Download Notes on ASA and AAS

Notes on ASA and AAS - Page 1
Name_________________________
Standard Addressed: MA1G3 Students will discover, prove, and apply properties of triangles, quadrilaterals, & other polygons.
c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL)
The included side is the side shared by two clearly stated angles.
C
For example, consider ABC to the left.
C
 A and  B are two interior angles inside
of the triangle.
A
B
A
B
Note that these two angles share side AB .
included side
between  A and  B
Thus, AB is the included side. See right.
--------------------------------------------------------------------------------------------------------------------Shade the included side between
Shade the included side between
D and J .
Q and W .
D
Q
J
W
M
R
--------------------------------------------------------------------------------------------------------------------Angle -Side-Angle Theorem (ASA) - If two angles and the included side of one triangle are congruent
to two corresponding angles and the included side of another triangle, then the two triangles are
congruent.
For example, the two triangles to the right have two pairs of
congruent angles:
D  G (one angle arc)
F  I (two angle arcs)
In addition, the sides located between the two angles (the
included sides) are also congruent:
G
D
H
E
I
F
DF  GI (one tick mark)
Thus, DFE  GIH by ASA Theorem.
--------------------------------------------------------------------------------------------------------------------In order for the triangles pictured to the left to be congruent
by ASA, mark the sides with tick marks that must be
congruent.
J
L
Given: JL is parallel to NO , and point K is the midpoint of JO .
K
N
Prove: JLK  ONK
O
Statements
Reasons
1.
JL is parallel to NO , and point K is the
midpoint of JO .
1. Given
2.
JK  KO
2. Since K is the midpoint of JO , it
divides JO into two congruent segments.
3.
J  O
3. Alternate Interior Angles' Theorem
4.
JKL  OKN
4. Vertical Angles' Theorem
5. JLK  ONK
5. ASA Theorem
--------------------------------------------------------------------------------------------------------------------Question:
If two triangles have two pairs of corresponding angles that are congruent, then
are the remaining two angles (the third pair) also congruent? To help answer this
question, let's consider an example:
In the diagram to the right, the two triangles have
two pairs of congruent angles ( 67 and 70 ).
To find the measure of the third angle in both triangles,
one should use the fact that the sum of the measures of the
interior angles of a triangle is always 180 .
?
70 
67 
70 
?
67 
Therefore, the unknown angle in each triangle will be found
using the same computation:
180  70  67  43 .
Since the third angle in both triangles measures 43 , the third pair of angles are congruent.
This will always occur if the first two pairs of angles are congruent.
--------------------------------------------------------------------------------------------------------------------Consider the case to the left in which two triangles have two
identical angles, as well as a pair of non-included sides that are
congruent.
Since two pairs of angles are congruent, the third pair must be
as well.
Notes on ASA and AAS - Page 2
Name_________________________
To the left, observe that the third pair of angles have been
marked as congruent.
third pair of angles
are also congruent
If the two angle arc angles are removed, you can just focus on
the information to the right.
As you can see, there are now two pairs of angles that are
congruent, and the included sides are congruent.
Hence, the two triangles are congruent by ASA Theorem.
--------------------------------------------------------------------------------------------------------------------Now, if you recall, the problem above actually started with two pairs of angles and the
non-included side. This leads to the following:
Angle-Angle-Side Theorem (AAS) - If two angles and a non-included side of one triangle are
congruent to two angles and the corresponding non-included side of another triangle, then the two
triangles are congruent.
--------------------------------------------------------------------------------------------------------------------For example, the two triangles to the left have two pairs of
R
T
congruent angles:
P
Q  U (one angle arc)
S
P  S (two angle arcs)
Q
In addition, the two triangles have a corresponding, non-included
side that is congruent:
U
PR  ST (one tick mark)
So, PRQ  STU by the AAS Theorem.
--------------------------------------------------------------------------------------------------------------------V
Z
B Consider the figures to the left. Write congruency
statements in order to answer the questions below.
What else be true in order to prove that the triangles
are congruent by ASA Theorem?
W
X
Y
What else be true in order to prove that the triangles
are congruent by AAS Theorem?
In order to use the AAS Theorem, the two triangles must have a pair of CORRESPONDING nonincluded sides that are congruent.
non-included side is
next to "two-arc" angles
The two triangles to the left are congruent by AAS.
