Transcript 4.4 Proving Triangles are Congruent: ASA and AAS

5.3 Proving Triangles are
Congruent – ASA & AAS
Objectives:
•
Show triangles are congruent
using ASA and AAS.
Key Vocabulary

Included Side
Postulates

14 Angle–Side–Angle (ASA)
Congruence Postulate
Theorems

5.1 Angle-Angle-Side (AAS)
Congruence Theorem
Definition: Included Side
An included side is the common side of two
consecutive angles in a polygon. The
following postulate uses the idea of an
included side.
Example: Included Sided
C
The side between 2
angles
A
B
Y
X
INCLUDED SIDE
Z
Postulate 14 (ASA): Angle-SideAngle 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.
Angle-Side-Angle (ASA)
Congruence Postulate
Two angles and the INCLUDED side
Example 1
Determine When To Use ASA Congruence
Based on the diagram, can you use the ASA Congruence
Postulate to show that the triangles are congruent? Explain your
reasoning.
a.
b.
SOLUTION
a.
You are given that C  E, B  F, and BC  FE.
You can use the ASA Congruence Postulate to show that
∆ABC  ∆DFE.
b. You are given that R  Y and S  X.
You know that RT  YZ, but these sides are not
included between the congruent angles, so you
cannot use the ASA Congruence Postulate.
Example 2: Applying ASA
Congruence
Determine if you can use ASA to prove the triangles
congruent. Explain.
Two congruent angle pairs are given, but the included sides
are not given as congruent. Therefore ASA cannot be used to
prove the triangles congruent.
Your Turn
Determine if you can use ASA to
prove NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL.
NL  LN by the Reflexive Property. No other congruence
relationships can be determined, so ASA cannot be
applied.
Theorem 5.1 (AAS): Angle-AngleSide Congruence Theorem

If two angles and a
non-included side
of one triangle are
congruent to two
angles and the
corresponding nonincluded side of a
second triangle,
then the triangles
are congruent.
AAS
A
B
D
C
F
E
OR
X
Y
H
Z
I
J
Angle-Angle-Side (AAS)
Congruence Theorem
Two Angles and One Side that is NOT included
Example 3
Determine What Information is Missing
What additional congruence is needed to show that
∆JKL  ∆NML by the AAS Congruence Theorem?
SOLUTION
You are given KL  ML.
Because KLJ and MLN are vertical angles,
KLJ  MLN. The angles that make KL and ML the
non-included sides are J and N, so you need to
know that J  N.
Example 4
Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the
triangles are congruent? If so, state the postulate or theorem you
would use.
b.
a.
c.
SOLUTION
a.
EF  JH
E  J
Given
FGE  HGJ
Vertical Angles Theorem
Given
Use the AAS Congruence Theorem to conclude that
∆EFG  ∆JHG.
Example 4
Decide Whether Triangles are Congruent
b.
c.
b.
Based on the diagram, you know only that
MP  QN and NP  NP. You cannot conclude that
the triangles are congruent.
c.
UZW  XWZ
Alternate Interior Angles Theorem
WZ  WZ
Reflexive Prop. of Congruence
UWZ  XZW
Alternate Interior Angles Theorem
Use the ASA Congruence Postulate to conclude that
∆WUZ  ∆ZXW.
Example 5:
Is it possible to prove these triangles are
congruent? If so, state the postulate or theorem
you would use. Explain your reasoning.
Example 6:
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.
Thus, you can use
the AAS
Congruence
Theorem to prove
that ∆EFG  ∆JHG.
Example 7:
Is it possible to
prove these
triangles are
congruent? If so,
state the postulate
or theorem you
would use. Explain
your reasoning.
Example 8:
In addition to the
congruent
segments that are
marked, NP  NP.
Two pairs of
corresponding
sides are
congruent. This is
not enough
information
(CBD) to prove the
triangles are
congruent.
Example 9
Prove Triangles are Congruent
A step in the Cat’s Cradle string game creates the triangles
shown. Prove that ∆ABD  ∆EBC.
A
SOLUTION
C
B
BD  BC, AD || EC
D
∆ABD  ∆EBC
Statements
Reasons
1.
BD  BC
1. Given
2.
AD || EC
2. Given
3.
D  C
3. Alternate Interior Angles Theorem
4. ABD  EBC
4. Vertical Angles Theorem
5. ∆ABD  ∆EBC
5. ASA Congruence Postulate
E
Your Turn:
1. Complete the statement: You can use the ASA Congruence
Postulate when the congruent sides are
_____
? between the corresponding congruent angles.
ANSWER
included
Does the diagram give enough information to show that the
triangles are congruent? If so, state the postulate or theorem you
would use.
3.
2.
ANSWER
no
ANSWER
4.
no
ANSWER
yes; AAS
Congruence
Theorem
Congruence Shortcuts
}
Ways To Prove
Triangles Are
Congruent
Congruence Shortcuts
AAA and SSA???

