__Postulates
& Theorems__

__Chapter 2
A Geometric System__

Postulate 2-1 Through any two points there is exactly one line.

Postulate 2-2 Through any three points not on the same line there is exactly one plane.

Postulate 2-3 A line contains at least two points.

Postulate 2-4 A plane contains at least three points not on the same line.

Postulate 2-5 If two points lie in a plane, then the entire line containing those two points lies in that plane.

Postulate 2-6 If two planes intersect, then their intersection is a line.

Theorem 2-1 If there is a line and a point not on the line, then there is exactly one plane that contains them.

Theorem 2-2 If two lines intersect, then exactly one plane contains both lines.

__Chapter 3 Measurement__

Postulate 3-1 Ruler
Postulate The points on any line can be paired with real numbers so that
given any two points P

and Q on the line, P corresponds to zero, and Q corresponds to a positive
number.

Postulate 3-2 Distance
Postulate For any two points on a line and a given unit of measure, there
is a unique positive

number called the measure of the distance between the two points.

Postulate 3-3 Segment Addition Postulate If line PQR, then PQ+RQ = PR

Theorem 3-1 Every segment has exactly one midpoint.

Theorem 3-2 Congruence of segments is reflexive, symmetric, and transitive.

Theorem 3-3 Midpoint Theorem If M is the midpoint of line PQ, then line PM is congruent to line MQ

Theorem 3-4 Bisector Theorem If line PQ is bisected at point M, then line PM is congruent to line MQ

**Chapter 4 Angles and Perpendiculars**

Postulate 4-1 Angle
Measure Postulate For every angle there is a unque positive number between 0
and 180 called

the degree measure of the angle

Postulate 4-2 Protractor
Postulate Given any ray on the edge of a half plane, gfor every positive number
r between 0

and 180 there is exactly one ray in the half plane such that the degree measure
of the angle formed by

the two rays is r.

Postulate 4-3 Angle
Addition Postulate If R is in the exterior of angle PQS, then the
measure of angle PQR + the

measure of angle RQS= the measure of angle PQS

Postulate 4-4 Supplement Postulate If two angles form a linear pair, then they are supplementary angles.

Theorem 4-1 Congruence of angles is reflexive, symmetric, and transitive.

Theorem 4-2 If two angles are supplementary to then same angle, the they are congruent.

Theorem 4-3 If two
angles are supplementary to two congruent angles, then the two angles are congruent
to each

other.

Theorem 4-4 If two angles are complementary to the same angle, then they are congruent to each other.

Theorem 4-5 If two
angles are complementary to two congruent angles, then the two angles are congruent
to each

other.

Theorem 4-6 If two angles are right angles, then the angles are congruent.

Theorem 4-7 If one angle in a linear pair is a right angle, then the other angle is a right angle.

Theorem 4-8 If two angles are congruent and supplementary, then each angle is a right angle.

Theorem 4-9 If two intersecting lines form one right angle, them they form four right angles.

Theorem 4-10 If two angles are vertical, then they are congruent.

Theorem 4-11 If two lines are perpendicular, then they form four right angles.

Theorem 4-12 If a point is on
a line in a given plane, then there is exactly one line in that plane perpendicular
to the

given line at the given point.

Theorem 4-13 Two intersecting lines are perpendicular if and only if they form congruent adjacent angles.

Theorem 4-14 Area of a Triagle
If a triangle has an area of *A* square units, a base of *B* units
and a corresponding

altitude of *h* units, then *A = 1/2bh*.

**Chapter 5 Paralleles**

Postulate 5-1 Parallel
Postulate If there is a line and a point not on a line, then there is
exactly one line through the

point that is parallel to the given line.

Theorem 5-1 If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

Theorem 5-2 If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Theorem 5-3 If two
parallel lines are cut by a transversal, then each pair of consecutive interior
angles is

supplementary.

Theorem 5-4 If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

Theorem 5-5 In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Theorem 5-6 In a
plane, if two lines are cut by a transversal so that a pair of alternate interior
angles is congruent,

then the two lines are parallel.

