POSTULATES:
Any
segment or angle is congruent to itself. (Reflexive Property)
If
there exists a correspondence between the vertices of two triangles such that
three sides of one triangle are congruent to the corresponding sides of the
other triangle, the two triangles are congruent. (SSS)
If
there exists a correspondence between the vertices of two triangles such that
two sides and the included angle of one triangle are congruent to the
corresponding parts of the other triangle, the two triangles are congruent. (SAS)
If
there exists a correspondence between the vertices of two triangles such that
two angles and the included side of one triangle are congruent to the
corresponding parts of the other triangle, the two triangles are congruent. (ASA)
Two
points determine a line (or ray or segment).
If
there exists a correspondence between the vertices of two right triangles such
that the hypotenuse and a leg of one triangle are congruent to the corresponding
parts of the other triangle, the two right triangles are congruent. (HL)
A
line segment is the shortest path between two points.
Through
a point not on a line there is exactly one parallel to the given line. (Parallel
Postulate)
Three
noncollinear points determine a plane.
If a
line intersects a plane not containing it, then the intersection is exactly one
point.
If
two planes intersect, their intersection is exactly one line.
If
there exists a correspondence between the vertices of two triangles such that
the three angles of one triangle are congruent to the corresponding angles of
the other triangle, then the triangles are similar. (AAA)
A
tangent line is perpendicular to the radius drawn to the point of contact.
If a
line is perpendicular to a radius at its outer endpoint, then it is tangent to
the circle.
Circumference
of a circle = 
• diameter.
The
area of a rectangle is equal to the product of the base and the height for that
base.
Every
closed region has an area.
If
two closed figures are congruent, then their areas are equal.
If
two closed regions intersect only along a common boundary, then the area of
their union is equal to the sum of their individual areas.
The
area of a circle is equal to the product of
and the square of the radius.
Total
area of a sphere =
, where r is the sphere's radius.
The
volume of a right rectangular prism is equal to the product of its length, its
width, and its height.
For
any two real numbers x and y, exactly one of the following statements is true x
< y, x = y, or x > y. (Law of Trichotomy)
If a
> b and b > c, then Q > c. Similarly, if x < y and y < z, then x
< z. (Transitive Property of Inequality)
If a
> b, then a + x > b + x. (Addition Property of Inequality)
If x
< y and a > 0, then a • x < a • y. (Positive Multiplication
Property of Inequality)
If x < y and a < 0, then a • x > a • y. (Negative Multiplication Property of Inequality)
The
sum of the measures of any two sides of a triangle is always greater than the
measure of the third side.
THEOREMS:
If two angles are
right angles, then they are congruent.
If two angles are
straight angles, then they are congruent.
If a conditional
statement is true, then the contrapositive of the statement is also true. (If
p, then q 
If ~q, then ~p.)
If angles are
supplementary to the same angle, then they are congruent.
If angles are
supplementary to congruent angles, then they are congruent.
If angles are
complementary to the same angle, then they are congruent.
If angles are
complementary to congruent angles, then they are congruent.
If a segment is added
to two congruent segments, the sums are congruent. (Addition Property)
If an angle is added
to two congruent angles, the sums are congruent. (Addition Property)
If congruent segments
are added to congruent segments, the sums are congruent. (Addition Property)
If congruent angles
are added to congruent angles, the sums are congruent. (Addition Property)
If a segment (or
angle) is subtracted from congruent segments (or angles), the differences are
congruent. (Subtraction Property)
If congruent segments
(or angles) are subtracted from congruent segments (or angles), the differences
are congruent. (Subtraction Property)
If segments (or
angles) are congruent, their like multiples are congruent. (Multiplication
Property)
If segments (or
angles) are congruent, their like divisions are congruent. (Division Property)
If angles (or
segments) are congruent to the same angle (or segment), they are congruent to
each other. (Transitive Property)
If angles (or
segments) are congruent to congruent angles (or segments), they are congruent
to each other. (Transitive Property)
Vertical angles are
congruent.
All radii of a circle
are congruent.
If two sides of a
triangle are congruent, the angles opposite the sides are congruent.
If two angles of a
triangle are congruent, the sides opposite the angles are congruent.
If A =
and B = 
, then the midpoint M =
of 
can be found by using
the midpoint formula:
If two angles are
both supplementary and congruent, then they are right angles.
