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Triangle Congruencies HOMEWORK: Proofs WS & Triangle Packet - evens Quiz Friday January11, 2012 Lessons 4.3 – 4.5 Triangle Sum, Triangle Inequalities, Side-Angle, Triangle Congruency
Triangle Congruencies
HOMEWORK: Proofs WS & Triangle Packet - evensQuiz Friday
January11, 2012Lessons 4.3 – 4.5
Triangle Sum, Triangle Inequalities, Side-Angle, Triangle Congruency
Two Triangles are congruent if they are the same shape and size.
To PROVE two triangles congruent means to show that all corresponding parts are congruent.
Each triangle has 6 parts: 3 sides & 3 angles
Congruent Triangles
However, shortcuts have been discovered and they make our job easier.
SSSAAASASSSAASASAA
Possible Triangle Congruency Conjecture
Combinations
S – SIDE
A – ANGLE
ORDER MATTERS!!The order of the
corresponding parts in the triangles must mimic the order of the letters in the Congruency Statements
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Three pairs of congruent sides
SSSAAASASSSAASASAA
Possible Triangle
Congruency Combinations
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Three pairs of congruent sides
YES!
SSS
√SSSAAASASSSAASASAA
90°
65°25°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Three pairs of congruent angles
SSSAAASASSSAASASAA
√
90°
65°
25° 25°90°
65°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Three pairs of congruent angles
NO!
All Equilateral/Equiangular triangles have three 60° angles.
60°
60°60°
60°
60°
60°
60°
60°60°
A Counterexample to an AAA congruence conjecture
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesAre these guarantees of triangle
congruence?
So far this is what has been
determined from the list Now let’s try
combinations with 2 sides and an angle
25°
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?One pair of congruent angles, two pairs of
congruent sides
25°
25°√
SSSAAASASSSAASASAA
√25°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and the included angle
YES! SAS!An included angle is between the two given sides.
A
BC
ABC DEF
92°
D
E
F
92°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and the included angle
√
SSSAAASASSSAASASAA
√
65°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and a non-included
angle
√
SSSAAASASSSAASASAA
√
65°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and a non-included
angle
√
SSSAAASASSSAASASAA
√
65°√
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and a non-included
angle
65°√
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and a non-included
angle
A
BC
D
E
F
ABC DEF
√
SSSAAASASSSAASASAA
√
NO!
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent sides and a non-included
angle
90°
25°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent angles and the included side
√
SSSAAASASSSAASASAA
√
90°25°
90°25°
√
SSSAAASASSSAASASAA
√
YES! ASA!
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent angles and the included side
65°55°
A
B
C
65°55°
D
E
F
ABC DEF
ASA!
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent angles and the included side
√
SSSAAASASSSAASASAA
√
√
90° 35°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent angles and the non-included
side
√
SSSAAASASSSAASASAA
√
√
35°
90°
35°
35°
Notes - Congruent TrianglesIs this a guarantee of triangle
congruence?Two pairs of congruent angles and the non-included
side
√
SSSAAASASSSAASASAA
√
√√ NO
NOYES!
√
SSSAAASASSSAASASAA
√
√√
Notes - Congruent TrianglesThese are the triangle congruencies.
SSS
SAS
ASASAA
Using Properties of Right triangles In Combination with Triangle Congruencies
Hypotenuse –Leg Congruence Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.
One More Congruency
HL (Hypotenuse - Leg)is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate.
HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!
SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.
Methods of Proving Triangles Congruent
Congruent TrianglesName the congruence
ASAS
F
R
120°
35° D
P
Q120°
35°
FRS PQD
Is ?
SSA
FRS QSR ?
S
F
R
42°
42° Q
Congruent TrianglesName the congruence
Shared Side – Reflexive Prop
Is
WHY NOT?SSA is NOT a valid Triangle congruence
SAS
FRS QSR
S
F
R
50°
50° Q
Congruent TrianglesName the congruence
Shared Side – Reflexive Prop
Is ?
SSS ?
MNR PTB
M
N
R
T
P
B
Names of the triangles in the congruence statement are not in corresponding order.
WHY NOT?
Congruent TrianglesName the congruence
Is ?
