Unit 3 Lessons 1 Properties, Definitions & Theorems
B
E
C
Reflexive Property of Equality: The property that a = a.
Ex: ECF  BCD
F


Symmetric Property of Equality: If a = b, then b = a.
Ex: BD  DB
A
D
Midpoint: the point in the center of a line segment. The midpoint breaks the line segment into two
congruent line segments.
AB  BC
Ex: If B is the midpoint then

A
Bisector of an Angle: a ray that begins at the vertex of the angle and divides
the angle into two angles of equal measure.
Ex: If BC bisec ts ABD then ABC  CBD
C
B

D
Perpendicular Bisector: a line (segment or ray) that is perpendicular to the
segment at its midpoint. Perpendicular lines form right angles.
Ex: If BD is the perpendicular bisector of AC then AB  BC & AC  DB
D
A
Angle Addition: Used in proofs to show that the sum of two or more
adjacent angles is congruent to the larger angle that is formed by them.
Ex: ABC  CBD  ABD


Segment Addition: Used in proofs to show that two or more collinear
segments are congruent to the larger segment that is formed by them.
Ex: DE  EF  DF
B
C
Similar Polygons: Polygons whose corresponding angles are congruent and corresponding sides are
proportional. They have the same shape but may be a different size.
Ex: ABC : A' B'C '
A
A  A , B  B , and C  C
'
AB
A' B'

BC
B 'C '
'

'
AC
4.25 cm
3.72 cm
A 'C '
A'
B
3.72 4.48 4.25


1.86 2.24 2.125
C
4.48 cm
B'
1
AB  A' B'
2
AB  2A' B'
C'
2.24 cm
Scale Factor: The constant you multiply a side of a similar
shape by to get its corresponding side’s length.
Ex:
2.125 cm
1.86 cm
1
3.72   1.86
2
3.72  2 1.86 
Corresponding Angles: In similar and congruent triangles the matching angles which are congruent.
Ex: A & A' , B & B' , and C & C '
Corresponding Sides: In similar triangles the matching sides which are proportional or in congruent
triangles the matching sides which are congruent.
Ex: AB & A' B' , BC & B'C ' , and AC & A'C ' .
E
Similar Triangles
H
Side-Side-Side Similarity Theorem: If three sides of one triangle are
proportional to the corresponding sides of another triangle then the
triangles are similar.
Ex: DEF ~ IHJ

D
F
I
Side-Angle-Side Similarity Theorem: If two sides of one triangle are
proportional to the two corresponding sides of another triangle and the
included angles are congruent then the triangles are similar.
Ex: LJK ~ YZX
J
L
Z
X
Y

Angle-Angle Similarity Theorem: If two angles of one triangle are
congruent to the corresponding two angles of another triangle then the
triangles are similar.
W
Ex: TSR ~ WXY
K
J
T
S
X

Y
R
C
Midpoint Connector Theorem for Triangles: If the midpoints of two consecutive
sides of any triangle are connected then the line segment that is formed is parallel
to the third side and is half the length of the third side.
1
Ex: If E and D are midpoint s then ED// AC and ED  AC.
2
E
A
D

B
3
Triangle Inequality Theorem: The sum of the lengths of two sides of a triangle is
greater than the length of the third.
Ex: 3 + 4 > 5
4+5>3
3+5>4
Congruent Polygons: Polygons whose corresponding angles and
sides are congruent. They are similar with a scale factor of one.
Ex: ABCD  A' B'C ' D'
B
C
4
5
A
A'
D
D'
B'
C'
Congruent Triangles
B
Side-Side-Side Congruency Theorem: If three sides of one
triangle are congruent to the corresponding sides of another
triangle then the triangles are congruent.
Ex: ABC  DEF
A
E
D
C
F

Side-Angle-Side Congruency Theorem: If two sides and the
included angle of one triangle are congruent to the corresponding
two sides and included angle of another triangle then the triangles
are congruent.
Ex: JLK  ZYX
L
Z
X
K
J
Y

Angle-Side-Angle Congruency Theorem: If two angles and
the included side of one triangle are congruent to the
corresponding two angles and included side of another
triangle then the triangles are congruent.
Ex: MNO  TSR
M
N
T
O
S

R
A
Angle-Angle-Side Congruency Theorem: If two angles and the non-included side of
one triangle are congruent to the corresponding two angles and non-included side of
another triangle then the triangles are congruent.
H
Ex: FGH  CAB
B
G

F
Hypotenuse-Leg: If the hypotenuse and leg of one right triangle are
congruent to the hypotenuse and leg of another triangle then the
triangles are congruent.
Ex: QRS  EFG
E
Q
G

F
R
S
Corresponding Parts of Congruent Triangles are Congruent
(CPCTC): Once you have proven that two triangles are congruent
you can then state that any of the corresponding parts are congruent.
Ex: if DAC  BAC then ADC  ABC

C
A
D
B
C
Isosceles Triangle: A triangle in which two sides are congruent. A property is
that its base angles are also congruent.
Ex: ABC is isosceles, AB  AC , and B  C .
A
B
C