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Geometry Tech
Postulates and Theorems
Postulates
Chapter 1 Reasoning In Geometry
1-1 Two points determine a unique line. (p. 18)
1-2 If two distinct lines intersect, then their
intersection is a point. (p. 18)
1-3 Three noncollinear points determine a unique
plane. (p. 18)
1-4 If two distinct planes intersect, then their
intersection is a line. (p. 20)
Chapter 2 Segment Measure and Coordinate
Graphing
2-1 Number Line Postulate Each real number
corresponds to exactly one point on a number line.
Each point on a number line corresponds to exactly
one real number. (p. 51)
2-2 Distance Postulate For any two points on a
line and a given unit of measure, there is a unique
positive real number called the measure of the
distance between the points. (p. 52)
2-3 Ruler Postulate The points on a line can be
paired with the real numbers so that the measure of
the distance between corresponding points is the
positive difference of the numbers. (p. 52)
2-4 Completeness Property for Points in the
Plane
Each point in a coordinate plane corresponds to
exactly one ordered pair of real numbers. Each
ordered pair of real numbers corresponds to exactly
one point in a coordinate plane. (p. 68)
Theorems
2-1 Congruence of segments is reflexive. (p. 63)
2-2 Congruence of segments is symmetric. (p. 63)
2-3 Congruence of segments is transitive. (p. 63)
2-4 If a and b are real numbers, a vertical line
contains all points (x, y) such that x = a, and a
horizontal line contains all points (x, y) such that
y = b. (p. 70)
2-5 Midpoint Formulas for a Number Line
On a number line, the coordinate of the midpoint of
a segment whose endpoints have coordinates a and
b is a + b (p. 77)
2
2-6 Midpoint Formulas for a Coordinate Plane
On a coordinate plane, the coordinates of the
midpoint of a segment whose endpoints have
coordinates (x1, y1) and (x2, y2) are
(x1 + x2 , y1 + y2)
(p. 77)
2
2
Chapter 3 Angles
3-1 Angle Measure Postulate For every angle,
there is a unique positive number between 0 and
180 called the degree measure of the angle. (p. 96)
3-1 Vertical Angle Theorem Vertical angles are
congruent. (p. 123)
3-2 Protractor Postulate On a plane, given AB and
a number r between 0 and 180, there is exactly one
ray with endpoint A, extending on each side of AB
such that the degree measure of the angle formed is
r. (p. 97)
3-3 Angle Addition Postulate For any angle PQR,
if A is in the interior of <PQR, then m<PQA +
m<AQR = m<PQR (p. 104)
3-4 Supplement Postulate If two angles form a
linear pair, then they are supplementary. (p. 119)
Chapter 4 Parallels
4-1 Corresponding Angles If two parallel lines are
cut by a transversal, then each pair of
corresponding angles is congruent. (p. 157)
4-2 In a plane, if two lines are cut by a transversal
so that a pair of corresponding angles is congruent,
then the lines are parallel. (p. 162)
4-3 Two nonvertical lines are parallel if and only if
they have the same slope. (p. 170)
4-4 Two nonvertical lines are perpendicular if and
only if the product of their slopes is -1. (p. 170)
3-2 If two angles are congruent, then their
complements are congruent. (p. 123)
3-3 If two angles are congruent, then their
supplements are congruent. (p. 123)
3-4 If two angles are complementary to the same
angle, then they are congruent. (p. 124)
3-5 If two angles are supplementary to the same
angle, then they are congruent. (p. 124)
3-6 If two angles are congruent and supplementary,
then each is a right angle. (p. 125)
3-7 All right angles are congruent. (p. 125)
3-8 If two lines are perpendicular, then they form
right angles. (p. 129)
3-9 If a line m is in a plane and point T is a point on
m, then there exists exactly one line in that plane
that is perpendicular to m at T. (p. 131)
4-1 Alternate Interior Angles If two parallel lines
are cut by a transversal, then each pair of alternate
interior angles is congruent. (p. 150)
4-2 Consecutive Interior Angles If two parallel
lines are cut by a transversal, then each pair of
consecutive interior angles is supplementary. (p.
