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Triangle Proofs in a Nutshell
5 Triangle Congruency Shortcuts:
SSS (Side, Side, Side)
B
A
E
C
D
F
SAS (Side, Included Angle, Side)
B
A
E
C
D
F
ASA (Angle, Included Side, Angle)
B
A
E
C
D
F
AAS (Angle, Angle, Non-included Side)
B
E
A
C
D
F
HL (Right Triangle, Hypotenuse, Leg)
E
B
A
C
D
F
*Note: We can NOT prove triangles congruent with AAA or SSA!!!!
10 Most Common Properties, Definitions & Theorems for Triangles
1. Reflexive Property : ̅̅̅̅
AB ≅ ̅̅̅̅
BA or 2. Vertical Angle Theorem (VAT)
Vertical angles are congruent, use when two
∠𝐸 ≅ ∠𝐸
Use when the triangles have an angle or side in
common.
lines are intersecting.
C
C
B
E
E
A
B
I
F
A
D
H
G
3. Right Angle Theorem (RAT)
D
4. AIAT (Alternate Interior Angles)
All right angles are congruent, use when you are Alternate interior angles are congruent, use
given right angles, squares, rectangles, or
when you are given parallel lines.
perpendicular lines.
A  C
A
B
C
B
D
A
C
D
5. Definition of Midpoint
6. Definition of Angle Bisector
Use when given a midpoint of a segment,
Use when given an angle bisector, showing an
showing the segment is divided into 2 congruent angle is divided into 2 congruent angles.
segments.
C
̅̅̅̅ bisects ABC
BD
̅̅̅̅
E is the midpoint of AB
B
B
E
A
D
A
D
C
7. Definition of Perpendicular
Lines
8. Definition of a Perpendicular
Bisector
Use when given line segments are
perpendicular, showing right angles.
Results in 2 congruent segments and 2 right
angles
̅̅̅̅
BD is a perpendicular bisector of ̅̅̅̅
AC
̅̅̅̅
BD  ̅̅̅̅
AC
B
B
A
D
A
C
9. Isosceles Triangle Theorem and
its Converse
C
D
10. 3rd Angle Theorem
If 2 angles of a triangle are congruent to 2
Use when given an isosceles triangle, 2 sides are angles of another triangle, then the third angles
are congruent.
congruent and the angles opposite them are
A  D and C  F then B  E
congruent.
B
E
If ̅̅̅̅
BA ≅ ̅̅̅̅
BC then ∠A ≅ ∠C or
̅̅̅̅ ≅ BC
̅̅̅̅
if ∠A ≅ ∠C then BA
B
A
A
C
C
**Note: DO NOT ASSUME ANYTHING IF IT IS NOT GIVEN
D
F