Nov 9-12:30 PM. Math Practices. Triangles. Triangles Similar Triangles. Throughout this unit, the Standards for Mathematical Practice are used.

Transcription

1 Triangles Triangles Similar Triangles Nov 9-12:30 PM 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: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. 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. Math Practices 1

2 efinition 13. boundary is that which is an extremity of anything. efinition 14. figure is that which is contained by any boundary or boundaries. efinition 19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 1 The letter on this triangle that corresponds to a side is: Oct 16-4:41 PM 2

3 2 The letter on this triangle that represents a vertex is: with a triangle symbol Δ in front followed vertices. Name the 3 sides of this triangle Oct 16-4:41 PM 3 4 Jul 8-4:07 PM Jul 8-4:07 PM 3

4 5 Δ Δ Δ Δ side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle. sides are to. Jul 8-4:07 PM 6 7 Jul 8-4:19 PM Jul 8-4:19 PM 4

5 8 9 & & Jul 8-4:19 PM Jul 8-4:19 PM Types of Triangles of all different measure. lengths and angles However, there are names given to triangles which have specific or special angles or some number of equal sides or angles. lassifying Triangles efinition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone equal, and a scalene triangle is that which has its three sides unequal uclid defined the names for a number of these in his definitions. ug 12-5:06 PM 5

6 lassifying Triangles lassifying Triangles efinition 21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has its three angles acute. efinition 21: "...an acute-angled triangle is that which has its three angles acute." º ug 12-5:06 PM lassifying Triangles lassifying Triangles efinition 21: "...a right-angled triangle is that which has a right angle..." º º efinition 20: "...an isosceles triangle is that which has two of its sides alone equal..." xº xº 6

7 lassifying Triangles lassifying Triangles The equal angles, of measure xº in this diagram, are called the base angles. The side between them is called the base. The other two sides, opposite the base angles and congruent to each other are called the legs. efinition 21: "...an obtuse-angled triangle is that which has an obtuse angle..." º xº xº This is a special case of an acute triangle. º º lassifying Triangles lassifying Triangles xº efinition 20: "...an equilateral triangle is that which has its three sides equal..." None of the sides or angles of a scalene triangle are congruent with one another. efinition 20: "...a scalene triangle is that which has its three sides unequal..." xº xº º 7

8 10 n isosceles triangle 11 is an isosceles triangle. 12 triangle can have more than one obtuse angle. 13 triangle can have more than one right angle. 8

9 14 ach angle in an equiangular triangle measures n equilateral triangle is also an isosceles triangle 16 This triangle is classified as. (hoose all that apply.) acute 17 This triangle is classified as. (hoose all that apply.) acute right isosceles º 8.6 right isosceles 57º obtuse equilateral 60º º obtuse equilateral 79º 44º 7.4 F equiangular F equiangular G scalene G scalene Oct 16-5:46 PM Oct 16-5:46 PM 9

10 This triangle is classified as. (hoose all that apply.) acute 19 This triangle is classified as. hoose all that apply. acute right isosceles obtuse equilateral right isosceles obtuse equilateral F equiangular F equiangular G scalene G scalene Oct 16-5:46 PM Oct 16-5:46 PM Measure and lassify the triangle by sides and angles Measure and lassify the triangle by sides and angles isosceles, acute

11 Measure and lassify the triangle by sides and angles 20 Side lengths: 3 cm, 4 cm, 5 cm quilateral Side lengths: 3 cm, 2 cm, 3 cm 22 Side lengths: 5 cm, 5 cm, 5 cm quilateral quilateral 11

12 23 24 quilateral quilateral 25 Side lengths: 3 cm, 4 cm, 5 cm quilateral 26 lassify the triangle by sides and angles quilateral 12

13 27 lassify the triangle by sides and angles quilateral 28 lassify the triangle by sides and angles quilateral M We can use what we learned about parallel lines to determine the sum of the measures of the angles of base of the triangle and the other through the opposite vertex. 13

14 29 between them? Is the same true for the pair of angles labeled y? nd extend to make it a transversal. Then, let's label some of the angles. Jul 8-5:11 PM Therefore, both angles labeled x are equal and can be called x, and x has the same measure as. Repeat the same process with side and find an angle along the upper parallel line equal to angle Let's just re-label the upper angles with, and. 14

