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? F E ongruence Slide 45 / 190 If two angles have the same measure, they are congruent since they can be rotated and moved to overlap at every point. E F

16 ongruence Slide 46 / 190 However, if their included angles do not have equal measure, they cannot be made to overlap at every point. For angles to be congruent, they need to have equal measures. re these two angles congruent? Explain why your answer is true. F E ongruence Slide 47 / 190 However, if their included angles do not have the same measure, they cannot be made to overlap at every point. For angles to be congruent, they need to have the same measure. Here you can see clearly when we rotate the two angles from the previous slide, they do not have the same angle measure. E F ongruent ngles Slide 48 / 190 One way to indicate that two angles have the same measure is to label them with the same variable. For instance, labeling both of these angles with xº indicates that they have the same measure. xº xº

17 ongruent ngles Slide 49 / 190 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. F E 14 Is congruent to E? Slide 50 / 190 Yes No F E 15 ongruent angles have the same measure. Slide 51 / 190 lways Sometimes Never

18 16 and are. Slide 52 / 190 ongruent Not ongruent annot be determined 17 E and F are. Slide 53 / 190 ongruent Not ongruent annot be determined F E 18 and are congruent. Slide 54 / 190 True False annot be determined

19 19 and are congruent. True False Slide 55 / 190 Slide 56 / 190 ngles & ngle ddition Postulate Return to Table of ontents djacent ngles Slide 57 / 190 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.

20 ngle ddition Postulate Slide 58 / 190 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 ngle ddition Postulate Slide 59 / 190 Further, 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 ngle ddition Postulate Example Slide 60 / 190 m PQS = 32 m SQR = 26 Q P S R What's the measure of PQR?

21 ngle ddition Postulate Example Slide 61 / 190 is in the interior of NJ. If m NJ = (7x +11), m N = (15x + 24), and m NJ = (9x + 204). Solve for x. (15x+24) N (7x+11) J 20 Given m = 22 and m = 46. Slide 62 / 190 Find m Given m OLM = 64 and m OLN = 53. Find m NLM. Slide 63 / O L N M

22 22 Given m = 95 and m = 48. Find m. Slide 64 / Given m KLJ = 145 and m KLH = 61. Find m HLJ. Slide 65 / 190 H K L J 24 Given m TRS = 61 and m SRQ = 153. Find m QRT. Slide 66 / 190 S 61 R 153 Q T

23 25 is in the interior of TUV. Slide 67 / 190 If m TUV = (10x + 72)⁰, m TU = (14x + 18)⁰and m UV = (9x + 2)⁰ Solve for x. 26 is in the interior of. Slide 68 / 190 If m = (11x + 66)⁰, m = (5x + 3)⁰and m = (13x + 7)⁰ Solve for x. 27 F is in the interior of QP. Slide 69 / 190 m QP = (3x + 44)⁰ m FQP = (8x + 3)⁰ m QF= (5x + 1)⁰ Solve for x.

24 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. r n p not to scale Slide 70 / 190 m 2 = 90 m 6 = 90 m 1 + m 6 = 90 E m 3 + m 4 = 90 m 3 = m 6 From EOY PR sample test F m 4 + m 5 = 90 Slide 71 / 190 Protractors Return to Table of ontents Protractors Slide 72 / 190 ngles are measured in degrees, using a protractor. Every angle has a measure from 0 to 180 degrees. ngles can be drawn in any size.

25 Protractors Slide 73 / 190 is a 23 degree angle The measure of is 23 degrees Protractors Slide 74 / 190 is a 118 angle. The measure of is 118. Protractors Slide 75 / 190 From our prior results we know that m = 118 and m = 23. So, the ngle ddition Postulate tells us that m must be what?

