Unit 4 Congruent Triangles Homework 1 Classifying Triangles Notes

Presentation on theme: "Chapter 4 Congruent Triangles."— Presentation transcript:

1 Chapter 4Congruent Triangles

2 4. 1 Congruent Figures 4. 2 Triangle Congruence by SSS and SAS 4
4.1 Congruent Figures 4.2 Triangle Congruence by SSS and SAS 4.3 Triangle Congruence by ASA and AASStudents Will be able toRecognize Congruent figures and their Corresponding PartsProve two triangles are congruent using SSS and SASProve two triangles are congruent using ASA and AASMA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

3 Notes forIn order to prove that two figures are congruent we need to make sure that all sides and all angles of one polygon are equal to all angles and sides of another polygon.In order to do this, we must first be able to decide which sides and angles on one polygon match with the sides and angles of another polygon… we call these matching pieces “corresponding parts”.If the polygons are congruent then the corresponding parts should be equal.

4 Notes forProving two polygons are congruent could take a lot of work. For example if we want to show that two triangles are congruent we would need to show that all 3 angles and all 3 sides of one triangle are equal to all 3 angles and all 3 sides of another triangle! This is 6 different pairs of congruent parts!!!We can use Logic and a few theorems to make some short cuts.

5 Notes forTheorem: 3rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle then the 3rd angles of both must be congruent.SSS Theorem: If 3 sides of one triangle are congruent to 3 sides of another triangle then the triangles are congruentSAS Theorem: If 2 sides of one triangle and the included angle of the triangle are congruent to 2 sides and the included angle of another triangle then the two triangles are congruent

6 Notes forASA Theorem: If 2 angles of one triangle and the included side of the triangle are congruent to 2 angles and the included side of another triangle then the two triangles are congruentAAS Theorem: If 2 Angles and 1 side of one triangle are congruent to 2 Angles and 1 side of another triangle then the triangles are congruentHypotenuse Leg Theorem: If two right triangles have congruent hypotenuses and another pair of equal sides then the two triangles are congruent

7 Classwork/Home Learning
Page 222 #10-19, 35, 36, 39, 40, 41Page 231 #11-14, 17, 24-26, 35-38Page 238 #13, 16-18, 25, 32-35

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9 4. 4: Corresponding Parts of Congruent Triangles are Congruent 4
4.4: Corresponding Parts of Congruent Triangles are Congruent 4.5: Isosceles and Equilateral TrianglesStudents will be able to:Use Triangle Congruence and CPCTS to prove that parts of two triangles are CongruentUse and apply the properties of isosceles and equilateral TrianglesMA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

10 Notes for 4.4 and 4.5Once you know that two triangles are congruent based on SSS, SAS, ASA, AAS and HL you can now make conclusions about specific corresponding parts of triangles.If you know that two shapes are exactly the same size and exactly the same shape (ie: They are congruent) then it makes sense that specific angles and specific sides that are corresponding should be the same too… this is what CPCTC means.

11 Notes for 4.4 and 4.5With Isosceles and Equilateral Triangles we know even more information because we know that sides are across from equal anglesThis means that in an Isosceles triangle we have 2 equal sides and the two angles across from them are also equal.In an equilateral triangle, all sides and all angels are equal and all angles measure 60 degrees.

12 Classwork/HomeworkPage 247 #6, 11-13, 23-26Page 254 #6-9, 16-19, 37-40

13 4. 6: Congruence in Right Triangles 4
4.6: Congruence in Right Triangles 4.7: Congruence in Overlapping TrianglesStudents will be able toProve right triangles are congruent using the Hypotenuse Leg TheoremIdentify congruent overlapping triangles and use congruent triangle theorems to prove triangles are congruent.MA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

14 Notes for 4.6 and 4.7Hypotenuse Leg Theorem: If two right triangles have congruent hypotenuses and another pair of equal sides then the two triangles are congruentWhen figures are overlapped it may be useful to separate the figures and identify the shared parts

15 Classwork / Home Learning
Page 262# 15, 29-31Page 268# 8-13, 17, 29-32

16 Office AidI am at a meeting in the main office… here’s your list of things to do:Come and see me in the office first!!!!At the back of the room there is a hole puncher and papers – please hole punchClean up my classroomThe baskets in the back have papers and folders please put the papers into the correct student folders. If the student has no folder, just leave out.