15 The sum of those angles along that upper parallel line equals 180º, so + + = 180º We made no special assumptions about this triangle, so this lick here to go to the lab titled, "Triangle º 30 What is m? J Find the measure of the missing angle. 32º K 20º L 15

16 31 What is the measurement of the missing angle? 32 Δ 33 In ΔF, if m is 63 and m is 12, find m F. Solve for x 16

17 34 Solve for x. 35 (3x-17) Then find: m Q = (x+40) (2x-5) The acute angles of a right triangle are complementary. Triangle Sum Theorem Given: Triangle is a right triangle Prove: Its acute angles, ngles and, are complementary ngle Relationships 17

18 36 Which reason applies to step 1? Subtraction Property of quality Substitution Property of quality Given efinition of right triangle efinition of a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary 37 Which reason applies to step 2? Subtraction Property of quality Substitution Property of quality Given efinition of right triangle efinition of a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary ug 16-3:04 M ug 16-3:04 M 38 Which reason applies to step 3? The measure of a straight angle is 180º m + m + m = 180º m + m = 90º m + m = 180º is a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary 39 Which reason applies to step 4? Subtraction Property of quality Substitution Property of quality Given efinition of right triangle efinition of a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary ug 16-3:04 M ug 16-3:04 M 18

19 40 Which reason applies to step 5? Subtraction Property of quality Substitution Property of quality Given efinition of right triangle efinition of a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary 41 Which reason applies to step 6? Subtraction Property of quality Substitution Property of quality Given efinition of right triangle efinition of a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary ug 16-3:04 M ug 16-3:04 M 42 Which reason applies to step 7? The measure of a straight angle is 180º The sum of the interior angles of a triangle is 180º The acute angles are complementary The acute angles are supplementary is a right angle Statement Reason 1 Triangle is a right triangle? Right triangles contain a right 2 angle.? 3? Interior ngles Theorem 4 m = 90º? 5 90º + m + m = 180º? 6 m + m = 90º? 7? efinition of complementary Triangle Sum Theorem Given: Triangle is a right triangle Prove: Its acute angles, ngles and, are complementary Statement Reason 1 Triangle is a right triangle Given Right triangles contain a right 2 angle. efinition of right triangle 3 m + m + m = 180º Interior ngles Theorem 4 m = 90º efinition of right angle 5 90º + m + m = 180º Substitution Property of quality 6 m + m = 90º Subtraction Property of quality The acute angles are efinition of 7 complementary complementary ug 16-3:04 M ug 16-3:04 M 19

20 The measure of one acute angle of a right triangle is five times the measure of the other acute angle. 43 In a right triangle, the two acute angles sum to 90. Find the measure of each acute angle. 44 What is the measurement of the missing angle? 45 Solve for x. 20

21 46 Solve for x m 2 = m 3 = 49 21

22 xterior angles are formed by extending any side of a triangle. xterior ngle Theorem The exterior angle is then the angle between that extended side and the nearest side of the triangle. One exterior angle is shown below. Take a moment and draw another. Return to Table xº xterior ngles xterior ngles Since a triangle has three vertices and two external angles can be drawn at each vertex, it is possible to draw six external angles to a triangle. The exterior angles at each vertex are congruent, since they are vertical angles. raw the other external angle at Vertex. xº xº xº xterior ngles xterior ngles 22

23 The interior angles of this triangle are, and. Once an exterior angle is drawn, one interior angle is adjacent, and the two others are remote. 50 Which are the remote interior angles in this instance? & & & Since you can draw exterior angles at any vertex, any interior angle can be the remote depending on at which vertex you draw the external angle. xº xº In this case, and are the remote interior angles and is the adjacent interior angle. xº xterior ngles ug 22-1:01 PM 51 of 2 and 1? 52 Jul 7-10:14 PM Jul 7-10:18 PM 23

