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Objectives:
- Identify & use the SSS, SAS, and ASA
Congruence Postulates and the AAS and HL
Congruence Theorems.
- Use counterexamples to prove that other
side and angle combinations cannot be used
to prove triangle congruence.
Warm-Up:
Which pair of
triangles could use
the ASA to prove
congruency?
Valid or Not?:
AAA combination – three angles
AAS combination – two angles and a side that
is not between them
SSA combination- two sides and an angle that
is not between them (that is,
an angle opposite one of the
two sides).
Valid or Not?: AAA combination – three angles
you can rule out the AAA combination by finding a counterexample
for the following conjecture:
Conjecture: If the three angles of one triangle are congruent to
the three angles of another triangle, then the triangles are
congruent.
Counterexample: In the triangles below there are three pairs
of congruent angles, but the two triangles are not congruent.
Therefore the conjecture is false.
Valid or Not?: SSA
combination- two sides & an angle that
is not between them
you can rule out the SSA combination by finding a counterexample
for the following conjecture:
Conjecture: If two sides and an angle that is not in between them
of one triangle are congruent to two sides and an angle that is not
in between them another triangle, then the triangles are congruent.
Counterexample: In the triangles below the sides and angles
are congruent, but the triangles are not congruent.
Valid or Not?: AAS
combination – two angles & a side that
is not between them
The AAS combination can be converted to the ASA combination:
The triangles below are an AAS combination. To convert this to
an ASA combination, find the measures of the third angle in each
triangle.
A
D
𝟎
𝟕𝟓
𝟕𝟓𝟎
B
𝟔𝟎𝟎
C
E
𝟔𝟎𝟎
A
B
𝟔𝟎𝟎
F
D
𝟒𝟓𝟎
C
E
𝟔𝟎𝟎
𝟒𝟓𝟎
F
Valid or Not?:
The two triangles below represent a version of
the AAS, but the triangles are not congruent.
There is an important difference between the
two triangles. What is it?
A
D
8
B
𝟔𝟎𝟎
8
𝟒𝟓𝟎
C
E
𝟔𝟎𝟎
𝟒𝟓𝟎
F
For an AAS combination to be used, the congruent parts must
correspond. Notice carefully the wording of the following theorem:
AAS (Angle-Angle-Side) Congruence Theorem:
If two angles and a non-included side of one
triangle are congruent to the corresponding
angles and non-included side of another
triangle, then the triangles are congruent.
Example:
Which pairs of congruent triangles can be proven to be
congruent by the AAS Congruence Theorem?
HL (Hypotenuse-Leg) Congruence Theorem
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of
another right triangle, then the two triangles
are congruent.
Examples:
Determine whether the given combination of angles and
sides determined a unique triangle. If so, identify the
theorem or postulate that supports your answer.
∆ABC; AB=6, m<B=𝟕𝟎𝟎 , and m<A=𝟒𝟎𝟎
∆DEF; DE=5, EF=7, and m<F=𝟑𝟎𝟎
∆JKL; m<J=𝟓𝟎𝟎 , m<K=𝟕𝟓𝟎 and m<L=𝟓𝟓𝟎
∆MNO; MN=8, MO=10, and m<N=𝟗𝟎𝟎
∆PQR; PQ=12, m<P=𝟒𝟓𝟎 , and m<R=𝟏𝟎𝟎𝟎
Homework:
4.3 Practice Worksheet
Textbook pages 230-232 Numbers 6-8, 10 -19
Objectives:
- Use definitions & postulates to prove a
pair of triangles are congruent.
Warm-Up:
Are the
lines
Parallel
or curved?
Match the Corresponding Parts
B
Given: AB ≅ DE
AC ≅ DF
<A ≅ <D
Therefore ∆BAC ≅ ∆EDF
<A
<D
<B
DE
<C
EF
AB
<E
BC
<F
AC
DF
A
C
F
D
E
Y
Given: YW bisects <XYZ
1 2
XY ≅ YZ
Prove: ∆YWX ≅ ∆YWZ
STATEMENTS
REASONS
1.
1.
2. <1 ≅ <2
2.
3. YW ≅ YW
3.
4. ∆YWX ≅ ∆YWZ
4.
X
3
5 4
W
6
Z
Given: AB || ED
A
CE ≅ CB
Prove: ∆ABC ≅ ∆DEC
STATEMENTS
REASONS
4
6
C
E
1
B
5
2
3
1.
1.
2.
2. If lines are || then alt int <‘s are ≅
3.
3. Vertical <‘s are ≅
4. ∆ABC ≅ ∆DEC
4.
D
H
Given: HJ bisects IK
6 5
HJ ⊥ IK
Prove: ∆HJI ≅ ∆HJK
STATEMENTS
REASONS
I
1
1.
1.
2.
2. Def of ⊥
3.
3. All right <‘s are ≅
4. J is the midpoint of IK 4.
5. IJ ≅ JK
5.
6. HJ ≅ HJ
6.
7. ∆HJI ≅ ∆HJK
7.
2 3
J
4
K
Given: AB || DC
AB ≅ DC
Prove: ∆ADC ≅ ∆CBA
STATEMENTS
A
2
D
1
3
4
6
B
5
C
REASONS
1.
1.
2. <3 ≅ <6
2. If lines are || then alt int <‘s are ≅
3.
3. ref.
4. ∆ADC ≅ ∆CBA
4.
Given: ME = TA
M
A
<M ≅ <T
Prove: ∆MEN ≅ ∆TAN
N
S
E
STATEMENTS
REASONS
1.
1.
2.
2.
3.
3.
4.
4.
T
Given: QU
& RT bisect eachother
T
Q
1
Prove: ∆QRS ≅ ∆TUS
3
2
STATEMENTS
1.
2.
REASONS
1.
2.
3.
3.
4.
4.
5
R
S
4
6
U
Homework: Practice
Worksheet