Geometry. Slide 1 / 183. Slide 2 / 183. Slide 3 / 183. Congruent Triangles. Table of Contents

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1 Slide 1 / 183 Slide 2 / 183 Geometry ongruent Triangles Table of ontents Slide 3 / 183 ongruent Triangles Proving ongruence SSS ongruence SS ongruence S ongruence S ongruence HL ongruence click on the topic to go to that section Triangle ongruence Proofs PT Isosceles Triangle Theorem PR Sample Questions

2 Throughout this unit, the Standards for Mathematical Practice are used. Slide 4 / 183 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. Slide 5 / 183 ongruent Triangles Return to Table of ontents Similar Triangles Slide 6 / 183 We learned in the Similar Triangles topic (Triangles unit) that if two triangles are similar: ll their angles are congruent ll their corresponding sides are in proportion We also learned how to identify the corresponding sides as being opposite to equal angles, or subtended by equal angles nd we learned that the constant of proportionality for the corresponding sides of one triangle to the other was called "k." If needed, go back to review that topic before proceeding.

3 ongruent Triangles Slide 7 / 183 ongruent triangles are a special case of similar triangles. The constant of proportionality is one, so the corresponding sides are of equal measure. For congruent triangles, all the angles are congruent N all the corresponding sides are congruent. Naming ongruent Triangles Slide 8 / 183 Just as in the case of similar triangles, the naming of congruent triangles is important: order matters. The statement: Δ is congruent to ΔF indicates that these triangles are congruent. N that these angle measures are equal: m = m m = m m = m F N these lengths are equal: = = F = F Proving Triangles ongruent Slide 9 / 183 We can prove triangles congruent by proving the measures of all three corresponding angles and the lengths of all three corresponding sides are equal. arlier we showed that we need to prove only two angles are congruent to show that triangles are similar, since the third angle must then be congruent. There are similar shortcuts to proving triangles congruent.

4 Third ngle Theorem Slide 10 / 183 Recall the proof showing if we know that two pairs of corresponding angles are congruent, then the third pair of corresponding angles are congruent as well. Statement 1 and Given Reason 2 m = m ; m = m efinition of angles 3 4 m + m + m = 180º m + m + m F = 180º m + m + m = 180º m + m + m F = 180º Triangle Sum Theorem Substitution Property of quality 5 m + m + m = m + m + m F 6 m = m F Substitution Property of quality Subtraction Property of quality orresponding Parts Slide 11 / 183 Let's review identifying the corresponding parts (angles and sides) of pairs of triangles. orresponding Parts Slide 12 / 183 Given that Δ is congruent to ΔF, identify all the congruent corresponding parts F

5 orresponding Parts Given that Δ is congruent to ΔF, the triangles are marked accordingly in this diagram. Slide 13 / 183 F Slide 14 / 183 Δ Δ Part Segment Segment Segment orresponding Part Segment Segment Segment xample Slide 15 / 183 Given that Δ ΔLMN, identify all the corresponding angles and sides. (raw a diagram) orresponding Sides orresponding ngles

6 1 What is the corresponding part to J? R K Q P Slide 16 / 183 J P K L R Q ΔJKL ΔPQR 2 What is the corresponding part to Q? Slide 17 / 183 J R K Q P P K L R Q ΔJKL ΔPQR 3 What is the corresponding part to QP? JL LK KJ PQ Slide 18 / 183 J P K L R Q ΔJKL ΔPQR

7 4 The congruence statement for the two triangles is: ΔV ΔX ΔX ΔX ΔV ΔX ΔV ΔX Slide 19 / 183 X V 5 omplete the congruence statement: ΔXY ΔXW ΔWX ΔWX ΔXW Slide 20 / 183 W X Y Properties of ongruence and quality Slide 21 / 183 We will be using the three properties of congruence we learned earlier Reflexive Property of ongruence Symmetric Property of ongruence Transitive Property of ongruence s well as the four properties of equality we learned earlier Reflexive Property of quality Symmetric Property of quality Transitive Property of quality Substitution Property of quality

8 Slide 22 / 183 Proving ongruence SSS (Side-Side-Side) Return to Table of ontents Proving ongruence Slide 23 / 183 ongruent triangles have all congruent sides and angles. However, congruence can be proven by showing less than that. We will prove some theorems which you can then use as shortcuts to proving two triangles congruent. It is not necessary to prove that all the angles and sides are congruent. Side-Side-Side Triangle ongruence Slide 24 / 183 If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. uclid - ook 1: Proposition 8

