Congruent TrianglesGeometry Chapter 4

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

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.

Chapter 4 Standards

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.

4-1 Congruent FiguresEQ: 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.

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

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

4-1 Congruent FiguresEQ: 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.

Theorem 4-1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

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

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

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

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

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

If you can prove that all sides of two triangles are congruent, then you know the triangles are congruent.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

The congruent angle must be the INCLUDED angle between the two sides.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

4-2 Triangle Congruence by SSS and SASEQ: Prove that triangles are congruent.

Warm Up:

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

What does the SAS Postulate say about triangle congruency?

4-3Triangle Congruence by ASA and AASEQ: 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.

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

What happened?

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

Which triangles are congruent?

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

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

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

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

homework: page 215 (1-15) all

Retake 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.

warm up

4-4: CPCTCEQ: 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

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

Before you can use CPCTC in a proof, you must first show that the triangles are congruent.

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

4-4: CPCTCEQ: Are all parts of congruent triangles congruent?

Warm Up

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

Construct an Isosceles Triangle

1. 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.

4-5 Isosceles and Equilateral TrianglesEQ: 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.

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

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

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

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

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

A corollary is a statement that follows directly from a theorem

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

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

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

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

Warm Up

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

4-6 Congruence in Right TrianglesEQ: What are the theorems about right triangles?

Homework: p237 (1-8)

Warm Up:

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.

4-7 Using Corresponding Parts of Congruent Triangles

4-7 Using Corresponding Parts of Congruent Triangles

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.

4-7 Using Corresponding Parts of Congruent Triangles

4-7 Using Corresponding Parts of Congruent Triangles

Worksheet 4-7, both sides

Chapter 4 test Tuesday – period 3 Chapter 4 test Wednesday – period 6

4-7 Using Corresponding Parts of Congruent Triangles

Draw RSTU congruent to GHIJ.

List all the congruent parts of the two figures.

Chapter 4 Review Questions

h

i

jg

What else would you need to have to prove these triangles congruent by SSS?

By SAS?

Chapter 4 Review Questions

What other piece of information do you need to prove these triangles are congruent?

By ASA? By SAS? By AAS?

Chapter 4 Review Questions

prove ∠P ≅∠Q

Chapter 4 Review Questions

prove ∠P ≅∠Q

Chapter 4 Review Questions

prove ∠P ≅∠Q

Chapter 4 Review Questions

Chapter 4 Review Questions

Chapter 4 Review Questions

Given: KM≅LJ, KJ ≅ LM Prove: OJ ≅ OM

Chapter 4 Review Questions