24 m = m + m or x = m + m xº xº Given: is an exterior angle of Δ and and are remote interior angles. Prove: m = m + m ngle Relationships 1 53 Which reason applies to step 2? ngles that form a linear pair are supplementary efinition of complementary Interior ngles Theorem Substitution Property of quality xº efinition of a right angle Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary? 3? efinition of supplementary 4 m + m + m = 180? 5 m + m = m + m + m 6?? Subtraction Property of quality 1 54 Which statement applies to step 3? m + m = 180 m = m + m m + m = 180 m + m = 90 x m + m = 180 Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary? 3? efinition of supplementary 4 m + m + m = 180? m + m = m + m + 5 m? 6? Subtraction Property of quality ug 16-3:04 M ug 16-3:04 M 24

25 1 55 Which reason applies to step 4? ngles that form a linear pair are supplementary efinition of complementary Interior ngles Theorem Substitution Property of quality x efinition of a right angle Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary? 3? efinition of supplementary 4 m + m + m = 180? m + m = m + m + 5 m? 6? Subtraction Property of quality 1 56 Which reason applies to step 5? ngles that form a linear pair are supplementary efinition of complementary Interior ngles Theorem Substitution Property of quality x efinition of a right angle Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary? 3? efinition of supplementary 4 m + m + m = 180? m + m = m + m + 5 m? 6? Subtraction Property of quality ug 16-3:04 M ug 16-3:04 M 1 57 Which statement applies to step 6? m + m = 180 m = m + m m + m = 180 m + m = 90 x m + m = 180 Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary? 3? efinition of supplementary 4 m + m + m = 180? m + m = m + m + 5 m? 6? Subtraction Property of quality Given: is an exterior angle of Δ and and are remote interior angles. Prove: m = m + m 1 Statement is an exterior angle of Δ and and are remote interior angles Reason Given 2 and are supplementary ngles that form a linear pair are supplementary 3 + m = 180 efinition of supplementary 4 m + m + m = 180 Interior ngles Theorem m + m = m + m + Substitution Property of 5 m quality 6 m = m + m Subtraction Property of quality x ug 16-3:04 M ug 16-3:04 M 25

26 58 m m 1 = m P Δ m 1 = m Q + m R m m 1 = m Q + m P 59 In this case, what must be the relationship between the interior angles of ΔPQR and 2? m 2 m 2 = m P m 2 = m Q + m R m m 2 = m Q + m P Jul 7-10:26 PM Jul 7-10:26 PM Solve for x and y. º º º xterior ngles 26

27 Solve for x and y. xº yº 75º 50º 60 Solve for x. 60º xº yº 55º xterior ngles 61 Solve for y. 62 Find the value of x. 60º xº yº 94º yº 55º 60º 2xº Sep 6-4:05 PM 27

28 63 Find the value of x. 100º yº 64 Find the value of x. (3x-5) y (2x+3)º 51º (x+2) 33 Sep 6-4:05 PM Sep 6-4:05 PM 65 Segment PS bisects RST, what is the value of w? S R wº P T Sep 6-4:05 PM ec 26-7:17 PM 28

29 66 Find the measure of º º º º Response Response 68 Find the measure of Find the measure of 4. º º º º Response Response 29

30 70 Find the measure of 5. º Inequalities in º Response Inequalities in one Triangle ngle Inequalities in a Triangle To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in one triangle" The longest side is always opposite the largest angle. Go to the sketch, "Inequalities in one triangle." Go to the worksheet, "Inequalities in one triangle." The shortest side is always opposite the smallest angle. Inequalities in 1 Triangle Sketch & Worksheet Sep 27-8:18 M 30

31 71 Name the longest side of this triangle. They are all equal 72 Name the shortest side of this triangle. They are all equal Sep 8-2:56 PM Sep 8-2:56 PM 73 Name the shortest side of this triangle. They are all equal 74 Name the largest angle of this triangle. They are all equal Sep 8-2:56 PM Sep 8-2:56 PM 31

32 75 Name the smallest angle of this triangle. They are all equal Name the smallest angle of this triangle. They are all equal Sep 8-2:56 PM Sep 8-2:56 PM Length Inequalities in a Triangle Length Inequalities in a Triangle No side can be longer than the sum of the other two sides. No side can be longer than the sum of the other two sides. No side can be less than the difference of the other two sides. This follows from the fact that if the two shorter sides cannot be placed at a 180º angle and exceed the length of the longest side, a triangle cannot be made. s shown below, if the blue side is longer than the sum of the red and the green side, it cannot form a triangle. Move the sides below and try to form a triangle. Sep 27-8:18 M Sep 27-8:18 M 32

