Download List of Theorems, Postulates and Definitions 4

Annalee Ith
Plis, 01
Important Postulates, Definitions, and Theorems!!!
Postulates
Distance Postulate
a. Uniqueness Postulate
On a line, there is a unique
distance between two points.
b. Distance Formula
If two points on a line have
coordinates x and y, the
distance between them is |xy|.
c. Additive Property
If B is on line segment AC,
then AB+BC=AC.
Angle Measure Postulate
a. Unique Measure
Assumption
Every angle has a unique
measure from 0 degrees to
180 degrees.
b. Unique Angle Assumption
Given any ray VA and any
Postulates of Equality
For any real numbers a, b,
and c:
1.Reflexive Property of
Equality: a=a
2.Symmetric Property of
Equality: If a=b, then b=a
3. Transitive Property of
Equality: If a=b and b=c,
then a=c.
Definitions
Parallel Lines- (Two
coplanar lines) If and only if
two lines have no points in
common or they are
identical.
Theorems
Line Intersection TheoremTwo different lines intersect
in at most one point.
Ray- With endpoint A and
containing a second point B
consists of the points on line
segment AB and all points
for which B is between each
of them and A.
Linear Pair Theorem- If
two angles form a linear pair,
then they are supplementary.
Ray AB and ray AC are
opposite rays if and only if
A is between B and C.
Vertical Angles TheoremIf two angles are vertical
angles, then they have equal
measures.
Postulates of Equality and
Operations
For any real numbers a, b,
and c:
Addition Property of
Equality: If a=b, then
a+c=b+c.
Multiplication Property of
Equality: If a=b, then ac=bc.
Postulates of Inequality
The midpoint of a segment
Parallel Lines and Slopes
AB is the point M on line AB Theorem
with AM=MB.
Two nonvertical lines are
parallel if and only if they
have the same slope.
Polygon- the union of
Transitivity of Parallelism
and Operations
For any real numbers a, b,
and c:
Transitive Property of
Inequality: If a<b and b<c,
then a<c.
Addition Property of
Inequality: If a<b, then
a+c<b+c.
Multiplication Properties
of Inequality: If a<b and
c>0, then ac<bc.
If a<b and c<0, then ac>bc.
Postulates of Equality and
Inequality
iFor any real numbers a, b,
and c:
Equation to Inequality
Property: If a and b are
positive numbers and a+b=c,
then c>a and c>b.
Substitution Property: If
a=b, then a may be
substituted for b in any
expression.
Cooresponding Angles
Postulate
Suppose two coplanar lines
are cut by a transversal.
a. If two corresponding
angles have the same
measure, then the
lines are parallel.
b. If the lines are
parallel, then
corresponding angles
have the same
measure.
Reflection Postulate
Under a reflection:
a. There is a 1-1
correspondence
between points and
their images.
b. Collinearity is
preserved
segments in the same plane
such that each segment
intersects exactly two others,
one at each of its endpoints.
Theorem
In a plane, if line L is parallel
to line m and line m is
parallel to line n, then line L
is parallel to line n.
Angle- the union of two rays
that have the same endpoint.
Sides- the two rays that form
an angle. Vertex- the
common endpoint of the two
rays.
Two Perpendicular
Theorem
If two coplanar lines L and m
are each perpendicular to the
same line, then they are
parallel to each other.
If m is the measure of an
angle, then the angle is
a. Zero if and only if m
=0
b. Acute if and only if
0 < m < 90
c. Right if and only if m
= 90
d. Obtuse if and only if
90<m<180
e. Straight if and only
if m=180
Perpendicular to Parallels
Theorem
In a plane, if a line is
perpendicular to one of two
parallel lines, then it is also
perpendicular to the other.
If the measures of two angles
are m1 and m2, then the angles
are
a.Complementary if and
only if m1+m2=90
b.Supplementary if and
only
if m1+m2=180
Perpendicular Lines and
Slopes Theorem
Two nonvertical lines are
perpendicular if and only if
the product of their slopes is
-1.
c. Betweenness is
preserved.
d. Distance is preserved.
e. Angle measure is
preserved.
f. Orientation is
preserved.
