Using Congruent Triangles

Report
Using Congruent
Triangles
Chapter 4
Objective
• List corresponding parts.
• Prove triangles congruent (ASA,
SAS, AAS, SSS, HL)
• Prove corresponding parts congruent
(CPCTC)
• Examine overlapping triangles.
Key Vocabulary - Review
•
•
•
•
Reflexive Property
Vertical Angles
Congruent Triangles
Corresponding Parts
Review: Congruence
Shortcuts
**Right triangles only: hypotenuse-leg (HL)
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
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
Your Turn: Name the
Congruence Shortcut or CBD
Your Turn: Name the
Congruence Shortcut or CBD
Example
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B 
For SAS:
AC 
For AAS:
A 
Your Turn:
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS:
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.
Example 1
Use Corresponding Parts
Statements
Reasons
1. AB and CD bisect
each other at M.
1. Given
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
The Proof Game!
Here’s your chance to play the
game that is quickly becoming a
favorite among America’s
teenagers: The Proof Game!
Rules:
1. Guys vs. Gals
2. Teams must take turns filling in the statements
and reasons in the proofs to come.
3. If the statement/reason combo is correct, team
gets 1 point. Next team continues.
4. If the statement/reason combo is incorrect, team
loses 1 point. Next team fixes mistake.
5. Teammates cannot help the person at the
board…he/she is on their own. Cheating loses all
points!!
Number One
Given: ∠ABD = ∠CBD, ∠ADB = ∠CDB
B
Prove: AB = CB
Statement
Reason
A
C
D
Number Two
Given: MO = RE, ME = RO
Prove: ∠M = ∠R
O
Statement
Reason
M
R
E
Number Three
Given: SP = OP, ∠SPT = ∠OPT
T
S
Prove: ∠S = ∠O
Statement
Reason
P
O
Number Four
Given: KN = LN, PN = MN
Prove: KP = LM
K
L
N
Statement
Reason
P
M
Number Five
Given: ∠C = ∠R, TY = PY
Prove: CT = RP
C
R
Y
Statement
Reason
T
P
Number Six
Given: AT = RM, AT || RM
Prove: ∠AMT = ∠RTM A
Statement
T
Reason
M
R
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
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.
Given KJ  KL and J  L, show
NJ  ML.
Your Turn:
Use Overlapping Triangles
3. Given SPR  QRP and Q  S, show ∆PQR  ∆RSP.
Joke Time
• What happened to the man who lost the whole
left side of his body?
• He is all right now.
• What did one eye say to the other eye?
• Between you and me something smells.
Upcoming Schedule
• Quiz on Friday…HL, proofs, CPCTC, Isosceles
Triangle Thm, overlapping triangles
•
•
•
•
Monday – vocabulary terms
Tues – Practice Day
Wednesday – Chapter 4 Test
**reminder – projects due Oct. 27!!!

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