4.3-4.6 proving triangles congruent
DESCRIPTION
4.3-4.6 Proving Triangles Congruent. Warm up: Are the triangles congruent? If so, write a congruence statement and justify your answer. Proving Triangles Congruent…. How can you prove sides congruent? (things to look for) How can you prove angles congruent?. - PowerPoint PPT PresentationTRANSCRIPT
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4.3-4.6 Proving Triangles Congruent
Warm up:
Are the triangles congruent? If so, write a congruence statement and justify your
answer.
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Proving Triangles Congruent…
• How can you prove sides congruent? (things to look for)
• How can you prove angles congruent?
Given Shared side(reflexive POE)Midpoints Segment Addition PropertySegment bisector Transitive POEothers?
Given Shared angle(reflexive POE)//→Alt. Int <s, . . . Angle Addition PropertyAngle bisector Vertical AnglesRight Angles(┴) Transitive POE
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Now you try!
GIVEN:
R M
MN = RS
MO = RT
PROVE:
ΔMNO ΔRST
N
M O
S
R T
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Now you try!
GIVEN:
R M
MN = RS
MO = RT
PROVE:
ΔMNO ΔRST
N
M O
S
R TSTEP 1 – DRAW IT AND MARK IT!
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Now you try!
GIVEN:
R M
MN = RS
MO = RT
PROVE:
ΔMNO ΔRST
N
M O
S
R TSTEP 1 – DRAW IT AND MARK IT!
STEP 2 – CAN YOU PROVE THE Δs =?
HOW?
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Now you try!
GIVEN:
R M
MN = RS
MO = RT
PROVE:
ΔMNO ΔRST
N
M O
S
R TYES, BY SAS FROM THE GIVENS!
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REAL LIFE EXAMPLES
Bridges – Golden Gate, Brooklyn Bridge, New River Bridge . . . .
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Real Life
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Real Life
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Types of Proofs
Traditional two-column: This looks like a T-chart and has the statements on the left and reasons on the right.
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Types of Proofs
Flow Chart: Starts from a “base line” and all information flows from the given. Great for visual learners.
Paragraph: Write it out! Tell me what you’re doing!
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Helpful Hints with Proofs…
• ALWAYS mark the given in your picture.
• Use different colors in your picture to see the parts better.
• ALWAYS look for a _______________________ which
uses the __________________ property.
• ALWAYS look for ______________ lines to prove mostly
that _____________________________________.
• ALWAYS look for ____________ angles which are always
___________.
common side/anglecommon side/angle
reflexivereflexive
parallelparallel
alternate interior angles are congruentalternate interior angles are congruent
verticalvertical
congruentcongruent
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Given: PQ PS; QR SR; 1 2
Prove: 3 4
Statements1. PQ PS; QR SR;
1 2
2. PR PR
3. ∆QPR ∆SPR
4. 3 4
Reasons1. Given
2. Reflexive Property
3. SAS Postulate
4. CPCTC
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Given: WO ZO; XO YO
Prove: ∆WXO ∆ZYO Statements1. WO ZO; XO YO
2. WOX ZOY
3. ∆WXO ∆ZYO
Reasons1. Given
2. Vertical angles are .
3. SAS Postulate
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Proof Practice
Given: PSU PTR; SU TR
Prove: SP TPHINT: draw the triangles separately!
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Proof Practice…
1. 1. PSU PSU PTR; SU PTR; SU TR TR 1. given1. given
2. <P 2. <P <P <P 2. Reflexive POE2. Reflexive POE
3. 3. ∆∆SUP SUP ∆∆TRP TRP 3. AAS Theorem3. AAS Theorem
4. SP 4. SP TP TP 4. CPCTC4. CPCTC
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Proof Practice…
PSU PTR SU TR <P <P
∆SUP ∆TRP
SP TP
CPCTCCPCTC
AAS Theorem
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Practice
Name the included side for 1 and 5.
Name a pair of angles in which DE is not included.
If 6 10, and DC VC, then
∆ DCA ∆ _______, by _________.
DCDC
<8, <9, for example<8, <9, for example
VCEVCE ASAASA
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More Proofs…Using 2 Column • Given: PQ RQ; S
is midpoint of PR.• Prove: P R
QS is an auxiliary line P
Q
RS
1. PQ PQ RQ; S is midpoint of PR RQ; S is midpoint of PR 1. givengiven
2. PS PS SR SR 2. Def midpointDef midpoint3. QS QS QS QS 3. Reflexive POEReflexive POE4. ∆∆PQS PQS ∆∆RQSRQS 4. SSS PostulateSSS Postulate
5. P P RR 5. CPCTCCPCTC
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More Proofs…Using Flow Chart • Given: PQ RQ;
S is midpoint of PR.
• Prove: P R
P
Q
RS
PQ RQ S is midpoint of PR
PS SR
QS QS
∆PQS ∆RQS
P P RR
CPCTC
SSS Postulate
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More Proofs…Using Paragraph • Given: PQ RQ; S is midpoint of PR.• Prove: P R
QS is an __________________.
We are given that ____________ and ___________________. Because S is the midpoint, we know that __________because of _____________.
We drew in QS so that we can use the reflexive property to prove that
_________. We now have enough information to prove that ∆PQS ∆RQS
by ____________. Therefore <P <R by __________________.
P
Q
RS
auxiliary lineauxiliary line
PQ PQ RQ RQ S is midpoint of PRS is midpoint of PR
PS PS SR SR def. of midptdef. of midpt
QS QS QSQS
SSS Post. SSS Post. CPCTCCPCTC
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Do the following proofs in whatever way
you feel comfortable
Given: AB EB; DEC B
Prove: ∆ABE is equilateral
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Group Work Time:• Group 4.3-4.6 proof
practice WS• Group presentations
• Next Class• Group presentations• More group practice
work