If two points are
each equidistant from the endpoints of a segment, then the two points determine
the perpendicular bisector of that segment.
If a point is on the
perpendicular bisector of a segment, then it is equidistant from the endpoints
of that segment.
If two nonvertical
lines are parallel, then their slopes are equal.
If the slopes of two
nonvertical lines are equal, then the lines are parallel.
If two lines are
perpendicular and neither is vertical, each line's slope is the opposite
reciprocal of the other's.
If a line's slope is
the opposite reciprocal of another line's slope, the two lines are
perpendicular.
The measure of an
exterior angle of a triangle is greater than the measure of either remote
interior angle.
If two lines are cut
by a transversal such that two alternate interior angles are congruent, the
lines are parallel. (Alt. int. 
|| lines)
If two lines are cut
by a transversal such that two alternate exterior angles are congruent, the
lines are parallel. (Alt. ext. 
|| lines)
If two lines are cut
by a transversal such that two corresponding angles are congruent, the lines
are parallel. (Corr. 
|| lines)
If two lines are cut
by a transversal such that two interior angles on the same side of the
transversal are supplementary, the lines are parallel.
If two lines are cut
by a transversal such that two exterior angles on the same side of the
transversal are supplementary, the lines are parallel.
If two coplanar lines
are perpendicular to a third line, they are parallel.
If two parallel lines
are cut by a transversal, each pair of alternate interior angles are congruent.
(|| lines 
alt. int. 
)
If two parallel lines
are cut by a transversal, then any pair of the angles formed are either
congruent or supplementary.
If two parallel lines
are cut by a transversal, each pair of alternate exterior angles are congruent.
(|| lines
alt. ext. 
)
If two parallel lines
are cut by a transversal, each pair of corresponding angles are congruent. (||
lines
corr. 
)
If two parallel lines
are cut by a transversal, each pair of interior angles on the same side of the
transversal are supplementary.
If two parallel lines
are cut by a transversal, each pair of exterior angles on the same side of the
transversal are supplementary.
In a plane, if a line
is perpendicular to one of two parallel lines, it is perpendicular to the
other.
If two lines are
parallel to a third line, they are parallel to each other. (Transitive Property
of Parallel Lines)
A line and a point
not on the line determine a plane.
Two intersecting
lines determine a plane.
Two parallel lines
determine a plane.
If a line is
perpendicular to two distinct lines that lie in a plane and that pass through
its foot, then it is perpendicular to the plane.
If a plane intersects
two parallel planes, the lines of intersection are parallel.
The sum of the
measures of the three angles of a triangle is 180º.
The measure of an
exterior angle of a triangle is equal to the sum of the measures of the remote
interior angles.
A segment joining the
midpoints of two sides of a triangle is parallel to the third side, and its
length is one-half the length of the third side. (Midline Theorem)
If two angles of one
triangle are congruent to two angles of a second triangle, then the third
angles are congruent. (No-Choice Theorem)
If there exists a
correspondence between the vertices of two triangles such that two angles and a
nonincluded side of one are congruent to the corresponding parts of the other,
then the triangles are congruent. (AAS)
The sum 
of the measures of the angles of a polygon with n sides is
given by the formula 
.
If one exterior angle
is taken at each vertex, the sum 
of the measures of the exterior angles of a polygon is given
by the formula 
.
The number d of
diagonals that can be drawn in a polygon of n sides is given by the formula
The measure E of each
exterior angle of an equiangular polygon of n sides is given by the formula
In a proportion, the
product of the means is equal to the product of the extremes. (Means-Extremes
Products Theorem)
If the product of a
pair of nonzero numbers is equal to the product of another pair of nonzero
numbers, then either pair of numbers may be made the extremes, and the other
pair the means, of a proportion. (Means-Extremes Ratio Theorem)
The ratio of the perimeters
of two similar polygons equals the ratio of any pair of corresponding sides.
If there exists a
correspondence between the vertices of two triangles such that two angles of
one triangle are congruent to the corresponding angles of the other, then the
triangles are similar. (AA~)
If there exists a
correspondence between the vertices of two triangles such that the ratios of
the measures of corresponding sides are equal, then the triangles are similar.
(SSS~)
If there exists a
correspondence between the vertices of two triangles such that the ratios of
the measures of two pairs of corresponding sides are equal and the included
angles are congruent, then the triangles are similar. (SAS~)
If a line is parallel
to one side of a triangle and intersects the other two sides, it divides those
two sides proportionally. (Side-Splitter Theorem)
If three or more
parallel lines are intersected by two transversals, the parallel lines divide
the transversals proportionally.