26 Protractors Slide 76 / 190 Without those prior results, we could just read the values of 118 and 23 from the protractor to get the included angle to be What is the m J? Slide 77 / E F G J H 30 What is the m JG Slide 78 / E F G J H

27 31 What is the m JE? Slide 79 / E F G J H 32 What is the m EJG? Slide 80 / E F G J H 33 What is the m JF? Slide 81 / E F G J H

28 34 m PJK = Slide 82 / 190 L M N K P J O 35 m PJM = Slide 83 / 190 L M N K P J O 36 m PJO = Slide 84 / 190 L M N K P J O

29 37 m PJL = Slide 85 / 190 L M N K P J O 38 m PJN = Slide 86 / 190 L M N K P J O 39 m NJM = Slide 87 / 190 L M N K P J O

30 40 m MJL = Slide 88 / 190 L M N K P J O 41 m LJK = Slide 89 / 190 L M N K P J O 42 m NJK = Slide 90 / 190 L M N K P J O

31 Slide 91 / 190 Special ngle Pairs Return to Table of ontents omplementary ngles Slide 92 / 190 omplementary angles are angles whose sum measures 90º. One such angle is said to complement the other. They may be adjacent, but don't need to be. 25 o 65 o 25 o omplementary adjacent 65 o omplementary nonadjacent omplementary ngles Slide 93 / 190 djacent complementary angles form a right angle. ngle and ngle are complementary since they comprise ngle, which is a right angle.

32 43 What is the complement of an angle whose measure is 72? Slide 94 / What is the complement of an angle whose measure is 28? Slide 95 / 190 Example Slide 96 / 190 Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles? Let x = the smaller angle and the larger angle = 2x.

33 45 n angle is 34 more than its complement. Slide 97 / 190 What is its measure? 46 n angle is 14 less than its complement. Slide 98 / 190 What is the angle's measure? Supplementary ngles Slide 99 / 190 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. 25 o 155o 25 o Supplementary adjacent a.k.a. Linear Pair 155 o Supplementary nonadjacent

34 Supplementary ngles Slide 100 / 190 ny two angles that add to a straight angle are supplementary. Or, two adjacent angles whose exterior sides are opposite rays, are supplementary. If ngle is a straight angle, its measure is 180. Then ngle and ngle are supplementary since their measures add to 180. omplementary vs. Supplementary ngles Slide 101 / 190 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 47 What is the supplement of an angle whose measure is 72? Slide 102 / 190

35 48 What is the supplement of an angle whose measure is 28? Slide 103 / The measure of an angle is 98 more than its supplement. Slide 104 / 190 What is the measure of the angle? 50 n measure of angle is 74 less than its supplement. Slide 105 / 190 What is the measure of the angle?

36 51 The measure of an angle is 26 more than its supplement. Slide 106 / 190 What is the measure of the angle? Vertical ngles Slide 107 / 190 Vertical ngles are two angles whose sides form two pairs of opposite rays Whenever two lines intersect, two pairs of vertical angles are formed. & E are vertical angles, and E & are vertical angles. E Vertical ngles Slide 108 / 190 & E are vertical angles E & are vertical angles. E E

37 ngle Pair Relationships Slide 109 / 190 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 ngle Pair Relationships Proof Slide 110 / 190 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 Equality 4) ombine Like 4) 18x + 10 = 11x + 66 Terms/Simplify 5) 7x + 10 = 66 5) Subtraction Property of Equality 6) 7x = 56 6) Subtraction Property of Equality 7) x = 8 7) ivision Property of Equality (5x + 3)⁰ (13x + 7)⁰ (11x + 66)⁰ Slide 111 / 190 Proofs Special ngles Return to Table of ontents

38 Two olumn Proofs Slide 112 / 190 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. ongruent omplements Theorem Slide 113 / 190 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 ongruent omplements Theorem Slide 114 / 190 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?

