Transcript Using Congruent Triangles

Using Congruent
Triangles
Section 5.5
Objective
• Show corresponding parts of
congruent triangles are congruent.
Key Vocabulary - Review
• Corresponding parts
Review: Congruence
Shortcuts
Congruent Triangles
(CPCTC)
Two triangles are congruent triangles
if and only if the corresponding parts
of those congruent triangles are
congruent.
• Corresponding
sides are
congruent
• Corresponding
angles are
congruent
Example: Name the
Congruence Shortcut or CBD
SAS
SSA
CBD
ASA
SSS
Your Turn: Name the
Congruence Shortcut or CBD
AAA
CBD
SAS
ASA
SSA
CBD
Your Turn: Name the
Congruence Shortcut or CBD
Reflexive
Property
SAS
Vertical
Angles
SAS
Vertical
Angles
SAS
Reflexive
Property
SSA
CBD
Your Turn: Name the
Congruence Shortcut or CBD
SSS
SSA
CBD
ASA
AAA
CBD
Your Turn: Name the
Congruence Shortcut or CBD
ASA
SSA
CBD
SAS
Example
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
Your Turn:
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
N  V
For SAS:
MK  UT
For AAS:
M  U
Using Congruent Triangles:
CPCTC
• If you know that two triangles are
congruent, then you can use CPCTC to
prove the corresponding parts in
whose triangles are congruent.
*You must prove that the triangles are
congruent before you can use CPCTC*
Example 1
Use Corresponding Parts
In the diagram, AB and CD bisect each
other at M. Prove that A  B.
SOLUTION
1. First sketch the diagram and label
any congruent segments and
congruent angles.
2. Because A and B are
corresponding angles in ∆ADM and
∆BCM, show that ∆ADM  ∆BCM to
prove that A  B.
Example 1
Use Corresponding Parts
Statements
Reasons
1. AB and CD bisect
each other at M.
1. Given
2. MA  MB
2. Definition of segment bisector
3. AMD  BMC
3. Vertical Angles Theorem
4. MD  MC
4. Definition of segment bisector
5. ∆ADM  ∆BCM
5. SAS Congruence Postulate
6. A  B
6. Corresponding parts of
congruent triangles are
congruent.
The Proof Game!
Here’s your chance to play the
game that is quickly becoming a
favorite among America’s
teenagers: The Proof Game!
Example: Using CPCTC
Given: ∠ABD = ∠CBD, ∠ADB = ∠CDB
B
Prove: AB = CB
Statement
∠ABD = ∠CBD, ∠ADB = ∠CDB
BD = BD
ΔABD = ΔCBD
AB = CB
Reason
A
C
Given
D
Reflexive Property
ASA (Angle-Side-Angle)
CPCTC (Corresponding Parts of
Congruent Triangles are Congruent)
Your Turn: Using CPCTC
Given: MO = RE, ME = RO
Prove: ∠M = ∠R
O
Statement
MO = RE, ME = RO
OE = OE
ΔMEO = ΔROE
∠M = ∠ R
R
Reason
M
Given
Reflexive Property
E
SSS (Side-Side-Side)
CPCTC (Corresponding Parts of Congruent Triangles
are Congruent)
Your Turn: Using CPCTC
Given: SP = OP, ∠SPT = ∠OPT
T
S
Prove: ∠S = ∠O
Statement
O
Reason
SP = OP, ∠SPT = ∠OPT
PT = PT
ΔSPT = ΔOPT
Given
Reflexive Property
∠S = ∠O
CPCTC (Corresponding Parts of Congruent
Triangles are Congruent)
SAS (Side-Angle-Side)
P
Your Turn: Using CPCTC
Given: KN = LN, PN = MN
Prove: KP = LM
K
L
N
Statement
KN = LN, PN = MN
∠KNP = ∠LNM
ΔKNP = ΔLNM
KP = LM
Reason
Given
Vertical Angles
P
SAS (Side-Angle-Side)
M
CPCTC (Corresponding Parts of Congruent Triangles
are Congruent)
Your Turn: Using CPCTC
Given: ∠C = ∠R, ∠T = ∠P, TY = PY
C
Prove: CT = RP
R
Y
Statement
Reason
∠C = ∠R, ∠T = ∠P,
TY = PY
ΔTCY = ΔPRY
Given
CT = RP
T
AAS (Angle-Angle-Side)
P
CPCTC (Corresponding Parts of Congruent Triangles
are Congruent)
Your Turn: Using CPCTC
Given: AT = RM, AT || RM
Prove: ∠AMT = ∠RTM A
Statement
Reason
AT = RM, AT || RM
Given
∠ATM = ∠RMT
Alternate Interior Angles
TM = TM
ΔTMA = ΔMTR
∠AMT = ∠RTM
T
M
R
Reflexive Property
SAS (Side-Angle-Side)
CPCTC (Corresponding Parts of Congruent
Triangles are Congruent)
Example 2
Visualize Overlapping Triangles
Sketch the overlapping triangles
separately. Mark all congruent angles and
sides. Then tell what theorem or postulate
you can use to show ∆JGH  ∆KHG.
SOLUTION
1. Sketch the triangles separately and mark any given
information. Think of ∆JGH moving to the left and
∆KHG moving to the right.
Mark GJH  HKG
and JHG  KGH.
Example 2
Visualize Overlapping Triangles
2. Look at the original diagram for shared sides, shared
angles, or any other information you can conclude.
In the original diagram, GH and HG are the same
side, so GH  HG.
Add congruence marks
to GH in each triangle.
3. You can use the AAS Congruence Theorem to show
that ∆JGH  ∆KHG.
Example 3
Use Overlapping Triangles
Write a proof that shows AB  DE.
ABC  DEC
CB  CE
AB  DE
SOLUTION
1. Sketch the triangles separately.
Then label the given information
and any other information you
can conclude from the diagram.
In the original diagram, C is the
same in both triangles
(BCA  ECD).
Example 3
Use Overlapping Triangles
Show ∆ABC  ∆DEC to prove that AB  DE.
Statements
Reasons
1. ABC  DEC
1. Given
2. CB  CE
2. Given
3. C  C
3. Reflexive Prop. of Congruence
4. ∆ABC  ∆DEC
4. ASA Congruence Postulate
5. AB  DE
5. Corresponding parts of
congruent triangles are
congruent.
Your Turn:
Use Overlapping Triangles
1. Tell which triangle congruence theorem or
postulate you would use to show that AB  CD.
ANSWER
SAS.
Your Turn:
Use Overlapping Triangles
Redraw the triangles separately and label all
congruences. Explain how to show that the triangles
or corresponding parts are congruent.
2. Given KJ  KL and J  L, show
NJ  ML.
ANSWER
Statements
1. KJ  KL
2. J  L
3. K  K
4. ∆KJN  ∆KLM
5. NJ  ML
Reasons
1. Given
2. Given
3. Reflexive Prop. of Congruence
4. ASA Congruence Postulate
5. Corresponding parts of  triangles
are .
Your Turn:
Use Overlapping Triangles
3. Given SPR  QRP and Q  S, show ∆PQR  ∆RSP.
ANSWER
Statements
1. SPR  QRP
2. Q  S
3. PR  RP
4. ∆PQR  ∆RSP
Reasons
1. Given
2. Given
3. Reflexive Prop. of Congruence
4. AAS Congruence Theorem
Joke Time
• What happened to the man who lost the whole
left side of his body?
• He is all right now.
• What do you find in an empty nose?
• Finger prints.
• What did one eye say to the other eye?
• Between you and me something smells.
Assignment
• Pg. 268 – 271:#1 – 29 odd