Theorem 5-7 In a
plane, if two lines are cut by a transversal so that a pair of corresponding
angles is congruent, then

the two lines are parallel.

Theorem 5-8 In a
plane, if two lines are cut by a transversal so that a pair of consecutive interior
angles is

supplementary, then the lines are parallel.

Theorem 5-9 In a
plane, if two lines are cut by a transversal so that a pair of alternate exterior
angles is congruent,

then the lines are parallel.

Theorem 5-10 In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.

Theorem 5-11 Two lines have the same slope if and only if they are parallel and nonvertical.

Theorem 5-12 Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

**Chapter 6 Triangles**

Postulate 6-1 SSS
If each side of one triangle is congruent to the corresponding side of another
triangle, then the

triangles are congruent.

Postulate 6-2 SAS
If two sides and the included angle of one triangle are congruent to the corresponding
sides and

included angle of another triangle, then the triangles are congruent.

Postulate 6-3 ASA
If two angles and the inclusded side of one triangle are congruent to the corresponding
angles and

included side of another triangle, then the triangles are congruent.

Theorem 6-1 Angle Sum Theorem The sum of the degree measures of the angles of a triangle is 180.

Theorem 6-2 If a triangle is a right triangle, then the acute angles are complementary.

Theorem 6-3 If a triangle is equiangular, then the degree measure of each angle is 60.

Theorem 6-4 Exterior
Angle Theorem If an angle is an exterior angle of a triagle, then its measure
is equal to the sum

of the measures of the two remote interior angles.

Theorem 6-5 Inequality
Theorem For any numbers a and b, a > b if and only if there is a positive
number c such that

a = b + c.

Theorem 6-6 If an
angle is an exterior angle of a triangle, then its measure is greater that the
measure of either remote

interior angle.

Theorem 6-7 Congruence of triangles is reflexive, symmetric, and transitive.

Theorem 6-8 AAS If
two angles and a nonincluded side of one triangle are congruent to the corresponding
angles

and nonincluded side of another triangle, then the triangles are congruent.

**Chapter 7 More About Triangles**

Postulate 7-1 HL
If the hypotenuse and a leg of one right triangle are congruent to the corresponding
sides of another

right triangle, then the triangles are congruent.

Theorem 7-1 Isosceles
Triangle Theorem If two sides of a triangle are congruent, then the angle
opposite those sides

are congruent.

Theorem 7-2 A triangle is equilateral if and only if it is equiangular.

Theorem 7-3 Each angle of an equilateral triangle has a degree measure of 60.

Theorem 7-4 If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Theorem 7-5 HA If
the hypotenuse and an acute angle of one right triangle are congruent to
the corresponding

hypotenuse and acute angle of another right triangle, then the triangles are
congruent.

Theorem 7-6 LL If
the legs of one right triangle are congruent to the corresponding legs to another
right triangle, then

the triangles are congruent.

Theorem 7-7 LA If
one leg and an acute angle of one right triangle are congruent to the corresponding
leg and acute

angle of another right triangle, then the triangles are congruent.

Theorem 7-8 If the
measures of two sides of a triangle are unequal, then the measures of the angles
opposite those

sides are unequal in the same order.

Theorem 7-9 If the
measure of two angles of a triangle are unequal, then the measures of the sides
opposite those

angles are unequal in the same order.

Theorem 7-10 A segment is the
shortest segment from a point to a line if and only if it is the segment perpendicular
to

the line.

Theorem 7-11 A segment is the
shortest segment from a point to a plane if and only if it is a segment perpendicular
to

the plane.

Theorem 7-12 Triangle Inequality
The sum of the measures of any two sides of another triangle and the measure
of the

included angles are unequal, then the measures of the third side are unequal
in the same order.

Theorem 7-13 Hidge Theorem
If two sides of one triangle are congruent to two sides of another triangle
and the

measures of the included angles are unequal, then the measures of the third
sides are uequal in the same

order.