33 Length Inequalities in a Triangle No side can be less than the difference of the other two sides. 77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6? This follows from the fact that if the longer sides cannot, when placed at a 0 angle, reach the end of the shorter side, a triangle cannot be made. s shown below, if the blue side is too short to reach the red line, even when the red line is at the smallest angle, it cannot form a triangle. Sep 27-8:18 M Sep 8-3:18 PM 78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7? 79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6? Sep 8-3:18 PM Sep 8-3:18 PM 33

34 80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8? Similar Sep 8-3:18 PM ongruence Recall that: Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps. ongruent Line Segments We learned earlier that: Only line segments with the same length are congruent. lso, all congruent segments have the same length. This is the symbol for congruence: a b c d If a is congruent to b, this would be shown as a b which is read as "a is congruent to b." ug 23-1:53 M ug 23-2:00 M 34

35 ongruent ngles Recall: Two angles are congruent if they have the same measure. Two angles are not congruent if they have different measures. If m = m If m m ongruent Triangles Triangles are made up of three line segments N three angles For one triangle to be congruent to another all three sides N all three angles must be congruent. ug 23-2:00 M ug 23-2:00 M Similar Triangles ongruent Triangles If all the sides of two triangles are congruent, we will soon show that all the angles are also congruent. Therefore, the triangles are congruent. However, two triangles can have all their angles congruent, with all or none of their sides being congruent. In that case, they are said to be Similar Triangles. ongruent Triangles are also Similar Triangles since their angles are all congruent. ongruent triangles are therefore a special case of similar triangles. We will focus on similar triangles first, and then work with congruent triangles in a later unit. Similar triangles represent a great tool to solve problems, and are the foundation of trigonometry. ug 23-2:00 M ug 23-2:00 M 35

36 Similar Triangles Have Proportional Sides Theorem Similar triangles have the same shape, but can have different sizes. If they have the same shape and are the same size, they are both similar and congruent. Similar Triangles This is the symbol for similarity So, the symbolic statement for Triangle is similar to Triangle F is: Δ ~ ΔF F ug 23-2:00 M ug 23-2:00 M Naming Similar Triangles Δ ~ ΔF This statement tells you more than that the triangles are similar. It also tells you which angles are equal. In this case, that m = m m = m m = m F Naming Similar Triangles Δ ~ ΔF So, when you are naming similar triangles, the order of the letters matters. They don't have to be alphabetical. ut they have to be named so that equal angles correspond to one another. nd, thereby which are the corresponding, proportional, sides. corresponds to corresponds to F corresponds to F ug 23-2:00 M ug 23-2:00 M 36

37 Proving Triangles Similar ngle-ngle Similarity Theorem If you can prove that all three angles of two triangles are congruent, you have directly proven that they are similar. However, there are shortcuts to proving triangles similar. We will explore three sets of conditions that imply the three angles of two triangles are congruent, meaning that the triangles must be similar. We know from the Triangle Sum Theorem that the sum of the interior angles of a triangle is always 180º. So, if two triangles have two pair of congruent angles which sum to x, then the third angle in both triangles must be (180 - x)º...forming three congruent pairs of angles. One way to prove that two triangles are similar is to prove that two of the angles in each triangle are congruent. ug 23-2:00 M ug 23-2:00 M ngle-ngle Similarity Theorem If two angles of a triangle are congruent to two angles of another triangle, their third angles are congruent and the triangles are similar. Here's the proof: Statement Reason and in Δ are to and 1 in ΔF 2 m = m ; m = m m + m + m = 180º 3 m + m + m F = 180º m =180º - (m + m ) 4 m F =180º- (m + m ) m =180º - (m + m ) 5 m F =180º- (m + m ) Given efinition of ongruent ngles Triangle Sum Theorem Subtraction Property of quality Substitution Property of quality 6 m = m F Substitution Property of quality 7 Δ and ΔF are similar efinition of Similarity Side-Side-Side Similarity Theorem If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which their corresponding sides subtend. uclid - ook Six: Proposition 5 quiangular triangles are similar, so this states that triangles with proportional sides are similar. This is a second way to prove triangles are similar: If you can prove that all three pairs of sides in two triangles are proportional, then you have proven the triangles similar. ug 23-2:00 M ug 23-2:00 M 37