9 Side-Side-Side Triangle ongruence Slide 25 / 183 uclid showed that: Having three equal sides requires having three equal angles. Therefore, having three pairs of equal sides verifies that two triangles are congruent since all their corresponding sides and angles must be congruent. Side-Side-Side Triangle ongruence Slide 26 / 183 uclid's argument of this (and for some of the following postulates/ theorems) was based on transposing one triangle on top of the other. He confirmed that if all the corresponding sides are equal, once you place one triangle atop the other in the correct orientation, all the sides have to line up and all the angles must as well. Recall the Fourth xiom: Things which coincide with one another are equal to one another. Side-Side-Side Triangle ongruence Slide 27 / 183 This is shown below. an you imagine a way that the corresponding angles could be of different measure without changing the length of one of the sides? This is often called the SSS Triangle ongruence for short. lick here to go to the lab titled, "Triangle ongruence SSS"

10 xample 1 Slide 28 / 183 Prove that ΔFK is congruent to ΔGK K Solution: F G The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other. y SSS, this proves congruence. Please note that this requires that all three sides are congruent. xample 2 Slide 29 / 183 Given: FG = JK, FH = JH, and H is the midpoint of GK Prove: ΔFGH ΔJKH F J G H K xample 2 Slide 30 / 183 F J G H K Statement FG = JK, FH = JH and H is 1 the midpoint of GK 2 FG JK, FH JH Given Reason efinition of congruent segments 3 GH HK efinition of midpoint 4 ΔFGH ΔJKH Side-Side-Side Triangle ongruence

11 6 Δ ΔHJK Slide 31 / 183 True False J H K 7 Δ ΔHJK Slide 32 / 183 True False J H K 8 ΔSRT ΔSUT True False S Slide 33 / 183 R T U

12 9 Provide the reason for the second step. Given Side-Side-Side Triangle ongruence Reflexive property of congruence Substitution property of congruence Transitive property of congruence S Slide 34 / 183 Statement 1 RS US, RT UT Given Reason 2 ST ST? 3 ΔSRT ΔSUT R Side-Side-Side Triangle ongruence T U nswer 10 Δ is congruent to Slide 35 / 183 ΔQRS S ΔSRQ Δ ΔRSQ Q R 4 4 Slide 36 / 183 Proving ongruence SS (Side-ngle-Side) Return to Table of ontents

13 Side-ngle-Side Triangle ongruence Slide 37 / 183 If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. - uclid's lements - ook One: Proposition 4 Side-ngle-Side Triangle ongruence Slide 38 / 183 s in Side-Side-Side Triangle ongruence, uclid verifies Side-ngle-Side Triangle ongruence by superposition (transposing one triangle atop the other.) He thereby indicates that if two sides of two triangles, and the angles contained by those sides, are equal, then all of the sides and angles must be equal...showing congruence. Side-ngle-Side Triangle ongruence Slide 39 / 183 Given that two triangles have two equal corresponding sides and equal angles contained by those two equal sides, the third sides must also be equal. Tap below and the third side of each triangle will become visible. Tap to reveal third side of the triangles

14 Side-ngle-Side Triangle ongruence Slide 40 / 183 It is clear that the third side of each triangle is completely defined by the two other sides and their included angle. So, the third sides must also be congruent. This is often called the SS Triangle ongruence for short. Side-ngle-Side Triangle ongruence Slide 41 / 183 So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent. lick here to go to the lab titled, "Triangle ongruence SS" xample Slide 42 / 183 Given: MP NP and LP OP Prove: ΔMLP ΔNOP M N 1 2 P L O

15 Onswer 11 Provide the reason for line 2. M Given Side-Side-Side Triangle ongruence Side-ngle-Side Triangle ongruence Vertical angles are congruent L lternate interior angles are congruent 1 2 P N O Slide 43 / 183 Given: MP NP and LP OP Prove: ΔMLP ΔNOP nswer Statement Reason 1 MP NP and LP OP Given 2 1 2? 3 ΔMLP ΔNOP? 12 Provide the reason for line 3. M Given Side-Side-Side Triangle ongruence Side-ngle-Side Triangle ongruence Vertical angles are congruent L lternate interior angles are congruent 1 2 P N Slide 44 / 183 Given: MP NP and LP OP Prove: ΔMLP ΔNOP Statement Reason 1 MP NP and LP OP Given 2 1 2? 3 ΔMLP ΔNOP? 13 What is the included angle of the given sides of the triangle? ΔJKL, sides KL and JK Slide 45 / 183 J K L Hint: raw the triangle!

