Proving Triangles are Congruent SSS, SAS; ASA; AAS
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Proving Triangles are Congruent
SSS, SAS; ASA; AASCCSS: G.CO7
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Standards for Mathematical Practices
• 1. Make sense of problems and persevere in solving them.
• 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the
reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated
reasoning.
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CCSS:G.CO 7
• USE the definition of congruence in terms of rigid motions to SHOW that two triangles ARE congruent if and only if corresponding pairs of sides and corresponding pairs of angles ARE congruent.
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ESSENTIAL QUESTION
• How do we show that triangles are congruent?
• How do we use triangle congruence to plane and write proves ,and prove that constructions are valid?
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Objectives:
1. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem
2. Use congruence postulates and theorems in real-life problems.
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Proving Triangles are Congruent:
SSS and SAS
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SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) arecongruent, then the triangles are congruent.
and thenIfSides are congruent
1. AB DE
2. BC EF
3. AC DF
Angles are congruent
4. A D
5. B E
6. C F
Triangles are congruent
ABC DEF
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SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate
Side MN QR
Side PM SQ
Side NP RS
If
If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.
then MNP QRSS
S
S
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SSS AND SAS CONGRUENCE POSTULATES
The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.
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POSTULATE
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate
Side PQ WX
Side QS XY
then PQS WXYAngle Q X
If
If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
A
S
S
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Congruent Triangles in a Coordinate Plane
AC FH
AB FGAB = 5 and FG = 5
SOLUTION
Use the SSS Congruence Postulate to show that ABC FGH.
AC = 3 and FH = 3
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Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 3 2 + 5
2
= 34
BC = (– 4 – (– 7)) 2 + (5 – 0 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 5 2 + 3
2
= 34
GH = (6 – 1) 2 + (5 – 2 )
2
Use the distance formula to find lengths BC and GH.
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Congruent Triangles in a Coordinate Plane
BC GH
All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.
BC = 34 and GH = 34
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SSS postulate SAS postulate
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T C
S G
The vertex of the included angle is the point in common.
SSS postulateSAS postulate
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SSS postulate
Not enough info
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SSS postulateSAS postulate
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Not Enough InfoSAS postulate
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SSS postulate
Not Enough Info
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SAS postulate SAS postulate
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Congruent Triangles in a Coordinate Plane
MN DE
PM FEPM = 5 and FE = 5
SOLUTION
Use the SSS Congruence Postulate to show that NMP DEF.
MN = 4 and DE = 4
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Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 4 2 + 5
2
= 41
PN = (– 1 – (– 5)) 2 + (6 – 1 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= (-4) 2 + 5
2
= 41
FD = (2 – 6) 2 + (6 – 1 )
2
Use the distance formula to find lengths PN and FD.
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Congruent Triangles in a Coordinate Plane
PN FD
All three pairs of corresponding sides are congruent, NMP DEF by the SSS Congruence Postulate.
PN = 41 and FD = 41
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Proving Triangles are Congruent
ASA; AAS
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Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate• If two angles and the
included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
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Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem• If two angles and a
non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
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Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremGiven: A D, C
F, BC EF
Prove: ∆ABC ∆DEF
B
C
A
F
D
E
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Theorem 4.5: Angle-Angle-Side (AAS) Congruence TheoremYou are given that two angles of
∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC ∆DEF.
B
C
A
F
D
E
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
G
E
JF
H
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Ex. 1 Developing Proof
A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG.
G
E
JF
H
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
N
M
Q
P
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Ex. 1 Developing Proof
B. In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.
N
M
Q
P
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Ex. 1 Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
UZ ║WX AND UW
║WX.
U
W
Z
X
12
34
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Ex. 1 Developing Proof
The two pairs of parallel sides can be used to show 1 3 and 2 4. Because the included side WZ is congruent to itself, ∆WUZ ∆ZXW by the ASA Congruence Postulate.
U
W
Z
X
12
34
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Ex. 2 Proving Triangles are CongruentGiven: AD ║EC, BD
BC
Prove: ∆ABD ∆EBC
Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC
. Use the fact that AD ║EC to identify a pair of congruent angles.
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1.
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
3. Alternate Interior Angles
B
A
ED
C
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Proof:
Statements:
1. BD BC
2. AD ║ EC
3. D C
4. ABD EBC
5. ∆ABD ∆EBC
Reasons:
1. Given
2. Given
3. Alternate Interior Angles
4. Vertical Angles Theorem
B
A
ED
C
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Proof:
Statements:1. BD BC2. AD ║ EC3. D C4. ABD EBC5. ∆ABD ∆EBC
Reasons:1. Given2. Given3. Alternate Interior
Angles4. Vertical Angles
Theorem5. ASA Congruence
Theorem
B
A
ED
C
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Note:
• You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D C and A E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.