If a ray bisects an
angle of a triangle, it divides the opposite side into segments that are
proportional to the adjacent sides. (Angle Bisector Theorem)
If an altitude is
drawn to the hypotenuse of a right triangle, then:
The square of the
measure of the hypotenuse of a right triangle is equal to the sum of the
squares of the measures of the legs. (Pythagorean Theorem)
If the square of the
measure of one side of a triangle equals the sum of the squares of the measures
of the other two sides, then the angle opposite the longest side is a right
triangle.
If 
and 
are any two points,
then the distance between them can be found with the formula:
or 
In a triangle whose
angles have the measures 30º, 60º, and 90º, the lengths of the sides opposite
these angles can be represented by x, x
, and 2x respectively. (30°-60°-90''-Triangle Theorem)
In a triangle whose
angles have the measures 45º, 45º, and 90º, the lengths of the sides opposite
these angles can be represented by x, x, and x
respectively. (45°-45°-90°-Triangle Theorem)
If a radius is
perpendicular to a chord, then it bisects the chord.
If a radius of a
circle bisects a chord that is not a diameter, then it is perpendicular to that
chord.
The perpendicular
bisector of a chord passes through the center of the circle.
If two chords of a
circle are equidistant from the center, then they are congruent.
If two chords of a
circle are congruent, then they are equidistant from the center of the circle.
If two central angles
of a circle (or of congruent circles) are congruent, then their intercepted
arcs are congruent.
If two arcs of
a circle (or of congruent circles) are congruent, then the corresponding
central angles are congruent.
If two central
angles of a circle (or of congruent circles) are congruent, then the
corresponding chords are congruent.
If two chords
of a circle (or of congruent circles) are congruent, then the corresponding
central angles are congruent.
If two arcs of
a circle (or of congruent circles) are congruent, then the corresponding chords
are congruent.
If two chords
of a circle (or of congruent circles) are congruent, then the corresponding
arcs are congruent.
If two tangent
segments are drawn to a circle from an external point, then those segments are
congruent. (Two-Tangent Theorem)
The measure of
an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half
the measure of its intercepted arc.
The measure of
a chord-chord angle is one-half the sum of the measures of the arcs intercepted
by the chord-chord angle and its vertical angle.
The
measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent
angle (vertex outside a circle) is one-half the difference of the measures of
the intercepted arcs.
If two
inscribed or tangent-chord angles intercept the same arc, then they are
congruent.
If two inscribed or tangent-chord angles intercept
congruent arcs, then they are congruent.
An angle
inscribed in a semicircle is a right angle.
The sum of the
measures of a tangent-tangent angle and its minor arc is 180º.
If a
quadrilateral is inscribed in a circle, its opposite angles are supplementary.
If a
parallelogram is inscribed in a circle, it must be a rectangle.
If two chords
of a circle intersect inside the circle, then the product of the measures of
the segments of one chord is equal to the product of the measures of the
segments of the other chord. (Chord-Chord Power Theorem)
If a tangent segment and a secant segment are drawn
from an external point to a circle, then the square of the measure of the tangent
segment is equal to the product of the measures of the entire secant segment
and its external part. (Tangent-Secant Power Theorem)
If two secant
segments are drawn from an external point to a circle, then the product of the
measures of one secant segment and its external part is equal to the product of
the measures of the other secant segment and its external part. (Secant- Secant
Power Theorem)
The length of
an arc is equal to the circumference of its circle times the fractional part of
the circle determined by the arc.
The area of a
square is equal to the square of a side.
The area of a
parallelogram is equal to the product of the base and the height.
The area of a
triangle is equal to one-half the product of a base and the height (or
altitude) for that base.
The area of a trapezoid equals one-half the product of the height and the sum of the bases
The measure of the median of a trapezoid equals the average of the measures of the bases.
The area of a trapezoid is the product of the median and the height.
The area of a kite equals half the product of its diagonals.
The area of an
equilateral triangle equals the product of one-fourth the square of a side and
the square root of 3.
The area of a regular
polygon equals one-half the product of the apothem and the perimeter.
The area of a sector
of a circle is equal to the area of the circle times the fractional part of the
circle determined by the sector's arc.