16 14 List the congruent parts of the triangles below. Is ΔPQR ΔSTV? Yes No Slide 46 / 183 P S 5 5 R Q V 4 T 15 Is ΔFGH ΔXY by Side-ngle-Side Triangle ongruence? Yes No Why? X Slide 47 / 183 F H G Y 16 Using SS Triangle ongruence, what information is needed to show Δ Δ? Slide 48 / 183

17 17 What type of congruence exists between the two triangles? SSS Triangle ongruence SS Triangle ongruence Not enough information Slide 49 / What type of congruence exists between the two triangles? SSS Triangle ongruence SS Triangle ongruence Not enough information Slide 50 / What type of congruence exists between the two triangles? Slide 51 / 183 SSS Triangle ongruence SS Triangle ongruence Not enough information

18 20 What type of congruence exists between the two triangles? Slide 52 / 183 SSS Triangle ongruence SS Triangle ongruence Not enough information 21 What type of congruence exists between the two triangles? Slide 53 / 183 SSS Triangle ongruence SS Triangle ongruence Not enough information 22 What type of congruence exists between the two triangles? Slide 54 / 183 SSS Triangle ongruence SS Triangle ongruence Not enough information

19 Slide 55 / 183 Proving ongruence S (ngle-side-ngle) Return to Table of ontents ngle-side-ngle Triangle ongruence Slide 56 / 183 nother way to prove two triangles are congruent makes use of uclid's Fifth Postulate. This illustration should look familiar from the unit on parallel lines. It shows non-parallel lines intersected by a transversal ngle-side-ngle Triangle ongruence Slide 57 / 183 We know from uclid's Fifth Postulate, that the non-parallel lines will intersect on the side of the transversal on which the sum of the interior angles is less than 180º. In this case, that's the side of the transversal with angles 2 and

20 ngle-side-ngle Triangle ongruence y extending the lines and decreasing angles 2 and 4, we can see where the non-parallel lines intersect. This forms a triangle in which the transversal is one side and the two non-parallel lines form the other two sides. Slide 58 / ngle-side-ngle Triangle ongruence Slide 59 / 183 You can see that we have formed a triangle on the right side of the transversal, with the transversal providing one side and the two non-parallel lines the other two sides. Let's examine that triangle ngle-side-ngle Triangle ongruence Slide 60 / 183 We can see that 2 and 4 lead to only one possible value for 5, since the angles must add to 180º. That means that two triangles with two corresponding angles which are congruent, must have their third angles equal, so they are similar...but we knew that from earlier

21 ngle-side-ngle Triangle ongruence Slide 61 / 183 Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size. The two triangles must be congruent. This is called S Triangle ongruence for short lick here to go to the lab titled, "Triangle ongruence S" 5 To see a visual representation of what we discussed for S Triangle ongruence, click the link below What is the included side between X and W? Slide 62 / 183 YX YW XW X Y W 24 What is the included side between X and Y? Slide 63 / 183 XW YX YW X Y W

22 25 What information is needed to have S Triangle ongruence between the two triangles? Slide 64 / 183 M N O P 26 What information is needed to have S Triangle ongruence between the two triangles? Slide 65 / Why is FM GMH? Slide 66 / 183 S Triangle ongruence vertical angles included angles congruent F M G H

23 28 What type of congruence exists between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information Slide 67 / 183 Q R U T S Strategy to Prove ongruence Slide 68 / 183 When you have overlapping figures that share sides and/or angles, marking the diagram with the given information and separating the triangles (when needed) make it easier to understand the problem. nother strategy that you could use is to look for repeating letters once you separate the two triangles. When 2 letters repeat, then you have a common side shared. When 1 letter repeats, then you have a common angle shared. 29 What type of congruence exists between ΔJLM and ΔNLK? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information J K Slide 69 / 183 L Hints: Pull click to the reveal triangles apart! Mark click to reveal the congruent parts! re there any common sides/angles (look for letters click to reveal that repeat)? N M

24 30 What type of congruence exists between ΔQ and ΔR? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information Slide 70 / 183 Hint Q Mark the diagram with the given information. e careful you don't always use all information click to reveal R 31 What type of congruence exists between ΔQR and ΔRQ? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information Slide 71 / 183 Hints: Pull click to the reveal triangles apart! Mark click to reveal the congruent parts! re there any common sides/angles (look for click to reveal letters that repeat)? Q R 32 What type of congruence exists between ΔSN and ΔT? Given: S SSS Triangle T ongruence N T S SS Triangle ongruence S Triangle ongruence Not enough information Hint N t lick the to Reveal intersection of two lines you always have vertical lick angles Slide 72 / 183

25 33 What type of congruence exists between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information Slide 73 / What type of congruence exists between the two triangles? P M SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence Not enough information Given: P M S P M Hint: Mark click to the reveal given information into your diagram. Identifying vertical angles plays an important part. Slide 74 / 183 Slide 75 / 183 Proving ongruence S (ngle-ngle-side) Return to Table of ontents