Does AAA and SSA provide enough
information to determine the exact
shape and size of a triangle?
AAA and SSA???

Does AAA and SSA provide enough
information to determine the exact
shape and size of a triangle? NO
Not Congruence Shortcuts
}
NO BAD
WORDS
Do Not prove
Triangle Congruence
NO CAR
INSURANCE
Triangle Congruence
Practice
Your Turn
Is it possible to prove the Δs are ?
(
No, there is no AAA !
CBD
Yes, ASA
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove that
they are congruent, write cannot be determined (CBD).
G
K
I
H
J
ΔGIH  ΔJIK by AAS
In ΔDEF and ΔLMN , D  N , DE  NL and
E  L. Write a congruence statement.
 DEF   NLM
ASA
by ____
What other pair of angles needs to be
marked so that the two triangles are
congruent by AAS?
D
L
E  N
M
F
E
N
What other pair of angles needs to be
marked so that the two triangles are
congruent by ASA?
D
L
 D  L
M
F
E
N
Determine if whether each pair of triangles is
congruent by SSS, SAS, ASA, or AAS. If it is not
possible to prove that they are congruent, write cannot
be determined (CBD).
E
A
C
B
D
ΔACB  ΔECD by SAS
Determine if whether each pair of triangles is
congruent by SSS, SAS, ASA, or AAS. If it is not
possible to prove that they are congruent, write
cannot be determined (CBD).
J
M
K
L
ΔJMK  ΔLKM by SAS or ASA
Determine if whether each pair of triangles is
congruent by SSS, SAS, ASA, or AAS. If it is not
possible to prove that they are congruent, write
cannot be determined (CBD).
J
T
L
K
V
U
Cannot Be Determined (CBD)
BC  YZ or AC  XZ
B  Y
A  X
Cannot Be Determined (CBD) – SSA is not a
valid Congruence Shortcut.
Yes, ∆TNS ≅ ∆UHS by AAS
Review
Remember!
SSS, SAS, ASA, and AAS use corresponding
parts to prove triangles congruent. CPCTC uses
congruent triangles to prove corresponding
parts congruent.
Example 10: Using CPCTC
A and B are on the edges of a
ravine. What is AB?
One angle pair is congruent,
because they are vertical angles.
Two pairs of sides are
congruent, because their lengths
are equal.
Therefore the two triangles are congruent by
SAS. By CPCTC, the third side pair is congruent,
so AB = 18 mi.
Your Turn
A landscape architect sets up the
triangles shown in the figure to find
the distance JK across a pond.
What is JK?
One angle pair is congruent,
because they are vertical angles.
Two pairs of sides are congruent, because their lengths are
equal. Therefore the two triangles are congruent by SAS.
By CPCTC, the third side pair is congruent, so JK = 41 ft.
Joke Time






Which one came first the egg or the
chicken?
I don't care I just want my breakfast
served.
What do you call a handsome intelligent
sensitive man?
A rumor.
What does a clock do when it's hungry?
Goes back 4 secounds!!!
Assignment

Pg. 253 - 256 #1 – 21 odd, 25 – 45
odd