If you notice, the non-included sides that are congruent
(noted with one tick mark) are both next to the angle
with two arc marks. Hence, they are in corresponding
locations.
However, one cannot use AAS Theorem to prove
the triangles to the right are congruent.
tick mark next to
"two-arc" angle
tick mark next to
"one-arc" angle
Note that the non-included sides are not in
corresponding locations. It is next to the one-arc
angle on the left triangle, and next to the two-arc
angle on the triangle to the right.
--------------------------------------------------------------------------------------------------------------------Place a tick mark on a side on
Is it possible to use either ASA Theorem
Triangle 2 so that one could use
or AAS Theorem to prove that Triangle 3
AAS Theorem to prove the two
and Triangle 4 below are congruent to
triangles are congruent.
each other?
2
1
3
4
--------------------------------------------------------------------------------------------------------------------Theorems which Prove Triangle Congruency
Do NOT Prove Triangle Congruency
1.
2.
3.
4.
5.
6.
SSS
HL
LL
SAS
ASA
AAS
1. SSA
2. Angle-Side-Side
3. AAA
will learn about
tomorrow
Homework on ASA and AAS
Name_________________________
Z
1. Given: A  Z , and R is the midpoint of TY .
T
R
Y
Prove: ART  ZRY by AAS Theorem
A
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
--------------------------------------------------------------------------------------------------------------------E
G
D
2. Given: BE bisects GED and GBD .
Prove: GEB  DEB by ASA Theorem
B
Statements
Reasons
1.
1.
2.
2.
3.
4.
5.
3.
4.
5.
---------------------------------------------------------------------------------------------------------------------
If KGB  OTA , then complete the congruence statement
3. TA  ___________
in the answer blank. Do not state that it is congruent to itself.
--------------------------------------------------------------------------------------------------------------------4.
Christina has proven that two sides of one triangle are
congruent to two corresponding sides of another triangle. 4.______________________
She wishes to establish their congruency by HL or LL.
What other piece of information does she need?
________________________
3.
5. Daniel has marked the figure to the left with
angle arcs. What theorem did he use in order
to make those markings?
5.________________
--------------------------------------------------------------------------------------------------------------------G
C
6. Consider the figures to the left. Write
a congruency statement for what else
6.________________
must be true in order to prove the
H
triangles are congruent by ASA Theorem.
D
E
F
--------------------------------------------------------------------------------------------------------------------For Questions 7-9, give the theorem by which it would be best to prove the triangles congruent if one
only uses the markings on the figure. If there is not enough information, write "NEI".
7.
8.
9.
7._____________
8._____________
9._____________
--------------------------------------------------------------------------------------------------------------------10.
Suppose you wish to prove that the two triangles to
10.
the right are congruent by HL. Add tick marks or
angle arcs as needed to the figure in order to have
enough information to prove the triangles are congruent by HL.
--------------------------------------------------------------------------------------------------------------------Y
B
11. Complete the triangle congruency
statement in the answer blank based on
11. WQY  __________
the figure to the left.
W
Q
Question 1-----------------------------------------------------------------------------------------------------------------------------------------------------------------1. A  Z , and R is the midpoint of TY .
1. Given
2. TRA  ZRY
2. Vertical Angles' Theorem
3. TR  RY
3. Since R is the midpoint of TY , it divides TY into 2 congruent segments.
4. ART  ZRY
4. AAS Theorem
NOTE THAT LINES 2 & 3 COULD BE SWITCHED.
Question2------------------------------------------------------------------------------------------------------------------------------------------------------------------1. BE bisects GED and GBD .
1. Given
2. GEB  DEB
2. An angle bisector divides an angle into two congruent angles.
3. GBE  DBE
3. An angle bisector divides an angle into two congruent angles.
4. EB  EB
4. Reflexive Property of Congruence
5. GEB  DEB
4. ASA Theorem
NOTE THAT LINES 2, 3, & 4 COULD BE SWITCHED.
Questions 3-11------------------------------------------------------------------------------------------------------------------------------------------------------------3. GB
4. That the two triangles are right triangles
5. Vertical Angles' Theorem
6. D  G
7. SSS
8. NEI
9. AAS
10. mark the hypotenuses (the sides across from the right angle) with two tick marks
11. BYQ