Theorem 7-14 Converse of the
Hinge Theorem If two sides of one triangle are congruent to two sides
of another

triangle and the measures of the third sides are unequal, then the measures
of the angles included

between the pairs of congruent sides are unequal in the same order.

**Chapter 8 Polygons**

Theorem 8-1 If a
convex polygon has *n* sides, and *S* is the sum of the degree measures
of its angles,

then* S* = (*n* - 2)180.

Theorem 8-2 If a
polygon is convex, then the sum of the degree measures of the exterior angles,
one at each vertex,

is 360.

Theorem 8-3 If a quadrilateral is a parallelogram, then a diagonal separates it into two congruent triangles.

Theorem 8-4 If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Theorem 8-5 If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Theorem 8-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Theorem 8-7 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 8-8 If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.

Theorem 8-9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 8-10 If a quadrilateral is a rectangle, then its diagonals are congruent.

Theorem 8-11 If a quadrilateral is a rhombus, then each diagonal bisects a pair of opposite angles.

Theorem 8-12 If a quadrilateral is a rhombus, then its diagonals are perpendicular.

Theorem 8-13 If a trapezoid is isosceles, then each pair of base angles is congruent.

Theorem 8-14 If a trapezoid is isosceles, then its diagonal are congruent.

Theorem 8-15 If a quadrilateral
is a trapezoid, then the median is parallel to then bases, and its measure is
one-half the

sum of the measures of the bases.

Theorem 8-16 If a segment is
an apothem of a regular polygon, then it is perpendicular to a side of the polygon
at the

point of tangency with the inscribed circle.

**Chapter 9 Similarity**

Postulate 9-1 AA
Similarity If two angles of one triangle are congruent to two corresponding
angles of another

triangle, then the triangles are similar.

Theorem 9-1 Equality
of Cross Products For any numbers a and c, and any nonzero numbers b and d,
a/b = c/d if

and only if ad=bc

Theorem 9-2 Addition and Subtraction Properties of Proportions

a/b = c/d if and only if a+b/b = c+d/d

a/b = c/d if and only if a-b/b =c-d/d

Theorem 9-3 Summation Property of Proportions a/b = c/d if and only if a/b = a+c/b+d or c/d a+c/b+d

Theorem 9-4 SSS Similarity
If there is a correspondence between the two triangles so that the measures
of their

corresponding sides are proportional, then the two triangles are similar.

Theorem 9-5 SAS Similarity
If the measures of two sides of a triangle are proportional to the measures
of two

corresponding sides of another triangle, and the included angles are congruent,
then the triangles are

similar

Theorem 9-6 If a
line is parallel to one side of a triangle and intersects the other two sides,
then it separates the sides

into segments of proportional lengths.

Theorem 9-7 If a
line intersects two sides of a triangle, and separates the sides into segments
of proportional lengths,

then the line is parallel to the third side.

Theorem 9-8 If a
segment has as its endpoints the midpoints of two sides of a triangle, then
it is parallel to the third

side and its length is one-half the length of the third side.

Theorem 9-9 If three parallel lines intersect two transversals, then they divide the transversal proportionally.

Theorem 9-10 If three parallel
lines cut off congruent segments on one transversal, then they cut off congruent

segments on any transversal.

Theorem 9-11 If two triangles
are similar, then the measures of corresponding perimeters are proportional
to the

measures of corresponding sides.

Theorem 9-12 If two triangles
are similar, then the measures of corresponding altitudes are proportional to
the

measures of corresponding sides.

Theorem 9-13 If two triangles
are similar, then the measures of corresponding angle bisectors of the triangles
are

proportional to the measures of corresponding sides.

Theorem 9-14 If two triangles
are similar, then the measures of corresponding medians are proportional to
the

measures of corresponding sides.

Theorem 9-15 If a dilation with
center *C* and a scale factor *k* maps *A* onto *E* and*
B* onto *D*, then *ED* = *k*(*AB*)

**Chapter 10 Right Triangles**

Theorem 10-1 If the altitude
is drawn from the vertex of the right angle to the hypotenuse of a right triangle,
then the

two triangles formed are similar to the given triangle and to each other.