Playfair’s Parallel
Postulate
(See Uniqueness of
Parallels Theorem)
Two non-straight and
nonzero angles are adjacent
angles if and only if a
common side is interior to
the angle formed by the noncommon sides.
Figure Reflection Theorem
If a figure is determined by
certain points, then its
reflection image is the
corresponding figure
determined by the reflection
images of those points.
Two adjacent angles form a
linear pair if and only if
their non-common sides are
opposite rays.
Two-Reflection Theorem
for Translations
If m ll L, the translation has
magnitude two times the
distance between L and m, in
the direction from
L
perpendicular to m.
Two-Reflections Theorem
for Rotations
If m intersects L, the rotation
has center at the point of
intersection of m and L and
has magnitude twice the
measure of the non-obtuse
angle formed by these lines,
in the direction from L to m.
CPCF Theorem
If two figures are congruent,
then any pair of
corresponding parts is
congruent.
A-B-C-D Theorem
Every isometry preserves
Angle measure,
Betweenness, Collinearity,
and Distance.
Two non-straight angles are
vertical angles if and only if
the union of their sides is
two lines.
The degree measure of a
minor arc of a circle is the
measure of its central angle.
The degree measure of a
major arc of a circle is 360
degrees- m(degree of minor
arc).
The slope of the line through
Segment Congruence
Theorem
(x1, y1) and (x2, y2) is y2-y1
divided by x2-x1.
Two segments are congruent
if and only if they have the
same length.
Angle Congruence
Two segments, rays, or lines Theorem
are perpendicular if and
Two angles are congruent if
only if the lines containing
and only if they have the
them form a 90 degree angle. same measure.
ll Lines
AIA
Bisector of a segment- its
Congruence Theorem
midpoint, or any line, ray, or If two parallel lines are cut
segment which intersects the by a transversal, then
segment only at its midpoint. alternate interior angles are
congruent.
AIA Congruence ll
Perpendicular Bisector- In Theorem
a given plane, one line that is If two lines are cut by a
a bisector and perpendicular transversal and form
to the segment.
congruent alternate interior
angles, then the lines are
parallel.
For a point P not on a line m, Theorem
If two lines are cut by a
the reflection image of P
over line m is the point Q if transversal and form
congruent alternate exterior
and only if m is the
angles, then the lines are
perpendicular bisector of line parallel.
segment PQ. For a point P
on m, the reflection image of
P over line m is P itself.
A transformation is a
Perpendicular Bisector
correspondence between two Theorem
sets of points such that
If a point is on the
1) Each point in the
perpendicular bisector of a
preimage set has a
segment, then it is
unique image
equidistant from the
2) Each point in the
endpoints of the segment.
image set has exactly
one preimage.
The composite of a first
Uniqueness of Parallels
transformaton S and a
Theorem
second trasnformation T is
Through a point not on a
the transformation that maps line, there is exactly one line
each point P onto T(S(P)).
parallel to the given line.
A translation is the
Triangle-Sum Theorem
composite of two reflections The sum of the measures of
over parallel lines.
A rotation is the composite
of two reflections over
intersecting lines.
A vector is a quantity that
can be characterized by its
direction and magnitude.
Let rm be a reflection and T
be a translation with positive
magnitude and direction
parallel to m. Then G=Torm is
a glide reflection.
Two figures F and G are
congruent figures if and
only if G is the image of F
under an isometry.
Circle- If A and B are on
cirlce O, then segment OA is
congruent to segment OB.
the angles of a triangle is 180
degrees.
Quadrilateral-Sum
Theorem
The sum of the measures of
the angles of a convex
quadrilateral is 360 degrees.
Polygon-Sum Theorem
The sum of the measures of
the angles of a convex n-gon
is (n-2) x 180 degrees.