26 ngle-ngle-side Triangle ongruence ased on that same logic, if NY two corresponding angles and one corresponding side of a pair of triangles are congruent, the triangles must also be congruent. Slide 76 / 183 This follows from the fact that the Triangle Sum Theorem tells us that once we know the measures of two angles, we know the measure of the third, since they must add to 180º S and S Triangle ongruence Slide 77 / 183 nother way of looking at these two theorems is that once you show that two corresponding angles in two triangles are congruent, you know that the third angles are congruent and that the two triangles are similar. That means that they have the same shape. If you can show a side in one of those triangles is congruent to the corresponding side of the other, you know that they are same size. Thus the scale factor, k, is 1. If they are the same size and shape, they are congruent. S and S Triangle ongruence It is really just a formality whether you use the term S or S, since all three angles must be congruent. Slide 78 / 183 However, to note the difference, if the angles are both adjacent to the side which has shown to be congruent, the reason for congruence is S ( F). If not, it is S ( GHI JKL). G J VS. S F H I K S L

27 xample Slide 79 / 183 Given: H HT T Is ΔT ΔHT? T H xample Slide 80 / 183 1) Mark the diagram: Given: H HT T T H xample Slide 81 / 183 2) y the reflexive property: T T Therefore, congruence ΔHT statement? ΔT by S Triangle ongruence. T H

28 35 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence Given: F SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information H G Slide 82 / What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Q Slide 83 / 183 S R 37 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Slide 84 / 183

29 38 What type of congruence exists, if any, between the two triangles? W R SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Slide 85 / 183 Q T 39 What type of congruence exists, if any, between the two triangles? SSS Triangle H F ongruence SS Triangle S ongruence S Triangle ongruence G S Triangle ongruence Not enough information Slide 86 / What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Slide 87 / 183

30 41 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Given: bisects, Slide 88 / What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence Not enough information Slide 89 / 183 Slide 90 / 183 HL ongruence Return to Table of ontents

31 Hypotenuse-Leg Triangle ongruence Slide 91 / 183 The final shortcut to proving congruence is Hypotenuse Leg Triangle ongruence, or the HL Triangle ongruence for short. This theorem states that if two right triangles have their hypotenuses and one of their legs congruent, then the triangles are congruent. The HL Triangle ongruence can be considered a corollary of the SSS Triangle ongruence. Hypotenuse-Leg Triangle ongruence Slide 92 / 183 In a right triangle, the sum of the squares of the lengths of the two legs must equal the square of the length of the hypotenuse. c 2 = a 2 + b 2 If we are given that for two right triangles the hypotenuse and one of the legs are equal (c 1 = c 2 and a 1 = a 2), then we know that the other leg but also be equal (b 1 = b 2). Thus, HL Triangle ongruence can be considered a special case, or corollary, of the Side-Side-Side Triangle ongruence. Hypotenuse-Leg Triangle ongruence Slide 93 / 183 c 1 2 = a b 1 2 c 2 2 = a b 2 2 Solving for b in both equations c a 1 2 = b 1 2 c a 2 2 = b 2 2 b 1 2 = c a 1 2 b 2 2 = c a 2 2 Substituting c 1 = c 2 and a 1 = a 2 b 1 2 = c a 2 2 b 2 2 = c a 2 2 b 1 2 = b 2 2 b 1 = b 2

32 xample re these two triangles congruent? T Slide 94 / 183 R S These are right triangles, so look for HL Triangle ongruence. xample Slide 95 / 183 T R S Recall that the side opposite the right angle is the hypotenuse, and the other two sides are called legs. Hypotenuse: RT Leg: TS y the HL Triangle ongruence, RST Postulates/Theorems to Prove Triangles ongruent To use the congruence postulates/theorems, we need to know or be able to show the following congruences between two triangles: Slide 96 / 183 Side-Side-Side (SSS): three sides Side-ngle-Side (SS): two sides and the included angle ngle-side-ngle (S): two angles and the included side ngle-ngle-side (S): two angles and one non-included side Hypotenuse-Leg (HL): hypotenuse and one leg (right triangles)

33 43 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence Given: Q X SS Triangle ongruence S Triangle ongruence S Triangle R S Y ongruence HL Triangle ongruence F Not enough information Mark the given Hint on the diagram. Note that it is a right triangle. lick to reveal Slide 97 / What type of congruence exists, if any, between the two triangles? SSS Triangle M P ongruence SS Triangle ongruence S Triangle ongruence L N O Q S Triangle ongruence HL Triangle ongruence F Not enough information If they are congruent what is the congruence statement? Slide 98 / What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S Triangle F ongruence S Triangle ongruence HL Triangle ongruence F Not enough information If they are congruent what is the congruence statement? Slide 99 / 183

