Download 4.4 Proving Triangles are Congruent: ASA and AAS

Proving Triangles are
Congruent
SSS, SAS; ASA; AAS
CCSS: G.CO7
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.
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.
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?
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.
Proving Triangles are
Congruent:
SSS and SAS
SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) are
congruent, then the triangles are congruent.
If
Sides are
congruent
and
Angles are
congruent
1. AB
DE
4.
A
D
2. BC
EF
5.
B
E
3. AC
DF
6.
C
F
then
Triangles are
congruent
 ABC
 DEF
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides
of a second triangle, then the two triangles are congruent.
If Side
S MN
QR
Side
S NP
RS
Side
S PM
SQ
then  MNP
 QRS
SSS AND SAS CONGRUENCE POSTULATES
The SSS Congruence Postulate is a shortcut for proving
two triangles are congruent without using all six pairs
of corresponding parts.
SSS AND SAS CONGRUENCE POSTULATES
POSTULATE
POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, then the two triangles are congruent.
If
Side
S
Angle
A
Side
S
PQ
WX
Q
X
QS
XY
then  PQS
WXY
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  ABC   FGH.
SOLUTION
AC = 3 and FH = 3
AC  FH
AB = 5 and FG = 5
AB  FG
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths BC and GH.
d=
BC =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(– 4 – (– 7)) 2 + (5 – 0 ) 2
d=
GH =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(6 – 1) 2 + (5 – 2 ) 2
=
32 + 52
=
52 + 32
=
34
=
34
Congruent Triangles in a Coordinate Plane
BC = 34 and GH = 34
BC  GH
All three pairs of corresponding sides are congruent,
 ABC   FGH by the SSS Congruence Postulate.
SSS  postulate
SAS  postulate
T
S
C
G
The vertex of the included angle is the point in common.
SAS  postulate
SSS  postulate
SSS  postulate
Not enough info
SSS  postulate
SAS  postulate
Not Enough Info
SAS  postulate
SSS  postulate
Not Enough Info
SAS  postulate
SAS  postulate
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  NMP   DEF.
SOLUTION
MN = 4 and DE = 4
MN  DE
PM = 5 and FE = 5
PM  FE
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths PN and FD.
d=
PN =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(– 1 – (– 5)) 2 + (6 – 1 ) 2
d=
FD =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(2 – 6) 2 + (6 – 1 ) 2
=
42 + 52
=
(-4) 2 + 5 2
=
41
=
41
Congruent Triangles in a Coordinate Plane
PN = 41 and FD = 41
PN  FD
All three pairs of corresponding sides are congruent,
 NMP   DEF by the SSS Congruence Postulate.
Proving Triangles are
Congruent
ASA; AAS
Postulate 21: Angle-Side-Angle
(ASA) Congruence Postulate
• If two angles and the
B
included side of one
triangle are
congruent to two
angles and the
C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
• If two angles and a
B
non-included side of
one triangle are
congruent to two
angles and the
corresponding non- C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
Given: A  D, C
 F, BC  EF
Prove: ∆ABC  ∆DEF
B
A
E
C
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
You 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 C
is the side included between
E and F. You can apply
the ASA Congruence
Postulate to conclude that
∆ABC  ∆DEF.
B
A
E
F
D
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.
H
E
G
F
J
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.
H
E
G
F
J
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
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
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
1
2
W
3
4
X
Z
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
1
2
W
3
4
X
Z
Ex. 2 Proving Triangles are
Congruent
Given: 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.
C
A
B
D
E
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1.
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
5. ASA Congruence
Theorem
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.