39 ongruent omplements Theorem Slide 115 / 190 Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Statement 2 m 1 + m 2 = 90 m 1 + m 3 = 90 Reason 2 efinition of complementary angles If both of the equations above equal 90 degrees, how are they related to each other? Explain how you know? ongruent omplements Theorem Slide 116 / 190 Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Statement 3 m 1 + m 2 = m 1 + m 3 Reason 3 Substitution 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? ongruent omplements Theorem Slide 117 / 190 Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Statement 4 m 2 = m 3 Reason 4 Subtraction property of equality Which is what we set out to prove

40 ongruent omplements Theorem Slide 118 / 190 Given: ngles 1 and 2 are complementary ngles 1 and 3 are complementary Prove: m 2 = m 3 Statement ngles 1 and 2 are complementary ngles 1 and 3 are complementary Reason Given 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 Equality Subtraction Property of Equality ongruent Supplements Theorem Slide 119 / 190 Theorem: ngles which are supplementary to the same angle are equal. Given: ngles 1 and 2 are supplementary ngles 1 and 3 are supplementary Prove: m 2 = m 3 This is so much like the last proof, that we'll do this by just examining the total proof. ongruent Supplements Theorem Slide 120 / 190 Given: ngles 1 and 2 are supplementary ngles 1 and 3 are supplementary Prove: m 2 = m 3 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 Reason Given efinition of supplementary angles Substitution property of equality Subtraction property of equality

41 Vertical ngles Theorem Slide 121 / 190 Vertical angles have equal measure E Given: line and line E 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 Vertical ngles Theorem Slide 122 / 190 E The first statement will focus on what we are given which makes this situation unique. In this case, it's just the Givens. Vertical ngles Theorem Slide 123 / 190 E Statement 1 line and line E 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?

42 52 We know that angles. Slide 124 / 190 E 1 & 4 are supplementary 1 & 2 are supplementary 2 & 3 are supplementary 3 & 4 are supplementary ll of the above E Vertical ngles Theorem Slide 125 / 190 E 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? Vertical ngles Theorem Slide 126 / 190 E 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 Let'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.

43 Vertical ngles Theorem Slide 127 / 190 E 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 Vertical ngles Theorem Given: and E 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 E Slide 128 / 190 Statement Reason line and line E 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 Vertical ngles Theorem Slide 129 / 190 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.

44 Vertical ngles Slide 130 / 190 Given: m = 55, solve for x, y and z. E y o x o 55 zo Vertical ngles Slide 131 / 190 Given: m = 55 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. E 125 o 55 o 55 o 125 o Example Slide 132 / 190 Find m 1, m 2 & m 3. Explain your answer m 1 = 180 m 1 = 144 Linear pair angles are supplementary m 2 = 36 ; Vertical angles are congruent (original angle & m 2) m 3 = 144 ; Vertical angles are congruent (m 1 & m 3)

45 53 What is the measure of angle 1? Slide 133 / none of the above What is the measure of angle 2? Slide 134 / none of the above What is the measure of angle 3? Slide 135 / none of the above

46 56 What is the measure of angle 4? Slide 136 / none of the above What is the measure of angle 5? Slide 137 / none of the above What is the m 6? Slide 138 / none of the above

47 Example Slide 139 / 190 Find the value of x. The angles shown are vertical, so they are congruent. (14x + 7) (13x + 16) Example Slide 140 / 190 Find the value of x. The angles shown are supplementary (2x + 8) (3x + 17) 59 Find the value of x. Slide 141 / o (2x - 5) o

48 60 Find the value of x. Slide 142 / o (6x + 3) o 61 Find the value of x. Slide 143 / o (9x - 4) o 62 Find the value of x. Slide 144 / (7x + 54) o 42o

49 Slide 145 / 190 ngle isectors Return to Table of ontents 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 Slide 146 / 190 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. Finding the missing measurement. Slide 147 / 190 Example: is bisected by ray. Find the measures of the missing angles. 52

50 63 EFG is bisected by FH. The m EFG = 56. Find the measures of the missing angles. Slide 148 / 190 E H 56 o F G 64 MO bisects LMN. Find the value of x. Slide 149 / 190 L M (x + 10) o O (3x - 20) o N 65 Ray NP bisects MNO Given that Slide 150 / 190 m MNP = 57, what is m MNO? Hint: click to reveal What does bisect mean? raw & label a picture.

