3.5 Overlapping Triangles3.5 Overlapping Triangles

Objective: After studying this lesson you will be able to

use overlapping triangles in proofs.

Let’s look at this problem. At first we might try to prove . We could then say that angles D and C are congruent because of CPCTC

Given:

Prove:

DB AC

AD BC

CC

AA BB

DD

D C EE

ADE BCE

But there is not enough information.

We have to find two other angles to which D and C belong. Then we can prove the triangles congruent and their corresponding parts.

We can make overlapping triangles easier.

CC

AA BB

DDEE

We can color the two triangles we are going to use.

We can draw the triangles separately.

BBBBAA AA

DD CC

Given:

Prove: CE BDStatement Reason

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

EE

CC

DD

AA

BB

AC ABAE AD

Given:

Prove:

G is the midpoint of FH

Statement Reason

1.

2.

3.

4.

5.

6.

7.

8.

1.

2.

3.

4.

5.

6.

7.

8.

MM

JJHH

GG

FH MJFF

KK

GHJ KJH

K is the midpoint of MJGJ HK

Summary:Summary:

Don’t let overlapping Don’t let overlapping triangles confuse you. triangles confuse you. Name ways that you can Name ways that you can help to see where the help to see where the overlapping triangles are.overlapping triangles are.

Homework: worksheetHomework: worksheet