Unit 4 Congruent Triangles Homework 2 Angles Of Triangles Equals

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Triangle Congruence - SSS and SAS

We have learned that triangles are congruent if their corresponding sides and angles are congruent. However, there are excessive requirements that need to be met in order for this claim to hold. In this section, we will learn two postulates that prove triangles congruent with less information required. These postulates are useful because they only require three corresponding parts of triangles to be congruent (rather than six corresponding parts like with CPCTC). Let's take a look at the first postulate.

SSS Postulate (Side-Side-Side)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

As you can see, the SSS Postulate does not concern itself with angles at all. Rather, it only focuses only on corresponding, congruent sides of triangles in order to determine that two triangles are congruent. An illustration of this postulate is shown below.

We conclude that ?ABC??DEF because all three corresponding sides of the triangles are congruent.

Let's work through an exercise that requires the use of the SSS Postulate.

Exercise 1

Solution:

The only information that we are given that requires no extensive work is that segment JK is congruent to segment NK. We are given the fact that A is a midpoint, but we will have to analyze this information to derive facts that will be useful to us.

In the two triangles shown above, we only have one pair of corresponding sides that are equal. However, we can say that AK is equal to itself by the Reflexive Property to give two more corresponding sides of the triangles that are congruent.

Finally, we must make something of the fact A is the midpoint of JN. By definition, the midpoint of a line segment lies in the exact middle of a segment, so we can conclude that JA?NA.

After doing some work on our original diagram, we should have a figure that looks like this:

Now, we have three sides of a triangle that are congruent to three sides of another triangle, so by the SSS Postulate, we conclude that ?JAK??NAK. Our two column proof is shown below.

We involved no angles in the SSS Postulate, but there are postulates that do include angles. Let's take a look at one of these postulates now.

SAS Postulate (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A key component of this postulate (that is easy to get mistaken) is that the angle must be formed by the two pairs of congruent, corresponding sides of the triangles. If the angles are not formed by the two sides that are congruent and corresponding to the other triangle's parts, then we cannot use the SAS Postulate. We show a correct and incorrect use of this postulate below.

Incorrect:

The diagram above uses the SAS Postulate incorrectly because the angles that are congruent are not formed by the congruent sides of the triangle.

Correct:

The diagram above uses the SAS Postulate correctly. Notice that the angles that are congruent are formed by the corresponding sides of the triangle that are congruent.

Let's use the SAS Postulate to prove our claim in this next exercise.

Exercise 2

Solution:

For this solution, we will try to prove that the triangles are congruent by the SAS Postulate. We are initially given that segments AC and EC are congruent, and that segment BC is congruent to DC.

If we can find a way to prove that ?ACB and ?ECD are congruent, we will be able to prove that the triangles are congruent because we will have two corresponding sides that are congruent, as well as congruent included angles. Trying to prove congruence between any other angles would not allow us to apply the SAS Postulate.

The way in which we can prove that ?ACB and ?ECD are congruent is by applying the Vertical Angles Theorem. This theorem states that vertical angles are congruent, so we know that ?ACB and ?ECD have the same measure. Our figure show look like this:

Now we have two pairs of corresponding, congruent sides, as well as congruent included angles. Applying the SAS Postulate proves that ?ABC??EDC. The two-column geometric proof for our argument is shown below.

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.

28

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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