If two figures are
similar, then the ratio of their areas equals the square of the ratio of
corresponding segments. (Similar-Figures Theorem)
A median of a
triangle divides the triangle into two triangles with equal areas
Area of a triangle =
, where a, b, and c are the lengths of the sides of the
triangle and s = semiperimeter =
. (Hero's formula)
Area of a cyclic
quadrilateral = 
, where a, b, c, and d are the sides of the quadrilateral and
s = semiperimeter =
. (Brahmagupta's formula)
The lateral area of a
cylinder is equal to the product of the height and the circumference of the
base.
The lateral area of a
cone is equal to one-half the product of the slant height and the circumference
of the base.
The volume of a right
rectangular prism is equal to the product of the height and the area of the
base.
The volume of any
prism is equal to the product of the height and the area of the base.
The volume of a
cylinder is equal to the product of the height and the area of the base.
The volume of a prism
or a cylinder is equal to the product of the figure's cross-sectional area and
its height.
The volume of a
pyramid is equal to one third of the product of the height and the area of the
base.
The volume of a cone
is equal to one third of the product of the height and the area of the base.
In a pyramid or a
cone, the ratio of the area of a cross section to the area of the base equals
the square of the ratio of the figures' respective distances from the vertex.
The volume of a
sphere is equal to four thirds of the product of 
and the cube of the
radius.
The y-form, or
slope-intercept form, of the equation of a nonvertical line is 
, where b is the y-intercept of the line and m is the slope
of the line.
The formula for an
equation of a horizontal line is y = b, where b is the y-coordinate of every
point on the line.
The formula for the
equation of a vertical line is x = a, where a is the x-coordinate of every
point on the line.
If P = 
and Q = 
are any two points,
then the distance between them can be found with the formula
The equation of a
circle whose center is (h, k) and whose radius is r is
The perpendicular
bisectors of the sides of a triangle are concurrent at a point that is
equidistant from the vertices of the triangle. (The point of concurrency of the
perpendicular bisectors is called the circumcenter of the triangle.)
The bisectors of the
angles of a triangle are concurrent at a point that is equidistant from the
sides of the triangle. (The point of concurrency of the angle bisectors is
called the incenter of the triangle.)
The lines containing
the altitudes of a triangle are concurrent. (The point of concurrency of the
lines containing the altitudes is called the orthocenter of the triangle.)
The medians of a
triangle are concurrent at a point that is two thirds of the way from any
vertex of the triangle to the midpoint of the opposite side. (The point of
concurrency of the medians of a triangle is called the centroid of the
triangle.)
If two sides of a
triangle are not congruent, then the angles opposite them are not congruent,
and the larger angle is opposite the longer side.
If two angles of a triangle
are not congruent, then the sides opposite them are not congruent, and the
longer side is opposite the larger angle.
If two sides of one
triangle are congruent to two sides of another triangle and the included angle
in the first triangle is greater than the included angle in the second
triangle, then the remaining side of the first triangle is greater than the
remaining side of the second triangle. 
If two sides of one
triangle are congruent to two sides of another triangle and the third side of
the first triangle is greater than the third side of the second triangle, then
the angle opposite the third side in the first triangle is greater than the
angle opposite the third side in the second triangle. 
The distance d from
any point P = 
to a line whose
equation is in the form 
can be found with the formula
The area A of a
triangle with vertices at 
, 
, and 
can be found with the
formula
In any triangle ABC,
with side lengths a, b, and c,
where D is the
diameter of the triangle's circumcircle.
In any triangle ABC,
with side lengths a, b, and c,
where d is the length of a segment from vertex C to the opposite side, dividing that side into segments with lengths m and n. (Stewart's Theorem)
If a quadrilateral is
inscribable in a circle, the product of the measures of its diagonals is equal
to the sum of the products of the measures of the pairs of opposite sides.
(Ptolemy's Theorem)
The inradius r of a
triangle can be found with the formula 
, where A is the triangle's area and s is the triangle's
semiperimeter.
The circumradius R of
a triangle can be found with the formula
, where a, b, and c are the lengths of the sides of the
triangle and A is the triangle's area.
lfABCis a triangle
with D on 
, E on 
, and F on 
, then the three segments 
, 
, and 
are concurrent if, and
only if,
If ABC is a triangle
and F is on 
, E is on 
, and D is on an extension of 
, then the three points D, E, and F are collinear if, and
only if,
