GOAL 1 PLANNING A PROOF

EXAMPLE 1

4.5 Using Congruent Triangles

By definition, we know that corresponding parts of congruent triangles are congruent. So once we have shown that two triangles are congruent by SSS, SAS, ASA, or AAS, we can conclude that all of the remaining pairs of corresponding parts are congruent.

We will abbreviate “corresponding parts of congruent triangles are congruent” as CPCTC.

Since , ____________HJ LK by the Alt. Int. Angles Thm.

Extra Example 1Given: ,

Prove:

HJ LK JK HL

LHJ JKL

HJL KLJ

H

L K

J

for the same reason.Also, since ,____________JK HL HLJ KJL

First, show that Then you can use CPCTC to prove that

.HJL KLJ .LHJ JKL

PLAN:

PROOF:

Then, because by the _______________,JL JL Reflexive Property we know by ____.HJL KLJ ASA Finally, by ______.LHJ JKL CPCTC

EXAMPLE 2

Extra Example 2Given: ,

Prove: is the midpoint of .

MS TR MS TR

A MT

M R

SA

TPLAN:Prove then show that ,MAS TAR .MA TAPROOF: Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

MS TR

,M T S R Alt. Int. s Thm.

MS TR Given

MAS TAR

MA TA

ASA

CPCTC

is the midpoint of A MT Def. of midpoint

EXAMPLE 3

Extra Example 3 Q

V

U T S

R

Given: is the bisector of ,

,

Prove:

QT US

QV QR VQU RQS

QUV QSR

PLAN:Show , then use CPCTC to show .QUT QST QU QS Then you can prove .QUV QSR

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

is the bisector of QT US

UT ST Def. of bisector

and are rt. s.QTU QTS Def. of

QTU QTS Rt. Thm.

QT QT Reflexive Prop. Continued on next slide

Extra Example 3 (cont.)Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

10. 10.

is the bisector of QT US

UT ST Def. of bisector

and are rt. s.QTU QTS Def. of

QTU QTS Rt. Thm.

QT QT Reflexive Prop.

Q

V

U T S

R

QUT QST SAS

QU QS CPCTC

VQU RQS Given

QV QR Given

QUV QSR SAS

Checkpoint

Given: bisects ,

Prove:

MP LMN LM NM

LP NP

P

N

M

LYour proof should include the following steps:

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

bisects MP LMN

LMP NMP Def. of bisector

MP MP Reflexive Prop.

LM NM Given

MNP MLP SAS

LP NP CPCTC

EXAMPLE 4

4.5 Using Congruent Triangles

GOAL 2 PROVING CONSTRUCTIONS ARE VALID

When proving a construction is valid:

• You may assume that any two segments constructed using the same compass setting are congruent.

• You may need to finish drawing a segment which is not part of the actual construction. (See segments BC and EF in Example 4.)

Extra Example 4Write a proof to verify that the construction (copying an angle) is valid.

X

Y

Z M

N

P

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

MN XY Given

MP XZ Given

NP YZ Given

MNP XYZ SSS

NMP YXZ CPCTC

Checkpoint

Prove that the construction of an angle bisector is valid.

A

DB

C

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

AB AC Given

BD CD Given

AD AD Reflexive Prop.

ABD ACD SSS

BAD CAD CPCTC