38 Side-Side-Side Similarity Theorem This follows from the way we constructed congruent angles. We made use of the fact that if angles are congruent, their sides are separating at the same rate as you move away from the vertex. Here's the drawing we used to construct so it would be congruent to FGH. Side-Side-Side Similarity Theorem If we draw the green line segments connecting the points where the blue arcs intersect the rays, we can see that the length of that segment would be the same for both angles. Since the angles are congruent, the line segment opposite those angles will also be congruent, if it intersects both sides of the angle at the same distance from the vertex in both cases. F F G H G H ug 23-2:00 M ug 23-2:00 M Side-Side-Side Similarity Theorem In this case the segments and will be congruent since segments G and G are also congruent to segments and. Therefore ΔG is congruent to Δ, since all the sides and angles are the same. hanging the scale of Δ won't change the angle measures. The sides would then be in proportion to those of ΔG, but not equal. Side-Side-Side Similarity Theorem The diagram below shows an expansion of Δ and we see that the measures of the angles are unchanged. They are still similar triangles. The corresponding sides are in proportion. F F G H G H ug 23-2:00 M ug 23-2:00 M 38

39 Side-Side-Side Similarity Theorem Removing the arcs and shifting the smaller triangle within the larger makes it clear that all angles are congruent and the sides are in proportion. So, the second way to prove triangles similar is to show that all their sides are in proportion. Side-ngle-Side Similarity Theorem If two triangles have one angle equal to another and the sides about the equal angle are in proportion, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. uclid's lements - ook Six: Proposition 6 F The third way to prove triangles are similar is to show they share an angle which is equal and the two sides forming that angle are proportional in the two triangles. G H ug 23-2:00 M ug 23-2:00 M Side-ngle-Side Similarity Theorem Side-ngle-Side Similarity Theorem This directly follows from the work we just did to show that Side-Side-Side proportionality can be used to prove triangles are similar. If you recall, the line segment which makes up the third side of a triangle is completely defined by its opposite angle and the lengths of the other two sides. If the angles are congruent and the two sides of the angle are in proportion, the third side must also be in proportion. If all three sides are in proportion, the triangles must be similar due to the Side-Side-Side Theorem. You can see that on the next page. ug 23-2:00 M ug 23-2:00 M 39

40 Side-ngle Side Similarity Theorem If and segments and are proportional to segments and F, then segment must also be proportional to segment F. Since all the sides are in proportion, the triangles are similar. ommon rror You NNOT prove triangles similar using Side-Side-ngle. This is not the same as Side-ngle-Side. s shown below, two triangles can have two corresponding sides and one corresponding angle congruent, but NOT be similar. lick here to see the third sides. F ug 23-2:00 M ug 23-2:00 M 81 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar 82 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar x x Sep 6-6:38 PM Sep 6-6:38 PM 40

41 83 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They are not similar They may not be similar 84 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar Sep 6-6:38 PM Sep 6-6:38 PM 85 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar 86 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar x x 3 x 6 x 3 6 Sep 6-6:38 PM Sep 6-6:38 PM 41

42 87 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar 88 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar Sep 6-6:38 PM Sep 6-6:38 PM 89 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar 90 Which theorem allows you to prove these two triangles are similar? ngle-ngle Side-ngle-Side Side-Side-Side They may not be similar They are not similar Note that is parallel to. Sep 6-6:38 PM Sep 6-6:38 PM 42

