Chapter 4

Congruent Triangles

4.1 Congruent Figures

Objectives:

Recognize congruent figures and their corresponding parts

NCTM 3,6,7,8,9,10

You will learn to identify corresponding parts of congruent polygons, including triangles.

If a polygon can be translated, rotated, or reflected onto another polygon, so that all of the vertices correspond, the polygons are _________________.

The parts of congruent polygons that “match” are called __________________.

The order of the ________ indicates the corresponding parts!

ΔABC ΔXYZ

A

C B

F

E D

In the figure, ΔABC ΔFDE.

As in a mapping, the order of the _______ indicates the corresponding parts.

Congruent Angles Congruent Sides

A F

B D

C E

AB FD

BC DE

AC FE

These relationships help define the congruent triangles.

Definition of

Congruent

Triangles

If the _________________ of two triangles are congruent, then the two triangles are congruent.

If two triangles are _________, then the corresponding parts of the two triangles are congruent.

ΔRST ΔXYZ. Find the value of n.

T

S

R

Z

X Y

40° (2n + 10)°

50°

90°

ΔRST ΔXYZ

S Y

50 = 2n + 10

40 = 2n

20 = n

identify the corresponding parts

corresponding parts are congruent

subtract 10 from both sides

divide both sides by 2

Homework:

p. 182 #1 - 42

4.2 Triangle Congruence by SSS and SAS Objectives:

To prove two triangles congruent using the SSS and SAS Postulates

SSS and SAS

1) Draw an acute scalene triangle on a piece of paper. Label its vertices A, B, and C, on the interior of each

angle.

A C

B

2) Construct a segment congruent to AC. Label the endpoints of the segment D and E.

D E

F

3) Construct a segment congruent to AB. 4) Construct a segment congruent to CB.

6) Draw DF and EF.

5) Label the intersection F.

This activity suggests the following postulate.

Postulate 5-1

SSS

Postulate

If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two

triangles are congruent.

A

B

C R

S

T

If AC RT and AB RS and BC ST

then ΔABC ΔRST

SSS and SAS

In two triangles, ZY FE, XY DE, and XZ DF.

Write a congruence statement for the two triangles.

Z Y F E

X D

Sample Answer:

ΔZXY ΔFDE

In a triangle, the angle formed by two given sides is called the ____________ of the sides.

A B

C

A is the included

angle of AB and AC

B is the included

angle of BA and BC

C is the included

angle of CA and CB

Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent.

You can also show their congruence by using two sides and the ____________.

Postulate 5-2

SAS

Postulate

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

A

B

C R

S

T

If AC RT and A R and AB RS

then ΔABC ΔRST

Determine whether the triangles are congruent by SAS.

If so, write a statement of congruence and tell why they are congruent.

If not, explain your reasoning.

P

R

Q

F E

D

Question:

Homework:

p. 189 #1-30

4.3 Triangle Congruence by ASA and AAS Objective:

To prove two triangles congruent using the ASA Postulate and the AAS Theorem

The side of a triangle that falls between two given angles is called the___________ of the angles.

It is the one side common to both angles.

A B

C

AC is the included side of A and C

CB is the included side of C and B

AB is the included side of A and B

You can show that two triangles are congruent by using _________ and the ___________ of the triangles.

R

S

T A

B

C

ASA and AAS

ASA

Postulate

If _________ and the ___________ of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent.

two angles included side

If A R and AC RT and

then ΔABC ΔRST

C T

A B

C

You can show that two triangles are congruent by using _________ and a ______________. two angles nonincluded side

CA and CB are the nonincluded sides of A and B

R

S

T A

B

C

AAS

Theorem

If _________ and a ______________ of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent.

two angles nonincluded side

If A R and CB TS

then ΔABC ΔRST

C T and

D

F

E

L

M

N

ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to show congruence.

What other pair of angles must be marked so that the two triangles are congruent by AAS?

However, AAS requires the nonincluded sides.

Therefore, D and L must be marked.

If F and M are marked congruent, then FE and MN would be included sides.

Homework:

p. 197 #1-26 or worksheet

4.4 Using Congruent Triangles: CPCTC Objective:

To use triangle congruence and CPCTC to prove that parts of two triangles are congruent

With SSS, SAS, ASA, and AAS, you know how to use three parts of triangles to show that the triangles are congruent.

Once the triangles are congruent, you can make conclusions about their other parts.

By definition of congruent triangles, the corresponding parts of congruent triangles are congruent.

We call this CPCTC

ONLINE VIDEO ABOUT CPCTC

Homework:

Worksheet

4.5 Isosceles and Equilateral Triangles Objective:

To use and apply properties of isosceles triangles

NCTM 2,3,6,7,8,9,10

Isosceles

Triangle

Theorem

Isosceles Triangle

Perpendicular Bisector

Thm.

If two sides of a

triangle are congruent,

then the angles

opposite those sides

are congruent.

The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle.

A

B C

A

B C D

Converse of

Isosceles

Triangle

Theorem

If two angles of a

triangle are

congruent, then the

sides opposite those

angles are

congruent.

A

B C

Equilateral/EquiangularTheorem

A triangle is equilateral if and only if it is equiangular.

Example: Find x.

Solution: If two angles of a triangle are congruent, the sides opposite them are congruent.

Example: Find the measures of angles 1, 2, 3, 4. Solution: If two sides of a triangle are congruent, the angles opposite them are congruent.

Homework:

p. 213 #1-27

4.6 Congruence in Right Triangles Objective:

To prove triangles congruent using the HL theorem

NCTM 3,6,7,8,9,10

In a right triangle, the side opposite the right angle is called the _________.

The two sides that form the right angle are called the _________.

leg

leg

Earlier in this chapter, we studied various ways to prove triangles to be congruent:

We studied two theorems

and

A

B

C R

S

T

A

B

C R

S

T

We also studied the following theorems to prove triangles to be congruent:

and

R

S

T A

B

C

R

S

T A

B

C

The theorems mentioned earlier in Chapter 4, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well.

LL Theorem

If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

A

C B

D

F E

same as

DEFABC

HA Theorem

If ______________ and an (either) __________ of one right triangle are congruent to the __________ and

_________________ of another right angle, then the

triangles are congruent.

same as

DEFABC

A

C B

D

F E

LA Theorem

If one (either) ___ and an __________ of a right triangle are congruent to the ________________________ of another right triangle, then the triangles are congruent.

same as

DEFABC

A

C B

D

F E

HL Theorem

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

A

C B

D

F E

DEFABC

Homework:

Right Triangle Congruence worksheet

4.7 Using Corresponding Parts of Congruent Triangles Objectives:

To identify congruent overlapping triangles

To prove two triangles congruent by first proving two other triangles congruent

NCTM 3,6,7,8,9,10

Using CPCTC worksheet

Homework:

p. 226 #1-22

Chapter 4 Test