34 46 What type of congruence exists, if any, between the two triangles? X SSS Triangle ongruence SS Triangle ongruence W Y S Triangle ongruence U S Triangle ongruence HL Triangle ongruence F Not enough information T V Slide 100 / 183 If they are congruent what is the congruence statement? 47 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence W SS Triangle ongruence S Triangle ongruence S Triangle ongruence HL Triangle ongruence Q Y F Not enough information Slide 101 / 183 If they are congruent what is the congruence statement? 48 What type of congruence exists, if any, between the two triangles? K SSS Triangle ongruence SS Triangle ongruence J L S Triangle ongruence M S Triangle ongruence HL Triangle ongruence F Not enough information O N Slide 102 / 183 If they are congruent what is the congruence statement?

35 49 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence H S Triangle ongruence S Triangle G ongruence HL Triangle ongruence F Not enough information If they are congruent what is the congruence statement? F Slide 103 / What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence H F S Triangle ongruence S Triangle G ongruence HL Triangle ongruence F Not enough information If they are congruent what is the congruence statement? Slide 104 / What type of congruence exists, if any, between the two triangles? K F SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence HL Triangle ongruence F Not enough information M Slide 105 / 183 If they are congruent what is the congruence statement?

36 52 What type of congruence exists, if any, between the two triangles? SSS Triangle What angles are congruent when ongruence parallel lines are cut by a transversal? alternate interior SS Triangle lick to Reveal ongruence P O S Triangle ongruence S Triangle ongruence HL Triangle ongruence Y U F Not enough information Slide 106 / 183 If they are congruent what is the congruence statement? 53 What type of congruence exists, if any, between the two triangles? SSS Triangle O K ongruence SS Triangle ongruence S Triangle ongruence S Triangle J M ongruence HL Triangle ongruence F Not enough information Slide 107 / 183 If they are congruent what is the congruence statement? 54 What type of congruence exists, if any, between the two triangles? SSS Triangle ongruence SS Triangle ongruence S S Triangle ongruence S Triangle ongruence HL Triangle ongruence F Not enough information X Slide 108 / 183 If they are congruent what is the congruence statement?

37 Slide 109 / 183 Triangle ongruence Proofs Return to Table of ontents Strategy to Prove ongruence Slide 110 / 183 First: identify given information Second: use a diagram that is marked with given information Third: review congruence postulates/theorems - what information is needed (sides/angles) to use one of these? SSS SS S S HL xample Slide 111 / 183 Given: MF = MH = 8 and m F = m H = 90º Prove: ΔFM ΔGHM F 90 8 M 8 90 H G

38 55 Provide the reason for line 3. ngle-side-ngle Triangle ongruence F Side-Side-Side Triangle ongruence Side-ngle-Side Triangle ongruence Vertical angles are congruent lternate interior angles are congruent Statement 90 8 M 8 H 90 Given: MF = MH = 8 and m F = m H = 90º Prove: ΔFM ΔGHM Reason G nswer Slide 112 / MF = MH = 8 and m F = m H = 90º Given 2 MF MH and F H efn. of congruence 3 FM HMG? 4 ΔFM ΔGHM? 56 Provide the reason for line 4. ngle-side-ngle Triangle ongruence F Side-Side-Side Triangle ongruence Side-ngle-Side Triangle ongruence Vertical angles are congruent lternate interior angles are congruent Statement 90 8 M 8 H 90 Given: MF = MH = 8 and m F = m H = 90º Prove: ΔFM ΔGHM Reason 1 MF = MH = 8 and m F = m H = 90º Given G nswer Slide 113 / MF MH and F H efn. of congruence 3 FM HMG? 4 ΔFM ΔGHM? # of congruent angles ongruent Reasons Summary postulate/ theorem (rag those that don't work out of the chart. Then put HL where it would belong.) Slide 114 / SSS SS SS S S HL

39 xample Given: F G, FK GK, and K K Prove: ΔFK ΔGK K Slide 115 / 183 Solution (two-column): F G Statements Reasons 1) F G, FK GK K K 1) Given 2) ΔFK ΔGK 2) SSS Triangle ongruence xample Slide 116 / 183 Given: bisects Prove: Statements Reasons 1., bisects Given click 2. efinition click of bisector Reflexive property click 4. SS Triangle ongruence click Problem Slide 117 / 183 Write a two-column proof. Given: FG FG Prove: G FG G F