Theorem 10-2 The measure
of the altitude drawn from the right angle to the hypotenuse of a right triangle
is the

geometric mean between the measures of the two segments of the hypotenuse.

Theorem 10-3 If the altitude
is drawn to the hypotenuse of a right triangle, then the measure of a leg of
the triangle is

the geometric mean between the measure of the hypotenuse and the measure of
the segment of the

hypotenuse adjacent to that leg.

Thorem 10-4
The Pythagorean Theorem If a triangle is a right triangle, then the sum of the
squares of the measures

of the legs equals the sqaure of the measure the hypotenuse.

Theorem 10-5 Converse of
the Pythagorean Theorem If the sum of the squares of the measures of two sides
of a

triangle equals the squaare of the measure of the longest side, then the triangle
is a right triangle.

Theorem 10-6 45-45-90 Theorem
In a 45-45-90 triangle the measure of the hypotenuse is the square root of 2
times

the measure of a leg.

Theorem 10-7 30-60-90 Theorem
In a 30-60-90 triangle the measure of the hypotenuse is 2 time the measure
of the

shorter leg and the measure of the longer leg is the square root of three
times the measure of the

shorter leg.

**Chapter 11 Cirlces**

Postulate 11-1 Arc Addition Postulate
If Q is a point on arc PQR, then the measure of arc PQ + the measure of arc

QR = the measure of arc PQR.

Theorem 11-1 All radii of a circle are congruent.

Theorem 11-2 In a circle of in
congruent circles, two central angles are congruent if and only if their minor
arcs are

congruent.

Theorem 11-3 In a circle or in
congruent circles, two minor arcs are congruent if and only if their corresponding
chords

are congruent.

Theorem 11-4 In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

Theorem 11-5 In a circle or in
congruent circles, two chords are congruent if and only if they are equidistant
from the

center.

Theorem 11-6 If an angle is inscribed
in a circle, then the measure of the angle equals one-half the measure of its

intercepted arc.

Theorem 11-7 If two inscribed
angles of a circle or congruent circles intercept congruent arcs, then the angles
are

congruent.

Theorem 11-8 If an angle is inscribed in a semicircle, then the angle is a right angle.

Theorem 11-9 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Theorem 11-10 In a plane, if a line is perpendicular
to a radius of a circle at its endpoint on the circle, then the line is
a

tangent.

Theorem 11-11 If two segments from the same exterior point are tangent to a circle, then they are congruent.

Theorem 11-12 If two secants intersect in
the interior of a circle, then the measures of an angle formed is one-half the

sum of the measures of the arcs intercepted by the angle and its vertical angle.

Theorem 11-13 If two secants intersect in
the exterior of a circle, then the measure of an angle formed is one-half the

positive difference of the measures of the intercepted arcs.

Theorem 11-14 If a secant and a tangent intersect
at the point of tangency, then the measure of each angle formed is

one-half the measure of its intercepted arc.

Theorem 11-15 If a secant and a tangent,
or two tangents, intersect in the exterior of a circle, then the measure of
the

angle formed is one-half the positive difference of the measures of the intercepted
arcs.

Theorem 11-16 If two chords intersect in
a circle, then the product of the measures of the segments of one chord equals

the product of the measures of the segments of the other chord.

Theorem 11-17 If two secant segments are
drawn to a circle from an exterior point, then the product of the measures of

one secant segment and its external secant segment equals the product of the
measures of the other

secant segments and its external secant segment.

Theorem 11-18 If a tangent segment and a
secant segment are drawn to a circle from an exterior point, then the square

of the measure of the tangent segment equals the product of the measures
of the secant segment and its

external secant segment.

Theorem 11-19 General Equation of a Circle
The equation of a circle with center at (*h, k*) and radius measuring *r*
units

is (*x - h*)^{2} + (*y - k*)^{2 }= *r*^{2}

Theorem 11-20 Circumference of a Circle If
a circle has an area of *A* square units and a radius of *r* units,

then C = 2 pi (r*)*.