150)
4-3 Alternate Exterior Angles If two parallel lines
are cut by a transversal, then each pair of alternate
exterior angles is congruent. (p. 150)
4-4 Perpendicular Transversal If a transversal is
perpendicular to one of two parallel lines; it is
perpendicular to the other. (p. 158)
4-5 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. (p. 163)
4-6 In a plane, if two lines are cut by a transversal
so that a pair of alternate exterior angles is
congruent, then the two lines are parallel. (p. 163)
4-7 In a plane, if two lines are cut by a transversal
so that a pair of consecutive interior angles is
supplementary, then the two lines are parallel. (p.
164)
4-8 In a plane, if two lines are perpendicular to the
same line, then the two lines are parallel. (p. 164)
Chapter 5 Triangles and Congruence
5-1 SSS Postulate If three sides of one triangle are
congruent to three corresponding sides of another
triangle, then the triangles are congruent. (p. 211)
5-2 SAS Postulate 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.
5-3 ASA Postulate If two angles and the included
side of one triangle are congruent to the
corresponding angles and included side of another
triangle, then the triangles are congruent. (p. 215)
5-1 Angle Sum Theorem The sum of the measures
of the angles of a triangle is 180. (p. 193)
5-2 The acute angles of a right triangle are
complementary. (p. 195)
5-3 The measure of each angle of an equiangular
triangle is 60. (p. 195)
5-4 AAS Theorem If two angles and a nonincluded
side of one triangle are congruent to the
corresponding two angles and nonincluded side of
another triangle, then the triangles are congruent.
(p. 216)
Chapter 6 More About Triangles
6-1 HL Postulate If the hypotenuse and a leg of
one right triangle are congruent to the hypotenuse
and corresponding leg of another right triangle,
then the triangles are congruent. (p. 252)
6-1 The length of the segment from the vertex to
the centroid is twice the length of the segment from
the centroid to the midpoint. (p. 230)
6-2 Isosceles Triangle Theorem If two sides of a
triangle are congruent, then the angles opposite
those sides are congruent. (p. 247)
6-3 The median from the vertex angle of an
isosceles triangle lies on the perpendicular bisector
of the base and the angle bisector of the vertex
angle. (p. 247)
6-4 Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent. (p. 248)
6-5 A triangle is equilateral if and only if it is
equiangular. (p. 249)
6-6 LL Theorem If two legs of one right triangle
are congruent to the corresponding legs of another
right triangle, then the triangles are congruent. (p.
249)
6-7 HA Theorem If the hypotenuse and an acute
angle of one right triangle are congruent to the
hypotenuse and corresponding angle of another
right triangle, then the triangles are congruent. (p.
252)
6-8 LA Theorem If one leg and an acute angle of
one right triangle are congruent to the
corresponding leg and angle of another right
triangle, then the triangles are congruent. (p. 252)
6-9 Pythagorean Theorem In a right triangle, the
square of the length of the hypotenuse c is equal to
the sum of the squares of the lengths of the legs a
and b. (p. 256)
6-10 Converse of the Pythagorean Theorem If c
is the measure of the longest side of a triangle, a
and b are the lengths of the other two sides, and
c^2 = a^2 + b^2, then the triangle is a right
triangle. (p. 258)
6-11 Distance Formula If d is the measure of the
distance between two points with coordinates
(x1, y1), and (x2, y2) then
d=
Chapter 7 Triangle Inequalities
7-1 Comparison Property For any two real
numbers a and b, exactly one of the following
statements is true: a < b< a = b, or a > b
7-1 If point C is between points A and B, and A, C
and B are collinear, then AB > AC and AB > AC.
(p. 277)
7-2 If EP is between ED and EF, then m<DEF is
greater than m<DEP and m<DEF is greater and
m<PEF
7-3 Exterior Angle Theorem The measure of an
exterior angle of a triangle is equal to the sum of
the measures of its two remote interior angles. (p.
283)
7-4 Exterior Angle Inequality Theorem
The measure of an exterior angles of a triangle is
greater than the measure of either of its two remote
interior angles. (p. 285)
7-5 If a triangle has one right angle, then the other
two angles must be acute. (p. 285)
7-6 If the measures of three sides of a triangle are
unequal, then the measure of the angles opposite
those sides are unequal in the same order. (p. 291)
7-7 If the measures of three angles of a triangle are
unequal, then the measures of the sides opposite
those angles are unequal in the same order. (p. 291)
7-8 In a right triangle, the hypotenuse is the side
with the greatest measure. (p. 291)
7-9 Triangle Inequality Theorem
The sum of the measures of any two sides of a
triangle is greater than the measure of the third side.