40 Problem Slide 118 / 183 Given: FG FG G F Statements 1. FG FG 2. G FG 3. G G 4. ΔG ΔFG Reasons 1. Given lick 2. lternate Interior ngles are lick 3. Reflexive Property of ongruence lick 4. SS Triangle ongruence lick Slide 119 / 183 Problem: complete the proof Given: and are right angles; T T Prove: ΔT ΔT T Statements Reasons 1. and are right angles T T 4. T T lick lick 5. ΔT ΔT lick 1. Given lick 2. right 's are congruent lick 3. Given 4. Vertical 's are congruent lick 5. S Triangle ongruence lick Problem: complete the proof Given: Prove: Δ Δ Slide 120 / 183 Statements 1., 2. and are right 's lick lick lick lick lick 6. Δ Δ Reasons 1. Given lick 2. efinition of lines lick 3. ll right angles are congruent 4. Given 5. reflexive property of lick 6. S Triangle ongruence lick

41 Problem Given:, is the midpoint of and Prove: Δ Δ Statements 1) 2) Reason s 1) 2) Slide 121 / 183 3) 3) 4) 4) 5) 5) SSS Triangle ongruence ef. of midpoint = ~ = ~ Given = ~ is the midpoint of and Slide 122 / 183 PT orresponding Parts of ongruent Triangles are ongruent Return to Table of ontents PT orresponding Parts of ongruent Triangles are ongruent Slide 123 / 183 Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true. PT states that if two or more triangles are congruent by one of the congruence postulates/theorems - SSS, SS, S, S, or HL, then all of their corresponding parts are also congruent.

42 Process for proving that two segments or angles are congruent Slide 124 / Find two triangles in which the two sides or two angles are corresponding parts 2. Prove that the two triangles are congruent (SSS, SS, S, S, HL) 3. State that the two parts are congruent, using as the reason: "orresponding Parts of ongruent Triangles are ongruent" 57 Which two triangles might you try to prove congruent in order to prove NM NO? ΔLO ΔNOL ΔLM ΔNM M N O L Slide 125 / Which two triangles might you try to prove congruent in order to prove O LM? Slide 126 / 183 ΔOL ΔNOL ΔLM ΔNM M N O L

43 59 Which two triangles might you try to prove congruent in order to prove 1 2? Slide 127 / 183 ΔLO ΔLM ΔNOL ΔNM M O N 1 2 L nswer 60 Which two triangles might you try to prove congruent in order to prove N LN? M O ΔLO N ΔNOL ΔLM ΔNM L Slide 128 / 183 Problem: complete the proof Slide 129 / 183 Given:,, is the midpoint of Prove: Statements Reasons is the midpoint of Δ Δ 6. lick 1. Given lick 2. Given lick 3. Given lick 4. efinition of midpoint lick 5. SSS Triangle ongruence lick 6. PT lick

44 Problem Given: and are right angles Prove: Slide 130 / 183 We are given that,, and and are lick right angles. Since all right angles are congruent,. With lick the congruent angles and segments, we can conclude that Δ Δ by S. Therefore, by PT. lick lick lick Problem: complete the proof W X Slide 131 / 183 Given: P is the midpoint of WY, P is the midpoint of X Prove: WX Y Statements Reasons P Y 1. P is the midpoint of WY 2. P is the midpoint of X 3. WP YP, P XP 4. WPX YP 5. ΔWPX ΔYP 6. X lick 7. WX Y lick lick 1. Given lick 2. Given lick 3. efinition of midpoint lick 4. Vertical angles are congruent lick 5. SS Triangle ongruence lick 6. PT lick 7. If alt. int. angles are congruent, then lines are parallel lick dditional Proof Practice Slide 132 / 183 Website link: Interactive Proofs

45 Using What You've Learned Slide 133 / 183 Learning mathematics is like climbing a ladder, one step leads to the next. No step is more difficult than the one before it, as long as you take them one step at a time. ongruent Triangles are an important step in geometry. They will be used through much of the rest of this course. For example, the following PR-type question looks like it's about parallelograms, but you can answer every part of this question with what you know already, before you even study quadrilaterals. Try it out. Given:, Slide 134 / 183 Prove: is a parallelogram Statements Reasons 1., Δ Δ 4. lick 5., 6. is a parallelogram 1. Given lick 2. Reflexive Property of ongruence lick 3. SS Triangle ongruence lick 4. PT lick 5. If alt. int. angles are congruent, then lines are parallel lick 6. efinition of a parallelogram Slide 135 / 183 Isosceles Triangle Theorems Return to Table of ontents

46 Isosceles Triangles Slide 136 / 183 In an isosceles triangle, the base is the side that is not necessarily congruent to the other two sides (legs). If an isosceles triangle has 3 congruent sides, it is also an equilateral triangle. leg leg base Isosceles Triangles Slide 137 / 183 The vertex angle is opposite the base, and is the included angle between the legs. vertex angle The base angles are the angles opposite the legs, and are included by a leg and the base. base angles ase ngles Theorem Slide 138 / 183 The base angles of an isosceles triangle are congruent. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. uclid: ook One Proposition 5 This says that the angles opposite equal sides of a triangle are of equal measure.

