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Module 5
Triangle Congruency
Criteria
10/6/19 115
How much do you need to know. . . . . . about two triangles to prove that they are congruent?
116 116
Do you need all six ?
NO !
5.4 SSS 5.3 SAS 5.2 ASA 6.2 AAS 6.3 HL
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1. AB ≅ DE 2. BC ≅ EF 3. AC ≅ DF
ΔABC ≅ Δ DEF
B
A C
Side
Side
Side
The triangles are congruent by
SSS.
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
SIDE-‐SIDE-‐SIDE (SSS)
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The angle between two sides
Included Angle
∠ HGI ∠ G
∠ GIH ∠ I
∠ GHI ∠ H
This combo is called side-angle-side, or just SAS.
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Name the included angle:
YE and ES
ES and YS
YS and YE
Included Angle
S Y
E
∠ YES or ∠E ∠ YSE or ∠S ∠ EYS or ∠Y The other two
angles are the NON-INCLUDED
angles.
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1. AB ≅ DE 2. ∠A ≅ ∠ D 3. AC ≅ DF
ΔABC ≅ Δ DEF
B
A C
included angle
Side
Angle Side
The triangles are congruent by
SAS.
If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
SIDE-‐ANGLE-‐SIDE (SAS)
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The side between two angles
Included Side
GI HI GH
This combo is called angle-side-angle, or just ASA.
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Name the included side:
∠Y and ∠E
∠E and ∠S
∠S and ∠Y
Included Side
S Y
E
YE ES SY
The other two sides are the
NON-INCLUDED sides.
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1. ∠A ≅ ∠ D 2. AB ≅ DE 3. ∠ B ≅ ∠ E
ΔABC ≅ Δ DEF
B
A C
included side
Angle
Side
Angle
The triangles are congruent by
ASA.
If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
ANGLE-‐SIDE-‐ANGLE (ASA)
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1. ∠A ≅ ∠ D 2. ∠ B ≅ ∠ E 3. BC ≅ EF
ΔABC ≅ Δ DEF
Non-included side
B
A C
Side Angle
Angle
The triangles are congruent
by AAS.
If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.
ANGLE-‐ANGLE-‐SIDE (AAS)
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Warning: No SSA Postulate
There is no such thing as an SSA
postulate!
The triangles are NOTcongruent!
Side
Side
Angle
126
Warning: No SSA Postulate
NOT CONGRUENT!
There is no such thing as an SSA
postulate!
127 127
BUT: SSA DOES work in one situation!
If we know that the two triangles
are right triangles!
Side Side
Side
Angle 128 128
We call this
These triangles ARE CONGRUENT by HL!
HL, for “Hypotenuse – Leg”
Hypotenuse Leg
Hypotenuse
RIGHT Triangles!
Remember! The
triangles must be RIGHT!
129 129
Hypotenuse-Leg (HL)
1. AB ≅ HL 2. CB ≅ GL 3. ∠C and ∠G
are rt. ∠ ‘s
ΔABC ≅ Δ DEF
The triangles are congruent
by HL.
Right Triangle
Leg
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
130 130
Warning: No AAA Postulate
A C
B
D
E
F
There is no such thing as an AAA
postulate!
NOT CONGRUENT!
Same Shapes!
Different Sizes!
131 131
Congruence Postulates and Theorems
• SSS • SAS • ASA • AAS • AAA? • SSA? • HL
132 132
Name That Postulate
SAS ASA
AAS SSA
(when possible)
Not enough info!
133 133
Name That Postulate (when possible)
SSS AAA
SSA
Not enough info!
Not enough info!
SSA HL 134 134
Name That Postulate (when possible)
SSA
AAA
Not enough info!
Not enough info!
HL
SSA
Not enough info!
135 135
Vertical Angles, Reflexive Sides and Angles
When two triangles touch, there may be additional congruent parts.
Vertical Angles
Reflexive Side
side shared by two
triangles
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Name That Postulate (when possible)
SAS
AAS
SAS Reflexive Property
Vertical Angles
Vertical Angles
Reflexive Property SSA
Not enough info!
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When two triangles overlap, there may be additional congruent parts.
Reflexive Side side shared by two
triangles Reflexive Angle angle shared by two
triangles
Reflexive Sides and Angles
138 138
Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
For SAS:
∠B ≅ ∠D
For AAS: ∠A ≅ ∠F AC ≅ FE
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140
141
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Both triangles are congruent. ∠V ≅ ∠Y, WV ≅ XY, and WX ≅ WX by the Reflexive Property of Congruence. Although at first instance it looks like SSA, which is not a congruence postulate, you then realize that it is a RIGHT TRIANGLE, and it can be seen that we have a hypotenuse and a leg. Therefore, both triangles are congruent by the HL congruence postulate.
ANSWER
5.2/5.3/5.4/6.2/6.3 Classwork PART 1
• GO ONLINE In Class: Due in TWO Days? Honors: 5.2: 3-‐6,14,20 5.3 2-‐7,12 5.4 4,6,14,15,16,17,23 6.2 1-‐6-‐9,21,27* 6.3 2,3,10-‐11 Regular: 5.2: 3-‐6, 5.3 2-‐6, 5.4 4,6,14,15, 6.2 1-‐6-‐9,21 6.3 2,3,10
Reminders: q Guess what?! A quiz is nearing.
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