51 66 Ray RT bisects QRS Given that Slide 151 / 190 m QRT = 78, what is m QRS? 67 Ray VY bisects UVW. Given that m UVW = 165o, Slide 152 / 190 what is m UVY? 68 Ray bisects. Find the value of x. Slide 153 / 190 (7x + 3) o (11x - 25) o

52 69 Ray FH bisects EFG. Find the value of x. Slide 154 / 190 H E (9x - 17) o (3x + 49) o F G 70 Ray JL bisects IJK. Find the value of x. Slide 155 / 190 I (7x + 1) o L (12x - 19) o J K Slide 156 / 190 Locus & ngle onstructions Return to Table of ontents

53 onstructing ongruent ngles Slide 157 / 190 Given: FGH onstruct: such that FGH 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. F G H 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. Slide 158 / 190 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. F G H onstructing ongruent ngles Slide 159 / raw a reference line with your straight edge. Place a reference point () to indicate where your new ray will start on the line. F G H

54 onstructing ongruent ngles Slide 160 / 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 FGH. (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.) F G H onstructing ongruent ngles Slide 161 / 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.) F G H onstructing ongruent ngles Slide 162 / 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.) F G H

55 onstructing ongruent ngles Slide 163 / 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.) F G H onstructing ongruent ngles Slide 164 / Now, use your straight edge to draw the second ray of the new angle which is congruent with the first angle. F G H onstructing ongruent ngles Slide 165 / 190 It should be clear that these two angles are congruent. Ray FG would have to be rotated the same amount to overlap Ray GH as would Ray to overlap Ray. Notice that where we place the points is not relevant, just the shape of the angle indicates congruence. F G H

56 onstructing ongruent ngles Slide 166 / 190 We can confirm that by putting one atop the other. F G H Try this! Slide 167 / 190 onstruct a congruent angle on the given line segment. 1) P Q R Try this! Slide 168 / 190 onstruct a congruent angle on the given line segment. 2) E L J K

57 Slide 169 / 190 Video emonstrating onstructing ongruent ngles using ynamic Geometric Software lick here to see video Slide 170 / 190 ngle isectors & onstructions Return to Table of ontents onstructing ngle isectors Slide 171 / 190 s we learned earlier, an angle bisector divides an angle into two adjacent angles of equal measure. 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 V W

58 onstructing ngle isectors Slide 172 / 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. U V W onstructing ngle isectors Slide 173 / 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 V W onstructing ngle isectors Slide 174 / 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 X V W

59 Try This! Slide 175 / 190 isect the angle 3) Try This! Slide 176 / 190 isect the angle 4) onstructing ngle isectors w/ string, rod, pencil & straightedge Slide 177 / 190 Everything 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.

60 onstructing ngle isectors w/ string, rod, pencil & straightedge Slide 178 / With the rod on the vertex, draw an arc across each side. U V W onstructing ngle isectors w/ string, rod, pencil & straightedge Slide 179 / Place the rod on the arc intersections of the sides & draw 2 arcs, one from each side showing an intersection point. U V W onstructing ngle isectors w/ string, rod, pencil & straightedge Slide 180 / With a straightedge, connect the vertex to the arc intersections. Label your point. U m UVX = m XVW X V W

61 Try This! Slide 181 / 190 isect the angle with string, rod, pencil & straightedge. 5) Try This! Slide 182 / 190 isect the angle with string, rod, pencil & straightedge. 6) onstructing ngle isectors by Folding Slide 183 / On patty paper, create any angle of your choice. Make it appear large on your patty paper. Label the points, &.

62 onstructing ngle isectors by Folding Slide 184 / Fold your patty paper so that ray lines up with ray. rease the fold. onstructing ngle isectors by Folding Slide 185 / Unfold your patty paper. raw a ray along the fold, starting at point. raw and label a point on your ray. Try This! Slide 186 / 190 isect the angle with folding. 7)

63 Try This! Slide 187 / 190 isect the angle with folding. 8) 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 PR Sample Test Questions Slide 189 / 190 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 Luck! Return to Table of ontents

64 Question 2/25 Slide 190 / 190 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 m 2 = 90 m 6 = 90 m 3 = m 6 m 1 + m 6 = 90 E m 3 + m 4 = 90 F m 4 + m 5 = 90