47 Proof of ase ngles Theorem Given: In Δ, Prove: Slide 139 / 183 There are several ways to prove this. uclid's way is pretty complicated. The link below shows two typical proofs and an alternate third one. The third proof uses the fact that order OS matter in making statements of congruence. It was supposedly generated by a computer. Proof of ase ngles Theorem elow are the arguments that could be used to explain the third proof from the link on the previous slide (computer generated). Slide 140 / 183 Given: In Δ, Prove: Statement Reason 1 In Δ, Given 2 Symmetric Property of 3 Reflexive Property of 4 Δ Δ SS Triangle ongruence 5 PT 61 What is the value of x in this triangle? Justify your answer. Slide 141 / 183 y x 44

48 62 What is the value of y in this triangle? Justify your answer. Slide 142 / 183 y x Solve for x and y. xplain your reasoning. Slide 143 / x y 64 What is the measure of each base angle? Slide 144 /

49 65 The vertex angle of an isosceles triangle is 38. What is the measure of each base angle? Slide 145 / onverse of ase ngles Theorem Slide 146 / 183 If two angles of a triangle are congruent, then the sides opposite them are congruent. If in a triangle two angles be equal to one another, the sides opposite those angles will also be equal to one another uclid: ook One Proposition 6 The sides opposite equal angles of a triangle are of equal length. 66 What is the length of F? Slide 147 / F 3

50 67 What is the length of? Slide 148 / F 7 quilateral Triangles Slide 149 / 183 If three sides of a triangle are equal, each of the three angles has a measure of 60º. n equilateral triangle is a special case of an isosceles triangle. The base and legs are all of equal length. Since all the angles are opposite sides of equal length, they all have equal measure. Since the three angles add to 180º and have equal measure, they each have a measure of 60º. onversely, if two angles of a triangle are each 60º, the third angle also has a measure of 60º and all the sides are of equal length. quilateral Triangles Slide 150 / 183 If three sides of a triangle are equal, each of the three angles has a measure of 60º. onversely, if the angles of a triangle each have a measure of 60º, all the sides are of equal length. lso, if two angles of a triangle each have a measure of 60º, the third angle must also has a measure of 60º since the Interior ngles Theorem indicates that the angles must add to 180º Then, all the sides must be of equal length.

51 68 lassify the triangle by sides and angles. Slide 151 / 183 equilateral isosceles F acute obtuse scalene G right equiangular 40º 7 69 lassify the triangle by sides and angles. Slide 152 / 183 equilateral isosceles scalene equiangular F G acute obtuse right lassify the triangle by sides and angles. Slide 153 / 183 equilateral isosceles G right acute scalene equiangular F obtuse 113º 3 3 5

52 71 lassify the triangle by sides and angles. Slide 154 / 183 equilateral isosceles scalene equiangular G F right acute obtuse lassify the triangle by sides and angles. Slide 155 / 183 equilateral isosceles scalene equiangular G F right acute obtuse 60º 60º xample Slide 156 / 183 Find the value of x and y. xplain your reasoning. x y

53 73 What is the value of y? Slide 157 / y 74 What is the value of x? Justify your answer. Slide 158 / x 75 Solve for x in the diagram. Slide 159 / / x

54 PR Sample Test Questions Slide 160 / 183 The remaining slides in this presentation contain questions from the PR Sample Test. fter finishing this unit, you should be able to answer these questions. Good Luck! Return to Table of ontents Question 18/25 Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction) Part 76 The first step of the construction is to draw an arc centered at point that intersects both sides of the given angle. What is established by the first step? Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Slide 161 / 183 PR Released Question (OY) Question 18/25 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Slide 162 / 183 Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction) Part omplete the sentence with the choices given below. 77 The construction creates congruent triangles. and are congruent because of the postulate/theorem. side-side-side angle-side-angle side-angle-side angle-angle-side

55 Question 18/25 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Slide 163 / 183 Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction) Part omplete the sentence with the choices given below. 78 It follows that must be the angle bisector of because. Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Marcella drew each step of a construction of an angle bisector. Slide 164 / 183 Step 1 Step 2 Step 3 Step 4 Step 5 Part ngle is given in Step 1. escribe the instructions for Steps 2 through 5 of the construction. This is a great problem and draws on a lot of what we've learned. Try it in your groups.then we'll work on it step by step together by asking questions that break the problem into pieces. PR Released Question (P) Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Marcella drew each step of a construction of an angle bisector. Step 1 Step 2 Step 3 Step 4 Step 5 79 ngle is given in Step 1. What would be the description used to get from Step 1 to Step 2? onstruct an arc located in the interior of angle using a compass centered at point with a radius length that is congruent to the radius length used to draw the arc centered at point. Label the intersection point of the 2 interior arcs point. onstruct an arc located in the interior of angle using a compass centered at point and a radius greater than half of angle. raw a ray, which is the angle bisector of angle. onstruct an arc using a compass centered at point and any radius length. Label the points where the arc intersects the angle and. Slide 165 / 183

