CHAPTER 4- PART 2Congruent Triangles

CPCTC (4-4)• Definition of Congruent Triangles

• Corresponding • Parts of • Congruent • Triangles are • Congruent

Using Congruent Triangles: CPCTC• CPCTC – Corresponding parts of congruent triangles are

congruent.

DEF ≅ DGF by _____

What other parts arecongruent by CPCTC?

G

D

E

F

Using Congruent Triangles: CPCTC

ACE BDE≅by _________

What other parts are congruent by CPCTC?

E

A

BC

D

Class work• 4-3 Part 1 worksheet (complete in groups)

Homework• 4-4 Practice worksheet #1-5

Homework• p. 246-247 #6, 7, 9, 10, and 16 (proofs using CPCTC)

Triangle Paper Activity (5 minutes!)

• Fold paper in half and crease the fold• Cut off a corner (not on the creased side) to make a triangle.• Open the triangle and lay it flat.• Label the triangle vertices P, R, and S and the bottom of the

fold Q.• Measure the lengths of the sides of the triangle using a ruler

and measure the angles using a protractor. Record your measurements on the triangle.

• Compare the lengths of the sides and the angle measures. Identify angle pairs and side pairs that are congruent.

• Write a conjecture (if… then…) regarding your findings.

Isosceles Triangles (4-5)Parts of an Isosceles Triangle1. Base2. Legs3. Vertex Angle4. Base Angles

Isosceles Triangles (4-5)Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Converse of Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Triangle Paper Activity• Using the same triangle as previously,

• Draw a line down the crease of the triangle.• Measure <PRQ and <SRQ. Compare these measurements.• Measure <PQR and <SQR. Compare these measurements.• Find PQ and QS using a ruler.• Compare these two angle pairs and lengths of sides with other

students’ pairs of angles and lengths of sides. Do you notice a pattern?

• Write a conjecture (conclusion based on observations). Write your conjecture as an “if-then” conditional statement.

Isosceles Triangles (4-5)

Corollary:

The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

In other words, the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

Isosceles Triangles (4-4)

Find the values

of x and y.

63

y

x

C

D

A

B

Equilateral Triangles (4-5)Corollaries involving equilateral triangles:

1. If a triangle is equilateral, then the triangle is equiangular.

2. If a triangle is equiangular, then the triangle is equilateral.

Classwork• 4-4 Handout (part 1)

Homework• Isosceles and Equilateral Triangles worksheet

Right Triangle Congruence (4-6)

Hypotenuse-Leg (HL) Theorem

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

Summary of Ways to Prove Two Triangles are CongruentAll triangles SSS SAS ASA AAS

Right Triangles

All ways listed above

HL

Example

Class work• Right Triangle Congruence worksheet

• Review answers in ~15 minutes!

Homework

• 4-5 Other Methods of Proving Triangles Congruent worksheet (1-5 all)

Section 4-7 Using More Than One Pair of Congruent Triangles

U

S

T

V

W

1

2

3 4

Given:

Prove: TU TW

How many triangles do you see here?

Which two triangles can we first prove are congruent?

Statements Reasons

Given

2. Reflexive

ASA

4. CPCTC

Reflexive

SAS

7. CPCTC

U

S

T

V

W

1

2

3 4