Presentation on theme: "Congruent Triangles Geometry Chapter 4."— Presentation transcript:

1 Congruent TrianglesGeometry Chapter 4

2 When you finish the test, please pick up a set of 9 index cards
When you finish the test, please pick up a set of 9 index cards. Copy the nine theorems and postulates from chapter four onto these index cards – including the drawings with them. Write the name of each theorem or postulate on the back of the card.see pages: 199, 205, 206, 213, 214, 228, 235

3 Chapter 4 Standards 2.0 Students write geometric proofs.
4.0 Students prove basic theorems involving congruence.5.0 Students prove that triangles are congruent, and are able to use the concept of corresponding parts of congruent triangles.6.0 Students know and are able to use the triangle inequality theorem.12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

4 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
What do you think makes figures congruent?They have the same size and shape.If you can slide, flip or turn a shape so that it fits exactly on another shape, then they are congruent.

5 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Congruent polygons have congruent corresponding parts – their matching sides and angles.Matching vertices are corresponding vertices.When you name congruent polygons, always list corresponding vertices in the same order.

6 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

7 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Two triangles are congruent if they have three pairs of congruent corresponding sides, and three pairs of congruent corresponding angles. Are the following triangles congruent? Justify your answer.

8 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Theorem 4-1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

9 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

10 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

11 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

12 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

13 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.
If you can prove that all sides of two triangles are congruent, then you know the triangles are congruent.

14 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

15 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

16 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.
The congruent angle must be the INCLUDED angle between the two sides.

17 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

18 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

19 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?Warm Up:

20 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?What does the SAS Postulate say about triangle congruency?

21 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?At your table, choose any two angle measures that add up to less than 120°. (No zeros)Agree on a segment length between 5 and 20 centimeters.Each of you: Draw the line segment, then construct the given angles on each end of the segment to form a triangle.Measure the two remaining sides and compare your answers.

22 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?What happened?

23 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?Which triangles are congruent?

24 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

25 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

26 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

27 Retake for Chapter 3 test:
homework:page 215 (1-15) allRetake for Chapter 3 test:Pick up a retake practice packet.Complete test corrections.You MUST have all chapter 3 homework completed and come in for at least 1 enrichment period before the retake next Thursday.

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29 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
warm up

30 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
Once you show that triangles are congruent using SSS, SAS, ASA or AAS, then you can make conclusions about the other parts of the triangles because, by definition, congruent parts of congruent triangles are congruent.Abbreviate this CPCTC

31 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
Before you can use CPCTC in a proof, you must first show that the triangles are congruent.

32 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

33 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

34 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

35 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

36 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?Warm Up

37 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?Construct an Isosceles Triangle1. Use a straight edge to make a line segment. Label the endpoints A and B.2. Set your compass to a length that is greater than half the length of the segment.3. Without changing the compass setting, make arcs from either end of the line segment.4. Connect the endpoints of the segment to the intersection point of the two arcs. Label this point C.5. Measure the sides of the triangle to confirm that they are equal.

38 Label the point where the fold intersects AB as point D.
4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?Fold your triangle carefully in half, so points A and B are exactly on top of each other.Label the point where the fold intersects AB as point D.What appears to be true of angles A and B?What appears to be true of the intersection of CD and AB?Write a conjecture about the angles opposite the congruent sides of an isosceles triangle.

39 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

40 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

41 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

42 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

43 A corollary is a statement that follows directly from a theorem
4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?A corollary is a statement that follows directly from a theorem

44 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

45 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

46 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

47 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?
Warm Up

48 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

49 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

50 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

51 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

52 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?
Homework: p237 (1-8)

53 4-7 Using Corresponding Parts of Congruent Triangles
Warm Up:

54 4-7 Using Corresponding Parts of Congruent Triangles
When a geometric drawing is complicated, it is sometimes helpful to separate it into more than one drawing.

55 4-7 Using Corresponding Parts of Congruent Triangles

56 4-7 Using Corresponding Parts of Congruent Triangles

57 4-7 Using Corresponding Parts of Congruent Triangles
Sometimes you can prove triangles are congruent and then use their corresponding parts to prove another pair congruent.

58 4-7 Using Corresponding Parts of Congruent Triangles

59 4-7 Using Corresponding Parts of Congruent Triangles
Worksheet 4-7, both sidesChapter 4 test Tuesday – period 3Chapter 4 test Wednesday – period 6

60 Chapter 4 Review Questions
Draw RSTU congruent to GHIJ.List all the congruent parts of the two figures.hijg

61 Chapter 4 Review Questions
What else would you need to have to prove these triangles congruent by SSS?By SAS?

62 Chapter 4 Review Questions
What other piece of information do you need to prove these triangles are congruent?By ASA?By SAS?By AAS?

63 Chapter 4 Review Questions
prove ∠P ≅∠Q

64 Chapter 4 Review Questions
prove ∠P ≅∠Q

65 Chapter 4 Review Questions
prove ∠P ≅∠Q

66 Chapter 4 Review Questions

67 Chapter 4 Review Questions

68 Chapter 4 Review Questions
Given: KM≅LJ, KJ ≅ LMProve: OJ ≅ OM

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