56 Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Marcella drew each step of a construction of an angle bisector. Step 1 Step 2 Step 3 Step 4 Step 5 80 What would be the description used to get from Step 2 to Step 3? onstruct an arc located in the interior of angle using a compass centered at point with a radius length that is congruent to the radius length used to draw the arc centered at point. Label the intersection point of the 2 interior arcs point. onstruct an arc located in the interior of angle using a compass centered at point and a radius greater than half of angle. raw a ray, which is the angle bisector of angle. onstruct an arc using a compass centered at point and any radius length. Label the points where the arc intersects the angle and. Slide 166 / 183 Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Marcella drew each step of a construction of an angle bisector. Step 1 Step 2 Step 3 Step 4 Step 5 81 What would be the description used to get from Step 3 to Step 4? onstruct an arc located in the interior of angle using a compass centered at point with a radius length that is congruent to the radius length used to draw the arc centered at point. Label the intersection point of the 2 interior arcs point. onstruct an arc located in the interior of angle using a compass centered at point and a radius greater than half of angle. raw a ray, which is the angle bisector of angle. onstruct an arc using a compass centered at point and any radius length. Label the points where the arc intersects the angle and. Slide 167 / 183 Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Marcella drew each step of a construction of an angle bisector. Step 1 Step 2 Step 3 Step 4 Step 5 82 What would be the description used to get from Step 4 to Step 5? onstruct an arc located in the interior of angle using a compass centered at point with a radius length that is congruent to the radius length used to draw the arc centered at point. Label the intersection point of the 2 interior arcs point. onstruct an arc located in the interior of angle using a compass centered at point and a radius greater than half of angle. raw a ray, which is the angle bisector of angle. onstruct an arc using a compass centered at point and any radius length. Label the points where the arc intersects the angle and. Slide 168 / 183

57 Question 2/11 Topic: ngle onstructions (Unit 2) & Triangle ongruence Proofs Part Marcella wants to explain why the construction produces and angle bisector. She makes a new step with line segments and added to the construction, as shown. Slide 169 / 183 Using the figure, prove that ray bisects angle. e sure to justify each statement of your proof. This is a great problem and draws on a lot of what we've learned. Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces. 83 What have we learned that will help solve this problem? Slide 170 / 183 onstruction of an angle bisector w/ a compass and straightedge Ways to prove triangles congruent The corresponding parts of congruent triangles are congruent (PT) ll of the above 84 What should be the first statement in our proof? Slide 171 / 183 bisects

58 85 Why can we say that these two segments in step #1 are congruent? PT efinition of an ngle isector Reflexive Property of ongruence oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. Slide 172 / What should be the second statement in our proof? bisects Slide 173 / Why can we say that these two segments in step #2 are congruent? PT efinition of an ngle isector Reflexive Property of ongruence oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. Slide 174 / 183

59 88 What should be the third statement in our proof? bisects Slide 175 / Why can we say that these two segments in step #3 are congruent? PT efinition of an ngle isector Reflexive Property of ongruence oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. Slide 176 / What should be the fourth statement in our proof? bisects Slide 177 / 183

60 91 Why can we say that these two triangles in step #4 are congruent? SSS Triangle ongruence SS Triangle ongruence S Triangle ongruence S Triangle ongruence HL Triangle ongruence Slide 178 / Since we know that the triangles are congruent, what should be the fifth statement in our proof? Slide 179 / 183 bisects 93 Why can we say that these two angles in step #5 are congruent? PT efinition of an ngle isector Reflexive Property of ongruence oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. Slide 180 / 183

61 94 Since we know that the angles are congruent, what should be the sixth statement in our proof? bisects Slide 181 / What is the final reason in our proof? PT efinition of an ngle isector Reflexive Property of ongruence oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. Slide 182 / 183 elow is a completed version of the proof that we just wrote. Given: The construction of the figure to the right Prove: bisects Slide 183 / 183 Statements Reasons 1) 1) oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. 2) 2) oth segments were drawn with the same compass setting, and all radii of a given circle are congruent. 3) 3) Reflexive Property of 4) 4) SSS Triangle 5) 5) PT 6) is bisects 6) efinition of an ngle isector