Download Triangle Congruence by ASA and AAS

Triangle Congruence:
ASA and AAS
Geometry (Holt 4-6)
K. Santos
Included Side
Included side---is the common side of two consecutive angles in
a triangle
A
B
C
𝐴𝐵 is the included side of < A and <B
Angle-Side-Angle (ASA) Congruence
Postulate (4-6-1)
If two angles and the included side of one triangle are
congruent to two angles and the included side of another
triangle, then the two triangles are congruent.
Given: <A ≅ <D
<B ≅ <F
𝐴𝐵 ≅ 𝐷𝐹
A
B
Then: ∆𝐴𝐵𝐶 ≅ ∆DFE
D
C
E
F
Angle-Angle-Side (AAS) Congruence
Theorem (4-6-2)
If two angles and a nonincluded side of one triangle are
congruent to the corresponding angles and nonincluded side of
another triangle, then the triangles are congruent.
Given: <A ≅ <D
<B ≅ <F
𝐴𝐶 ≅ 𝐷𝐸
A
B
D
C
E
F
Then: ∆𝐴𝐵𝐶 ≅ ∆DFE
Another way the triangles could have been congruent by AAS
would be to use the same angles with 𝐵𝐶 ≅ 𝐹𝐸
Hypotenuse-Leg (HL)
Congruence Theorem (4-6-3)
If the hypotenuse and a leg of a right triangle are congruent to
the hypotenuse and a leg of another triangle, then the triangles
are congruent.
B
C
D
A
F
E
Given: ∆ABC and ∆DEF are right triangles
𝐵𝐶 ≅ 𝐸𝐹
𝐶𝐴 ≅ 𝐹𝐷
Then: ∆𝐴𝐵𝐶 ≅ ∆DEF
Methods to prove two triangles
are congruent
Five ways to prove two triangles are congruent:
SSS Postulate
SAS Postulate
ASA Postulate
AAS Theorem
HL Theorem
Example
Write a congruence statement for each pair of triangles. Name
the postulate or theorem that justifies your statement.
M
1.
Given: < P ≅ < N
< PMO ≅ < NMO
P
O
N
So, ∆𝑃𝑀𝑂 ≅ ∆NMO by AAS
2. V
W
∆𝑉𝑊𝑌 ≅ ∆YZV by ASA
Z
Y
Example
Determine if you can use ASA or AAS to prove ∆𝑁𝑂𝑄 ≅ ∆POM
M
O is a midpoint of 𝑁𝑃
N
O
Q
Since O is a midpoint of 𝑁𝑃
then 𝑁𝑂 ≅ 𝑃𝑂
You know: <Q ≅ <M
And you know vertical angles are congruent <NOQ ≅ <POM
So, ∆𝑁𝑂𝑄 ≅ ∆POM by AAS
P
Proof 1:
Given: < S≅ < Q
𝑅𝑃 bisects <SRQ
Prove: ∆𝑆𝑅𝑃 ≅ ∆QRP
P
S
Q
R
Statements
1. < S≅ < Q
2. 𝑅𝑃 bisects <SRQ
3. < PRS ≅ < PRQ
4. 𝑃𝑅 ≅ 𝑃𝑅
1.
2.
3.
4.
5. ∆𝑆𝑅𝑃 ≅ ∆QRP
5.
Reasons
Given
Given
definition of an angle bisector
Reflexive Property of
congruence
AAS Theorem (1, 3, 4)
Proof 2:
A
Given: 𝐴𝐶 ⟘ 𝐵𝐷
<B≅<D
Prove: ∆𝐴𝐵𝐶 ≅ ∆ADC
B
Statements
1. 𝐴𝐶 ⟘ 𝐵𝐷
2. <ACB and <ACD are right
angles
3. <ACB ≅ < 𝐴𝐶𝐷
4. < B ≅ < D
5. 𝐴𝐶 ≅ 𝐴𝐶
6. ∆𝐴𝐵𝐶 ≅ ∆ADC
C
D
Reasons
1. Given
2. definition of perpendicular
lines
3. all right angles are congruent
4. Given
5. Reflexive Property of
congruence
6. AAS Theorem (3, 4, 5)
Proof 3:
Given: 𝐴𝐶||𝐵𝐷
𝐴𝐷|| 𝐵𝐸
D is a midpoint of 𝐶𝐸
Prove: ∆𝐴𝐷𝐶 ≅ ∆BED
A
B
C
E
D
Reasons
Statements
1. 𝐴𝐶||𝐵𝐷
2. <ACD ≅ <BDE
1. Given
2. Corresponding angles postulate
3. 𝐴𝐷|| 𝐵𝐸
4. <ADC ≅ <BED
3. Given
4. Corresponding angles postulate
5. D is a midpoint of 𝐶𝐸 5. Given
6. 𝐶𝐷 ≅ 𝐸𝐷
6. definition of a midpoint
7. ∆𝐴𝐷𝐶 ≅ ∆BED
7. ASA Postulate (2, 6, 4)
Proof 4:
Given: 𝑃𝑅 ⟘ 𝑆𝑇
𝑃𝑆 ≅ 𝑃𝑇
Prove: ∆ 𝑃𝑅𝑆 ≅ ∆PRT
Statements
1. 𝑃𝑅 ⟘ 𝑆𝑇
2. < PRS and < PRT are right
angles
3. ∆ 𝑃𝑅𝑆 𝑎𝑛𝑑 ∆PRT are right
triangles
4. 𝑃𝑆 ≅ 𝑃𝑇
5. 𝑃𝑅 ≅ 𝑃𝑅
6. ∆ 𝑃𝑅𝑆 ≅ ∆PRT
S
R
T
P
Reasons
1. Given
2. definition of perpendicular
lines
3. definition of a right triangle
4. Given
5. Reflexive Property of congruence
6. HL Theorem (3-4, 5)