Download Segment Addition Postulate

Theorem
Pythagorean Theorem
c
a
b
a² + b² = c²
p. 20
Theorem
Distance Formula
A( x1 , y1 )
B ( x2 , y 2 )
AB  ( x2  x1 )  ( y2  y1 )
2
2
p. 19
Postulate
Segment Addition Postulate

A
C
B
If B is between A and C,
then AB + BC = AC.
p. 18
Postulate
Angle Addition Postulate
A
P
B

C
If P is in the interior of ABC, then
mABP + mPBC = mABC
p. 27
Definition
midpoint

M
C
A
The midpoint is a point that
divides or bisects a segment into
two equal segments.
If M is a midpoint, then AM = MC.
p. 34
Definition
segment bisector
k

M
C
A
A segment bisector is a line, ray,
segment or plane that intersects a
segment at its midpoint.
p. 34
Definition
angle bisector
1
2
1  2
An angle bisector is a ray that
divides an angle into two
congruent adjacent angles.
p. 36
Theorem
Midpoint Formula
M

A( x1 , y1 )
B( x2 , y2 )
 x1  x2 y1  y2 
M 
,

2 
 2
p. 35
Definition
complementary
angles
1
2
m1 + m2 = 90
A pair of angles whose sum is
90° are complementary.
p. 46
Definition
supplementary
angles
1
2
m1 + m2 = 180
A pair of angles whose sum is
180° are supplementary.
p. 46
Definition
right angle
90°
An angle whose measure is 90° is
a right angle.
p. 28
Definition
perpendicular
lines
Two lines are called perpendicular if
they intersect to form a right angle.
p. 79
Property
Reflexive
B
BD  BD
A
D
C
For any real number, a = a.
p. 96
Property
Transitive
If AB = CD and CD = EF, then AB = EF.
.
A
.B
.
C
.D
.
.F
E
If a = b and b = c, then a = c.
p. 96
Property
Addition Property
of Equality
If AB = CD, then AC = BD.
.
.
.
.
A
B
C
D
If a = b, then a + c = b + c.
p. 96
Property
Subtraction Property
of Equality
If AC = BD, then AB = CD.
.
.
.
.
A
B
C
D
If a = b, then a  c = b  c.
p. 96
Property
Substitution
Example:
If AB = 5 + x and x = 3, then AB = 8.
If a = b, then a can be substituted
for b in any equation or expression.
p. 96
Theorem
Right Angle Congruence
Theorem
1  2
1
2
All right angles are congruent.
p. 110
Theorem
Congruent Supplements
Theorem
If m1 + m2 = 180 and m2 + m3 = 180,
then 1  3.
2
1
3
Two angles supplementary to the
same angle (or  ’s) are congruent.
p. 111
Theorem
Congruent Complements
Theorem
If m1 + m2 = 90 and m2 + m3 = 90,
then 1  3.
1
2
3
Two angles complementary to the
same angle (or  ’s) are congruent.
p. 111
Postulate
Linear Pair Postulate
m1 + m2 = 180
1
2
If two angles form a linear pair,
then they are supplementary.
p. 111
Theorem
Vertical Angles
Theorem
1  2 and 3  4
1
3
4
2
Vertical angles are congruent.
p. 112
Theorem
Linear Pair of  s
h
g
h
g
If two lines intersect to form a linear
pair of congruent angles, then the
lines are perpendicular.
p. 137
Postulate
Corresponding Angles
Postulate
1  2
1
2
If two parallel lines are cut by a transversal,
then corresponding ’s are .
p. 143
Theorem
Alternate Interior Angles
Theorem
1  2
1
2
If two parallel lines are cut by a transversal,
then alt. int. ’s are .
p. 143
Theorem
Alternate Exterior Angles
Theorem
1  2
1
2
If two parallel lines are cut by a transversal,
then alt. ext. ’s are .
p. 143
Theorem
Consecutive Interior Angles
Theorem
m1 + m2 = 180
1
2
If two parallel lines are cut by a
transversal, then consecutive int. ’s
are supplementary.
p. 143
Theorem
Perpendicular Transversal
Theorem
jm
j
k
m
If a transversal is perpendicular to one of
two parallel lines, then it is perpendicular
to the other.
p. 143
Theorem
Two Lines Perpendicular to
Same Line
j // m
j
m
k
In a plane, two lines perpendicular to
the same line are parallel to each other.
p. 157
Theorem
Two Lines Parallel to the
Same Line
m // n
m
k
n
If two lines are parallel to the same line,
then they are parallel to each other.
p. 157
Theorem
Triangle Sum Theorem
B
A
C
mA + mB + mC = 180
The sum of the measures of the interior
angles of a triangle is 180°.
p. 196
Theorem
Exterior Angle Theorem
B
A
1
m1 = mA + mB
The measure of an exterior angle of a
triangle is equal to the sum of the two
remote interior angles.
p. 197
Theorem
Third Angles Theorem
B
E
A
If A  D and B  E,
then C  F
C
D
F
If two angles of one  are  to two angles
of another , the third angles are .
p. 203
Postulate
SSS
Side-Side-Side Congruence
B
A
E
C
D
F
If AB  DE , BC  EF , and CA  FD ,
then ABC  DEF.
If three sides of one  are  to three
sides of another , then the ’s are .
p. 212
Postulate
SAS
Side-Angle-Side Congruence
B
A
E
C
D
F
If AB  DE , CA  FD and A  D,
then ABC  DEF.
If two sides of one  are  to two sides
of another , and the included s are ,
then the ’s are .
p. 213
Theorem
Perpendicular/Right  Theorem
(Meyers Theorem)
j
m
1
2
k
If j  k and m  k,
then 1  2.
Perpendicular lines form  right s.
p. 157
Postulate
ASA
Angle-Side-Angle Congruence
B
A
E
C
D
F
If A  D, C  F and CA  FD ,
then ABC  DEF.
If two s of one  are  to two s of
another , and the included sides are ,
then the ’s are .
p. 220
Postulate
AAS
Angle-Angle-Side Congruence
B
A
E
C
D
F
If A  D, C  F and BC  EF ,
then ABC  DEF.
If two s of one  and a non-included side are
 to two s of another  and the corresponding
non-included side, then the ’s are .
p. 220
Theorem
Base Angles Theorem
If two sides of a  are , then the s
opposite those sides are .
p. 236
Theorem
Base Angles Converse
Theorem
If two s of a  are , then the sides
opposite those s are .
p. 236
Theorem
Hypotenuse-Leg Theorem
H-L
A
B
D
C
E
F
If the hypotenuse and a leg of one right 
are  to a hyp. and a leg of another rt. ,
the two s are .
p. 238
Theorem
Perpendicular Bisector
Theorem
k
AC = BC
C
B
A
P
If a point is on the  bisector of a
segment, then it is equidistant from
the endpoints of that segment.
p. 265
Theorem
Angle Bisector Theorem
AP = CP
A
P
B
C
If a point is on the bisector of an
angle, then it is equidistant from the
sides of the angle.
p. 266
Theorem
Circumcenter
The perpendicular bisectors of a triangle
intersect in a point that is equidistant from
the vertices of the triangle.
p. 273
Theorem
Incenter
The angle bisectors of a triangle intersect
in a point that is equidistant from the sides
of the triangle.
p. 274
Theorem
Centroid
B
E
A
P
F
D
C
The medians of a triangle (E, D, and F are
midpoints) intersect in a point called a centroid.
AP = 2/3 AD, BP = 2/3 BF, CP = 2/3 CE
p. 279
Theorem
Orthocenter
The altitudes of a triangle intersect in a
point of concurrency called an orthocenter.
p. 281