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November 09, 2016
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4.5 ASA and AAS2 ink.notebook
Lesson Objectives
Standards
November 09, 2016
Lesson Notes
Lesson Objectives
Standards
Lesson Notes
4.5 ASA and AAS
After this lesson, you should be able to successfully use ASA and AAS to prove triangles are congruent. Press the tabs to view details.
Press the tabs to view details.
ANGLE­SIDE­ANGLE (ASA)
Lesson Objectives
Standards
Lesson Notes
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
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 two triangles are congruent.
and Angle ∠C ≅ ______,
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
E
B
G.CO.10 Prove theorems about triangles.
A
C
D
F
2
4.5 ASA and AAS2 ink.notebook
November 09, 2016
For each diagram, determine which pairs of triangles can be proved congruent by the ASA Postulate.
b) H
a)
G
E
F
D
1. Given: AB Ç CD, ÚCBD ¤ ÚADB Prove: ÆABD ¤ ÆADB A
D
Statements
B
C
Reasons
1. AB Ç CD
2. ÚCBD ¤ ÚADB 1.
3. ÚABD ¤ ÚBDC 3. 4. BD = BD 4.
5. ÆABD ¤ ÆADB 5.
2.
3
4.5 ASA and AAS2 ink.notebook
November 09, 2016
2. Given: ÚS ¤ ÚV and T is the midpoint of SV
Prove: ÆRTS ¤ ÆUTV R
U
S
Statements
V
T
Reasons
1. 1.
2. 2.
3. 3. 4. 4.
5. 5.
ANGLE­ANGLE­SIDE (AAS)
CONGRUENCE THEOREM
If two angles and a non­included side of one triangle are congruent to two angles and the corresponding non­included side of a second triangle, then the two triangles are congruent.
If Angle ∠A ≅ _________, Angle ∠C, ≅ ________, B
E
A
D
C
F
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4.5 ASA and AAS2 ink.notebook
November 09, 2016
For each diagram, determine which pairs of triangles can be proved congruent by the AAS Postulate.
c)
d)
B
D
A
C
3. In the diagram, ∠BCA ¤ ∠DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem?
B
A
1
2
C
D
5
4.5 ASA and AAS2 ink.notebook
November 09, 2016
Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use.
4.
6.
5.
Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence.
7.
A
C
T
9.
8.
M
N
O
P
V
U
B
E
D
Q
P
R
Q
6
4.5 ASA and AAS2 ink.notebook
November 09, 2016
Flow Chart Proof:
10. Given: CF bisects ÚACE and ÚBFD Prove: ÆCBF ¤ ÆCDF C
3 D
4
B 2
1
A
CF bisects ÚACE
ÚACF ¤ ÚECF F
E
CF bisects ÚBFD
ÚBCF ¤ ÚDCF CF ¤ CF
ÆCBF ¤ ÆCDF 11. Given: BC Ç EF, AB = ED, ÚC ¤ ÚF Prove: ÆABC ¤ ÆDEF Statements
A
Reasons
B
C
D
E
F
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4.5 ASA and AAS2 ink.notebook
November 09, 2016
Term/Postulate Abbreviation
Included Side
Angle-Side-Angle
Picture
The side between two
angles. It is in the middle
of the angles.
Included Angle
The angle formed by two
sides. It is in the middle
of the two sides.
Side-Side-Side
If 3 sides of 2 è's are ¤,
then the 2 è are ¤
Side-Angle-Side
If 2 sides & the included Ú
are ¤ in 2 è's, then the
2 è are ¤
Picture
Definition/Explanation
Term/Postulate Abbreviation
Definition/Explanation
If 2 Ú's and the included side
are ¤ in 2 è's, then the 2 è
are ¤
Angle-Angle-Side
If 2 Ú's and the NON-included
side are ¤ in 2 è's, then the 2 è
are
Parts of a Right
Triangle
¤
Hypotenuse: Side opposite the right Ú
Leg
Hypotenuse
Leg: Sides that form a right Ú
Leg
HypotenuseLeg Congruence
Corresponding
Parts of Congruent
Triangles are
Congruent
If hypotenuse and a leg of one
RIGHT
è's,
If 2
è are ¤
to the other RIGHT
then the 2 rt
è' s
è
are
¤
are ¤, then the
corresponding parts are also
¤
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4.5 ASA and AAS2 ink.notebook
November 09, 2016
State if the two triangles are congruent. If they are, state how you know. 1.
2.
5.
6.
PRACTICE
3.
4.
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4.5 ASA and AAS2 ink.notebook
November 09, 2016
Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence.
8.
7.
Name the triangle congruence postulate you can use to prove each pair of triangles congruent. Then state the triangle congruence.
Q
9.
K
J
W
T
L
M
P
S
R
Y
X
10
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November 09, 2016
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Answers:
Answers page 1: 1. ASA 3. ASA 5. ASA 7. SSS, èKJM ¤ èKLM 9. ASA, èXYT ¤ èTWX
Answers page 2: Answers page 3 & 4: 1. C 3. D 5. D 7. C 9. A 11. D 15