Objectives: - Identify & use the SSS, SAS, and ASA Congruence Postulates and the AAS and HL...
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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. 4.3 Analyzing Triangle Congruence Warm-Up: Which pair of triangles could use the ASA to prove congruency?
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Transcript of Objectives: - Identify & use the SSS, SAS, and ASA Congruence Postulates and the AAS and HL...
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.
4.3 Analyzing Triangle Congruence
Warm-Up:
Which pair of triangles could use the ASA to prove congruency?
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 anglesValid or Not?: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.
SSA combination- two sides & an angle that is not between them
Valid or Not?: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.
AAS combination – two angles & a side that is not between them
Valid or Not?:
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
B C E
D
F
𝟕𝟓𝟎𝟕𝟓𝟎
𝟔𝟎𝟎
𝟔𝟎𝟎
A
B C E
D
F𝟒𝟓𝟎𝟒𝟓𝟎𝟔𝟎𝟎
𝟔𝟎𝟎
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
B C E
D
F𝟒𝟓𝟎𝟒𝟓𝟎
𝟔𝟎𝟎𝟔𝟎𝟎
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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.
AAS (Angle-Angle-Side) Congruence Theorem:
For an AAS combination to be used, the congruent parts must correspond. Notice carefully the wording of the following theorem:
Example: Which pairs of congruent triangles can be proven to be congruent by the AAS 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.
HL (Hypotenuse-Leg) Congruence Theorem
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= and m<A=
∆DEF; DE=5, EF=7 and m<F=
∆MNO; MN=8, MO=10, and m<N=
∆JKL; m<J= m<K=and m<L=
∆PQR; PQ=12, m<P= and m<R=
Textbook pages 230-232 Numbers 6-8, 10 -19
Homework:
4.3 Practice Worksheet
Objectives:- Use definitions & postulates to
prove a pair of triangles are congruent.
4.3 Triangle Congruence Proofs
Warm-Up:
Are the lines Parallel or curved?
Given:
Match the Corresponding Parts AB DE
Therefore ∆BAC ∆EDF
B
A
D
CAC DF<A <D
E
F
<A
<B
<C
AB
BC
AC
<D
DE
EF
<E
<F
DF
YW bisects <XYZ
Given: Prove:
XY YZ
∆YWX ∆YWZ
STATEMENTS REASONS
1. 1.
2. <1 <2
3. YW YW
4. ∆YWX ∆YWZ
2.
3.
4.
Y
X ZW
1 2
3 45 6
AB EDGiven: Prove:
CE CB
∆ABC ∆DEC
STATEMENTS REASONS
1. 1.
2.
3.
4. ∆ABC ∆DEC
2. If lines are || then alt int <‘s are
3. Vertical <‘s are
4.
A B
C
E D
4 6
5
1 3
2
HJ bisects IKGiven: Prove:
HJ IK
∆HJI ∆HJK
STATEMENTS REASONS
1. 1.
2.
3.
4. J is the midpoint of IK
2. Def of ⊥
3. All right <‘s are
4.
H
I KJ
6 5
1 2 3 4
5. IJ JK
6. HJ HJ
7. ∆HJI ∆HJK
5.
6.
7.
AB DCGiven: Prove:
AB DC
∆ADC ∆CBA
STATEMENTS REASONS
1. 1.
2. <3 <6
3.
4. ∆ADC ∆CBA
2. If lines are || then alt int <‘s are
3. ref.
4.
A B
CD
4
65
1
32
ME = TAGiven: Prove:
<M <T
∆MEN ∆TAN
STATEMENTS REASONS
1. 1.
2.
3.
4.
2.
3.
4.
A
S N
M
T
E
QU & RT bisect eachother
Given: Prove:
∆QRS ∆TUS
STATEMENTS REASONS
1. 1.
2.
3.
4.
2.
3.
4.
Q T
S
R U
4
6
51
3
2
Homework:Practice Worksheet
