Proving Triangles Congruent - Mrs....
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Transcript of Proving Triangles Congruent - Mrs....
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Proving Triangles Congruent
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Two geometric figures with exactly the same size and shape.
The Idea of Congruence
A C
B
DE
F
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How much do you need to know. . . !
. . . about two triangles to prove that they are congruent?
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Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.
!
Corresponding Parts
ΔABC ≅ Δ DEF
B
A C
E
D
F
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
4. ∠ A ≅ ∠ D
5. ∠ B ≅ ∠ E
6. ∠ C ≅ ∠ F
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Do you need all six ?
NO !
SSS SAS ASA AAS
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Side-Side-Side (SSS)
1. AB ≅ DE
2. BC ≅ EF
3. AC ≅ DF
ΔABC ≅ Δ DEF
B
A
C
E
D
F
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.
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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
SY
E
∠ YES or ∠E
∠ YSE or ∠S
∠ EYS or ∠Y The other two angles are the
NON-INCLUDED angles.
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Side-Angle-Side (SAS)
1. AB ≅ DE
2. ∠A ≅ ∠ D
3. AC ≅ DF
ΔABC ≅ Δ DEF
B
AC
E
D
F
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.
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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
SY
E
YE
ES
SY
The other two sides are the NON-
INCLUDED sides.
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Angle-Side-Angle (ASA)
1. ∠A ≅ ∠ D
2. AB ≅ DE
3. ∠ B ≅ ∠ E
ΔABC ≅ Δ DEF
B
AC
E
D
F
!included
sideAngle
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.
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E
D
F
Angle-Angle-Side (AAS)
1. ∠A ≅ ∠ D
2. ∠ B ≅ ∠ E
3. BC ≅ EF
ΔABC ≅ Δ DEF
Non-included side
B
AC
SideAngle
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.
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Warning: No SSA Postulate
There is no such thing as an SSA
postulate!
The triangles are NOTcongruent!
Side
Side
Angle
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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!
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Congruence Postulates and Theorems
• SSS • SAS • ASA • AAS • AAA? • SSA?
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Name That Postulate
SAS ASA
AASSSA
(when possible)
Not enough info!
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Name That Postulate(when possible)
SSSAAA
SSA
Not enough info!
Not enough info!
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Name That Postulate(when possible)
SSA
AAA
Not enough info!
Not enough info!
SSA
Not enough info!
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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
SASReflexive 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 Angles22
Let’s PracticeIndicate 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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