Triangle Congruence Packet #3

Transcription

1 Triangle Congruence Packet #3 Name Teacher 1

2 Warm-Up Day 1: Identifying Congruent Triangles Five Ways to Prove Triangles Congruent In previous lessons, you learned that congruent triangles have all corresponding sides and all corresponding angles congruent. Do we need to show all six parts congruent to conclude that two triangles are congruent? The answer is no. We can show triangles are congruent by showing few than all three sides and angles congruent, so long as these congruent sides and angles are in the correct order. The arrangements that prove triangles congruent are as follows: Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA) Angle-Angle-Side (AAS) Hypotenuse-Leg (HL) for right triangles only We will take a look at each of these in turn. 2

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4 Example 1: Identifying Congruent Triangles ** Challenge** 4

5 Example 2: The pair of triangles below has two corresponding parts marked as congruent Answer: Answer: Answer: Answer: Answer: Answer: 5

6 Example 3: Using the tick marks for each pair of triangles, name the method {SSS, SAS, ASA, AAS} that can be used to prove the triangles congruent. If not, write not possible. (Hint: Remember to look for the reflexive side and vertical angles!!!!) The Reflexive Side Vertical Angles 6

7 The Two That DON T Work So far, we have seen that there are four ways to prove a triangle congruent: SSS, SAS, ASA, and AAS. You might be wondering if there are any configurations that don t prove triangles congruent. There are two: AAA and SSA. Why AAA doesn t work Given: 1 Can we prove that Answer: NO. Having all three angles congruent without any congruent corresponding sides will ensure the triangles are similar (same shape), but not necessarily congruent. Why SSA doesn t work (the Donkey Postulate) Given: Can we prove that Answer: NO. It is possible to draw two different triangles given two congruent corresponding sides and a nonincluded angle. Therefore, we cannot guarantee that, given SSA, we will have two congruent triangles. 7

8 SUMMARY The Two ThaT DoN T work: aaa and SSa 8

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11 Day 2 Proving Triangles Congruent by SSS and SAS Warm-Up What other corresponding parts would have to be proven congruent in order to prove by SAS? Ans: Model Problem #1 Given: D is the midpoint of Prove: 11

12 2) 3) 12

13 Exercise 1) 2) 13

14 Altitude and Median Proofs An altitude of a triangle is a line segment that is drawn from any vertex of a triangle and is PERPENDICULAR to the opposite side. How can we use altitudes in proofs? Given: is an altitude to Conclusion: Chain of Reasoning: Example: Given: and Prove: BAD CAD 14

15 A median to a triangle is a line segment that joins any vertex of a triangle to the MIDPOINT of the opposite side. Given: is a median to Conclusion: Chain of Reasoning: Example Given: is a median to,. Prove: 15

16 Summary Tell which method you would use to prove each set of triangles congruent: 1) 2) Homework Level A: 1) 16

17 2) Given:, Prove: LEVEL B: 3) 17

18 4) Level C: 5) Given: Prove: NOT RSP 18

19 (Hint: Redraw the overlapping triangles so you can see more clearly! ) 6) 19

20 Day 3 Proving Triangles Congruent by ASA and AAS Warm-Up What additional information would you need to prove these triangles congruent by ASA? Model Problem #1 20

21 Model Problem #2 Model Problem #3 21

22 You Try! Prove: 22

23 Model Problem #4 SUMMARY 23

24 Homework Level A:

25 Level B: 3. Prove: JGH PMO 4. Given:, T is the midpoint of. Prove: RST UVT 25

26 Day 4 CPCTC Warm - Up 26

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28 Example 1: Example 2: 28

29 You Try! Z 29

30 Summary 30

31 Homework 31

32 Proofs Level A: Level B: 32

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34 Day 5 Overlapping Triangles Warm Up 34

35 4. Draw the triangles separately here: 5. Draw the triangles separately here: 35

36 You Try! 36

37 Homework

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39 Day 6 ISOSCELES Triangles Theorem Warm - Up Using Isosceles Triangles in Proofs When we studied triangles in earlier chapters, we learned about the properties of the isosceles triangles. We can use these ideas to help us when constructing proofs. 39

40 Example 1: Given: XYZ is isosceles with base. Prove: Z You Try It! 40

41 Example 2: Example 3: 41

42 Level A Homework 42

43 B is the midpoint of 43

44 Level B

45 Day 7 CPCTC and Beyond Warm - Up 45

46 Example 1: Example 2: Prove: B is the midpoint of 46

47 Ex 3: Ex 4: Given: Prove: 47

48 Proving Perpendicular Segments Theorem: If two angles are congruent and supplementary each angle is a right angle Ex 5: Prove: 48

49 Homework

50 3. Given: Prove: 4. 50

51 5. 51

52 Day 8 HL 52

53 Example 1: 53

54 You Try It! Given: Prove: 54

55 Summary Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. What is the first thing you have to prove when you plan to use Hypotenuse-Leg? 55

56 Homework Level A 56

57 Level B Day 9 Indirect Proofs 57

58 So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you prove a statement indirectly by showing that its opposite can not be true. Normally, a presence of the word not or the symbol in a problem indicates the needs for Indirect Proofs. 58

59 When is an Indirect Proof Needed? Generally, the word "not" or the presence of a "not symbol" (such as the not equal sign ) in a problem indicates a need for an Indirect Proof. Direct Proof Example Indirect Proof Given: BDis an altitude to AC. Prove: BD AC Given: BDis an altitude to AC. Prove: BD is AC How do I write an Indirect Proof? Exercise 1: Identifying Contradictions A. In each question, information is given regarding a proof. On the line provided, write the assumption that must appear in an indirect proof of the statement. 1. Given: In Δ RST, R T Prove: RS = RT R Assume S T 2. Given: In Δ ABC, AB BC Prove: m C m A C Assume A B 3. Lines l and m are cut by transversal k; 1 2 Prove: l m Assume

60 Exercise 2: Format of an Indirect Proof Given: is not isosceles. is an altitude. Prove: does not bisect. S T A T E M E N T S 1. is not isosceles. is an altitude. 2. Assume Given. R E A S O N S 2. Assumption leading to a contradiction is isosceles

61 Exercise Given: DB AC Prove: AB and CD do not bisect each other Statements Reasons Summary To start an indirect proof. Assume the opposite of the PROVE statement. Try to contradict the GIVEN statement. 61

62 Homework 62

63 4. Given: is scalene Prove: 5. Given: BE is the median of AC, Prove: ΔABC is not Isosceles ABE CBE 63

64 Day 10 REVIEW 64

65 Addition and Subtraction Use ADDITION when: Use SUBTRACTION when: a) you have a gap. b) you need larger pieces. a) you have overlap. b) you need smaller pieces. CPCTC Stands for: Corresponding Parts of Congruent Triangles are Congruent Use when you are asked to prove SEGMENTS or ANGLES congruent. Isosceles Triangle Theorem Or Overlapping Triangles Draw triangles separately. Mark up both the together and separate diagrams. Indirect Proofs 65