4.4 - Prove Triangles Congruent by SAS and HL. Included Angle: Angle in-between two congruent sides.
Using Congruent Triangles Chapter 4. Objective List corresponding parts. Prove triangles congruent...
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Transcript of Using Congruent Triangles Chapter 4. Objective List corresponding parts. Prove triangles congruent...
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 corresponding parts of those congruent triangles are congruent.• Correspondin
g sides are congruent
• Corresponding angles are congruent
Example: Name the Congruence Shortcut or CBD
SASASA
SSSSSACBD
Name the Congruence Shortcut or CBD
SAS
SAS
SAS
Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SS
ACBD
Your Turn: Name the Congruence Shortcut or CBD
Your Turn: Name the Congruence Shortcut or CBD
Your Turn: Name the Congruence Shortcut or CBD
ExampleIndicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
B
For AAS: A
AC
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.
Given1.
2.2.
3. 3.
5. 5.
6. 6.
4.4.
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. Gals2. 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 OneGiven: ∠ABD = ∠CBD, ∠ADB = ∠CDBProve: AB = CB
A
B
C
D
Statement Reason
Number TwoGiven: MO = RE, ME = ROProve: ∠M = ∠R
O R
EMStatement Reason
Number Three
Given: SP = OP, ∠SPT = ∠OPTProve: ∠S = ∠O S
P
OT
ReasonStatement
Number Four
Given: KN = LN, PN = MNProve: KP = LM K
N
L
MP
Statement Reason
Number Five
Given: ∠C = ∠R, TY = PYProve: CT = RP C
Y
R
PT
ReasonStatement
Number Six
Given: AT = RM, AT || RMProve: ∠AMT = ∠RTMA T
RM
Statement Reason
Example 2 Visualize Overlapping Triangles
SOLUTION
Sketch the triangles separately and mark any given information. Think of ∆JGH moving to the left and ∆KHG moving to the right.
1.
Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show ∆JGH ∆KHG.
Mark GJH HKG and JHG KGH.
Example 2 Visualize Overlapping Triangles
Look at the original diagram for shared sides, shared angles, or any other information you can conclude.
2.
Add congruence marks to GH in each triangle.
In the original diagram, GH and HG are the same side, so GH HG.
You can use the AAS Congruence Theorem to show that ∆JGH ∆KHG.
3.
Example 3 Use Overlapping Triangles
SOLUTION
Write a proof that shows AB DE.
ABC DEC
AB DECB CE
Your Turn: Use Overlapping Triangles
Given KJ KL and J L, show NJ ML.
Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent.
Your Turn: Use Overlapping Triangles
Given SPR QRP and Q S, show ∆PQR ∆RSP.3.
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!!!