43 Side Splitter Theorem Proof of Side Splitter Theorem ny line parallel to a side of a triangle will form a triangle which is similar to the first triangle. Given: is parallel to Prove: Δ ~ Δ. s we will learn later, it also makes all the sides proportional, splitting them...hence the name of the theorem. Sep 6-6:38 PM Sep 6-6:38 PM 91 What is the reason for step 2? 92 What is the reason for step 3? ngle-ngle Similarity Theorem Side-Side-Side Similarity Theorem Reflexive Property of ongruence When two parallel lines are intersected by a transversal, the corresponding angles are congruent. When two parallel lines are intersected by a transversal, the alternate interior angles are congruent. Statement Reason 1 is parallel to Given 2 ;? 3? 4 Δ ~ Δ? ngle-ngle Similarity Theorem Side-Side-Side Similarity Theorem Reflexive Property of ongruence When two parallel lines are intersected by a transversal, the corresponding angles are congruent. When two parallel lines are intersected by a transversal, the alternate interior angles are congruent. Statement Reason 1 is parallel to Given 2 ;? 3? 4 Δ ~ Δ? Sep 8-12:12 PM Sep 8-12:12 PM 43

44 93 What is the reason for step 4? ngle-ngle Similarity Theorem Side-Side-Side Similarity Theorem Reflexive Property of ongruence When two parallel lines are intersected by a transversal, the corresponding angles are congruent. When two parallel lines are intersected by a transversal, the alternate interior angles are congruent. Statement Reason 1 is parallel to Given 2 ;? 3? 4 Δ ~ Δ? Given: is parallel to Prove: Δ ~ Δ Statement Reason 1 is parallel to Given ; When two parallel lines are 2 intersected by a transvesal, the corresponding angles are congruent 3 Reflexive Property of ongruence 4 Δ ~ Δ ngle-ngle Similarity Theorem Sep 8-12:12 PM ug 16-3:04 M Similar Triangles Have Proportional Sides Theorem Similar triangles have the same shape, but can have different sizes. If they have the same shape and are the same size, they are congruent. If they have the same shape and are different sizes, they are similar and their sides are in proportion. F Similar Triangles Have Proportional Sides Theorem The converse is also true, and will prove very useful. If two triangles are similar, all of their corresponding sides are in proportion. *While uclid does prove this theorem, his proof relies on other theorems which would have to be proven first and would take us beyond the scope of this course. So, we'll just rely on this theorem and note that the proof is available in The lements by uclid - ook Six: Proposition 5. ug 23-2:00 M ug 23-2:00 M 44

45 Similar Triangles and Proportionality In the triangles below, if we know that m = m, m = m, and m = m F, then we know that the triangles are similar. Similar Triangles and Proportionality We also then know that the corresponding sides are proportional. The symbol for proportional is the Greek letter, alpha:α α, since corresponds to α F, since corresponds to F α F, since corresponds to F F F ug 23-2:00 M ug 23-2:00 M orresponding Sides Our work with similar triangles and our future work with congruent triangles requires us to identify the corresponding sides. One way to do that is to locate the sides opposite congruent angles. If we know triangles and F are similar and that angle is congruent to angle, then the sides opposite and are in proportion: α F orresponding Sides nother way of identifying corresponding sides is to use uclid's description "...those angles [are] equal which their corresponding sides subtend." elow, since angle is equal to angle and angle is equal to angle, then sides and are in proportion. F F ug 23-2:00 M ug 23-2:00 M 45

46 orresponding Sides ither approach works; use the one you find easiest. Identify corresponding sides as the sides connecting equal angles or the sides opposite equal angles...you'll get the same result. Similar Triangles and Proportionality nother way of saying two sides are proportional is to say that one is a scaled-up version of the other. If you multiply all the sides of one triangle by the same scale factor, k, you get the other triangle. In this case, if Δ is k times as big as ΔF, then: = k = kf = kf F F ug 23-2:00 M ug 23-2:00 M Similar Triangles and Proportionality Or, dividing the corresponding sides yields: = F = F = k This property of proportionality is very useful in solving problems using similar triangles, and provides the foundation for trigonometry. 94 If m = m, m = m, and m = m F, identify which side corresponds to side. F FG F F ug 23-2:00 M ug 25-2:33 PM 46

