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.

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

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

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 = r2

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 + 2b.

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 = 1/2pl.

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) + pir)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 = 1/3Bh.

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 = 1/3Bh.

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(x1,y1,z1) and B(x2,y2,z2) in space, the distance between A and B is given by the
following equation. AB - the square root of (x2-x1)2+(y2-y1)2+(z2+z1)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.