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 Y The side between 2 angles A B X
INCLUDED SIDE
Z
Postulate 14 (ASA): Angle-Side Angle 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
You can use the ASA Congruence Postulate to show that
∆
ABC
∆ DFE.
FE
.
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 NL
LN
MNL
by the Reflexive Property. No other congruence . relationships can be determined, so ASA cannot be applied.
Theorem 5.1 (AAS): Angle-Angle Side 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 A Y X
AAS
D C
OR
E H Z I F 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 know that
J
N. J and N, so you need to
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.
a. b. c. SOLUTION a.
EF
JH
E
J
FGE
HGJ
Given Given Vertical Angles Theorem Use the AAS Congruence Theorem to conclude that
∆
EFG
∆ JHG.
Example 4 b. Decide Whether Triangles are Congruent 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 WZ
WZ
UWZ
XZW
Alternate Interior Angles Theorem Reflexive Prop. of Congruence 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 segments that are marked, EGF JGH by the Theorem. Two pairs of corresponding angles and one pair sides are congruent. Thus, you can use the
AAS Congruence
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
B
BD
BC
,
AD || EC
∆
ABD
∆
EBC
Statements 1.
BD
BC
Reasons 1.
Given 2.
Given 2.
3.
4.
5.
AD || EC
D
C
ABD
EBC
∆
ABD
∆
EBC
D
3.
Alternate Interior Angles Theorem 4.
Vertical Angles Theorem 5.
ASA Congruence Postulate
C 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.
2.
3.
4.
ANSWER no ANSWER 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
Δ
s 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
,
L
. Write a congruence statement.
and
F
What other pair of angles needs to be marked so that the two triangles are congruent by AAS?
E
N
D L M N E
F D
What other pair of angles needs to be marked so that the two triangles are congruent by ASA?
D
L
L M N E
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)
.
A E 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 K L M
ΔJMK
ΔLKM by SAS or ASA
J
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)
.
T K L V U
Cannot Be Determined (CBD)
BC
YZ
or
AC
XZ
Y
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 – odd 21 odd, 25 – 45