47 95 If m I = m M, m H = m N, and m J = m L, identify xample - Proportional Sides which side corresponds to side IJ. MN NL ML L Given that Δ is similar to ΔF, and given the indicated lengths, find the lengths and. 5 7 M 8 4 F J N ug 25-2:33 PM ug 25-2:55 PM xample - Proportional Sides xample - Proportional Sides F Since the triangles are similar we know that the following relationship holds between all the corresponding sides. = F = F = k First, let's find the constant of proportionality, k, by using the two sides for which we have values: and F. What ratio could I write to determine the value of k? 8 = = k = 2 F 4 8 That means that the other two sides of Δ will also be twice as large as the corresponding sides of ΔF = F = F = k = 2 How can we write the proportions required to calculate and? 4 F ug 25-2:55 PM ug 25-2:55 PM 47

48 xample - Proportional Sides 96 Given that m = m, m = m, and m = m F. If = 8, = 6, and = 4, F =? = 2 5 = 2 = 10 F = 2 7 = 2 = 14 4 F F ug 25-2:55 PM ug 25-2:55 PM 97 Given that ΔJIH is similar to ΔLMN; find the length of LM. 98 Given that ΔJIH is similar to ΔLMN; find the length of LN. 10 L 10 L M 14 M N 5 N 5 ug 25-2:55 PM ug 25-2:55 PM 48

49 99 Given that is parallel to and the given lengths, find the length of. 100 Given that is parallel to and the given lengths, find the length of ug 25-2:55 PM ug 25-2:55 PM xample - Similarity & Proportional Sides If they are similar write a similarity statement. If they are not similar, explain why. xample - Similarity & Proportional Sides ug 25-2:55 PM ug 25-2:55 PM 49

50 xample - Similarity & Proportional Sides If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero. Side of ΔPK Length Side of ΔRL Length Ratio K 9 R P 15 RL PK 18 L ΔKP is similar to ΔRL. heck to make sure that all the sides are in the correct order. ug 25-2:55 PM ug 25-2:55 PM 102 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero. 103 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero. 2.8 ug 25-2:55 PM ug 25-2:55 PM 50

51 onverse of Side Splitter Theorem 104 Find the value of x to prove that is parallel to R. If a line divides the two sides of a triangle proportionally, then the line is parallel to the third side. R Sep 6-6:38 PM ug 25-2:55 PM 105 Find the value of x to prove that F is parallel to MN. 106 Find the value of y. ug 25-2:55 PM ug 25-2:55 PM 51

52 107 Find the value of y. 108 Find the value of y. ug 25-2:55 PM Proportions of Similar Triangles 109 The figure Δ ~ ΔF with side lengths as indicated. What is the value of x? PR Sample Question and pplications F x 21 From PR OY sample test PR Questions PR Question 52

53 How can we find the distance across the Grand anyon? Using similar triangles and indirect measurement, we can find large distances and the heights of trees, flagpoles, and buildings. indirect measurement direct measurement Solve Problems using Similar Triangles Solve Problems using Similar Triangles First, construct right triangle. Then, construct right triangle Solve Problems using Similar Triangles Solve Problems using Similar Triangles 53

54 How can you prove that ~? How can you find the distance across the Grand anyon? ~ Why? Why? Why? Solve Problems using Similar Triangles Solve Problems using Similar Triangles How do you find d? Write a statement of proportionality that uses d. click there are no shadows? Solve Problems using Similar Triangles Solve Problems using Similar Triangles 54

55 mirror trick Place a mirror with cross hairs (an X) drawn on it flat on the ground Monument lining up with the mirror's cross hairs. In Physics, angle of reflection = angle of incidence angle of reflection angle of incidence and back up to your eye form equal angles. reflected ray incident ray surface Solve Problems using Similar Triangles Solve Problems using Similar Triangles Measure the distance How can you prove that ~ F? ~ Why? Why? Why? Solve Problems using Similar Triangles Solve Problems using Similar Triangles 55

56 How do you find h? proportionality that uses h. click 110 Your little sister wants to know the height of the giraffe. You place a mirror on the ground and stand where you can see the top of the giraffe as shown. How tall is the giraffe? 189 in 15 ft 5 ft 5 ft 3 in Solve Problems using Similar Triangles Solve Problems using Similar Triangles 111 To find the width of a river, you use a surveying technique as shown. Set up the proportion to find the distance across the river. w w w w Solve Problems using Similar Triangles 56