Transcript Using Congruent Triangles
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 c orresponding p arts of those c ongruent t riangles are c ongruent.
• Corresponding sides are congruent • Corresponding angles are congruent
Example: Name the Congruence Shortcut or CBD
SAS ASA SSA CBD SSS
Name the Congruence Shortcut or CBD Reflexive Property
SAS
Vertical Angles
SAS
Vertical Angles
SAS
Reflexive Property
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: For SAS: For AAS:
B
AC
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 4.
5.
6.
Statements 1.
AB
and
CD
bisect each other at
M
.
2.
3.
Reasons 1.
2.
3.
4.
5.
6.
Given
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 Prove: AB = CB B
Statement Reason
A C D
Number Two
Given: MO = RE, ME = RO
Prove:
∠ M = ∠ R O
Statement Reason
M E R
Number Three
Given: SP = OP, ∠ SPT = ∠ OPT
Prove:
∠ S = ∠ O S T
Statement Reason
P O
Number Four
Given: KN = LN, PN = MN Prove: KP = LM K N
Statement Reason
P L M
Number Five
Given:
∠ C = ∠ R, TY = PY Prove: CT = RP C
Statement Reason
T Y R P
Number Six
Given: AT = RM, AT || RM
Prove:
∠ AMT = ∠ RTM A
Statement Reason
M R T
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
∆
KHG
moving to the right.
moving to the left and Mark and
GJH
HKG
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
side, so
GH
HG
.
and
HG
are the same 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
AB
CE 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
NJ
KJ ML
.
KL
and
J
L
, show
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!!!