Theorem 11-21 Area of a Circle If a circle
has an area of *A* square units and a radius of *r* units, then
*A* = pi(*r*)^{2}.

**Chapter 12 Area and Volume**

Postulate 12-1 Volume Postulate
For any solid region and a given unit of measure, there is a unique positive
number

called the measure of the volume of the region.

Postulate 12-2 If two solid regions are congruent, then they have equal volumes.

Postulate 12-3 Volume Addition
Postulate If a solid region is separated into nonoverlapping regions, then the
sum of

the volumes of these equals the volume of the given region.

Postulate 12-4 If a right prism
has a volume of *V* cubic units, a base with an area of *B* square
units, and a height of *h*

units, then *V* = *Bh*.

Postulate 12-5 Cavalieri's Principle
If two solids have the same cross-sectional area at every level, and the same

height, then they have the same volumes.

Theorem 12-1 Lateral Area of
a Right Prism If a right prism has a lateral area of *L* square units,
a heights of *h* units,

and each base has a perimeter of *p* units, then* L *=* ph*.

Theorem 12-2 Total Surface Area
of a Right Prism If the total surface area of a right prism is *T* square
units, each base

has an area of* B* square units, a perimeter of *p* units, and a height
of *h* units, then *T = ph* + 2*b*.

Theorem 12-3 Lateral Area of
a Right Cylinder If a right cylinder has a lateral area of* L* square units,
a height of *h*

*
*units, and the bases have radii of* r* units, then* L* = 2pi(*r*)(*h*).

Theorem 12-4 Total Surface Area
of a Right Cylinder If a right cylinder has a total surface area of *T*
square units, a

height of *h *units, and the bases have radii of* r* units, then *T
*= 2pi(*r*)(*h*) + 2pi(*r*)^{2}.

Theorem 12-5 Lateral Area for
a Regular Pyramid If a regular pyramid has a lateral area of *L* square
units, a slant

height of *l* units, and its base has a perimeter of *p* units, then*
L* = 1/2*pl*.

Theorem 12-6 Lateral and Total
Surface Area of a Right Circular Cone If a right circular cone has a lateral
area of *L*

square units, a total surface area of *T* square units, a slant height
of *l* units, and the radius of the base is

*
r* units, then *L* = pi(*r*)(*l*) + pi*r*)^{2}.

Theorem 12-7 Surface Area of
a Sphere If a sphere has a surface area of *A* square units and a radius
of *r* units, then

*
A* = 4pi(*r*)^{2}.

Theorem 12-8 Volume of a Right
Cylinder If a right pyramid has a volume of *V* cubic units, a height of
*h* units, and the

area of the base is *B* square units, then *V* = pi(*r*)^{2}(*h*).

Theorem 12-9 Volume of a Right
Pyramid If a right pyramid has a volume of *V* cubic units, a height of*
h *units, and the

area of the base is *B* square units, then *V* = 1/3*Bh*.

Theorem 12-10 Volume of a Right Circular
Cone If a right circular cone has a volume of *V* cubic units, a height
of *h*

*
*units, and the area of the base is *B* square units, then* V*
= 1/3*Bh*.

Theorem 12-11 Volume of a Sphere If a sphere
has a volume of* V *cubic units and a radius of *r* units, then

*
V *= 4/3pi(*r*)^{2}.

Theorem 12-12 Given two points *A*(*x*_{1},*y*_{1},*z*_{1})
and *B*(*x*_{2},*y*_{2},*z*_{2})
in space, the distance between *A* and *B* is given by the

following equation. *AB* - the square root of (*x*_{2}-*x*_{1})^{2}+(*y*_{2}-*y*_{1})^{2}+(*z*_{2}+*z*_{1})^{2}.

**Chapter 13 Loci**

Postulate 13-1 In a given rotation,
if *A* is the preimage, *P* is the image, and *W* is the center
of rotation, then the

measure of the angle of rotation, angle *AWP*, equals twice the measure
of the angle between the

intersecting lines of reflection.

Adapted from: http://phs.mesa.k12.co.us/math/quickfacts/postulatestheorems.html