(p. 296)
Chapter 8 Quadrilaterals
8-1 The sum of the measures of the angles of a
quadrilateral is 360. (p. 312)
8-2 Opposite angles in a parallelogram are
congruent. (p. 317)
8-3 Opposite sides of a parallelogram are
congruent. (p. 317)
8-4 The consecutive angles of a parallelogram are
supplementary. (p. 317)
8-5 The diagonals of a parallelogram bisect each
other. (p. 318)
8-6 A diagonal of a parallelogram separates it into
two congruent triangles. (p. 319)
8-7 If both pairs of opposite sides of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram. (p. 323)
8-8 If one pair of opposite sides of a quadrilateral is
parallel and congruent, then the quadrilateral is a
parallelogram. (p. 324)
8-9 If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram. (p.
324)
8-10 The diagonals of a rectangle are congruent. (p.
329)
8-11 The diagonals of a rhombus are perpendicular.
(p. 329)
8-12 Each diagonal of a rhombus bisects a pair of
opposite angles. (p. 329)
8-13 The median of a trapezoid is parallel to the
bases, and the length of the median equals one-half
the sum of the lengths of the bases. (p. 334)
8-14 Each pair of base angles in an isosceles
trapezoid is congruent. (p. 334)
Chapter 9 Proportions and Similarity
9-1 AA Similarity If two angles of one triangle are
congruent to corresponding angles of another
triangle, then the triangles are similar. (p. 361)
9-1 Property of Proportions For any numbers a
and c and any nonzero numbers b and d, if a/b =
c/d, then ad = bc. (p. 351)
9-2 SSS Similarity If the measures of the sides of a
triangle are proportional to the measures of the
corresponding sides of another triangle, then the
triangles are similar. (p. 363)
9-3 SAS Similarity If the measures of two sides of
a triangle are proportional to the measures of two
corresponding sides of another triangle and their
included angles are congruent, then the triangles are
similar. (p. 363)
9-4 If a line is parallel to one side of a triangle and
intersects the other two sides, then the triangle
formed is similar to the original triangle. (p. 368)
9-5 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. (p.
370)
9-6 If a line intersects two sides of a triangle and
separates the sides into corresponding segments of
proportional lengths, then the line is parallel to the
third side. (p. 374)
9-7 If a segment joins the midpoints of two sides of
a triangle, then it is parallel to the third side and its
measure equals one-half the measure of the third
side. (p. 375)
9-8 If three or more parallel lines intersect two
transversals, they divide the transversal
proportionally. (p. 383)
9-9 If three or more parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal. (p. 384)
9-10 If two triangles are similar, then the measures
of the corresponding perimeters and proportional to
the measures of the corresponding sides. (p. 388)
Chapter 10 Polygons and Area
10-1 Area Postulate For any polygon and a given
unit of measure, there is a unique number A called
the measure of the area of the polygon. (p. 413)
10-2 Congruent polygons have equal area. (p. 413)
10-3 Area Addition Postulate The area of a given
polygon equals the sum of the areas of the nonoverlapping polygons that form the given polygon.
(p. 413)
Chapter 11 Circles
11-1 Arc Addition Postulate The sum of the
measures of two adjacent arcs is the measure of the
arc formed by the adjacent arcs. (p. 463)
10-1 If a convex polygon has n sides, then the sum
of the measures of its interior angles is (n – 2)180.
p. (409)
10-2 In any convex polygon, the sum of the
measures of the exterior angles, one at each vertex,
is 360. (p. 410)
10-3 Area of a Triangle If a triangle has an area of
A square units, a base of b units, and a
corresponding altitude of h units, then A = ½bh.
(p. 419)
10-4 Area of a Trapezoid If a trapezoid has an
area of A square units, bases of b1, and b2 units,
and an altitude of h units, then A = ½h(b1 + b2). (p.
421)
10-5 Area of a Regular Polygon If a regular
polygon has an area of A square units, an apothem
of a units, and a perimeter of P units, then
A = ½aP (p. 426)
11-1 All radii of a circle are congruent. (p. 455)
11-2 The measure of the diameter d of a circle is
twice the measure of the radius r of the circle.
(p. 455)
11-3 In a circle or in congruent circles, two minor
arcs are congruent if and only if their corresponding
central angles are congruent. (p. 464)
11-4 In a circle or in congruent circles, two minor
arcs are congruent if and only if their corresponding
chords are congruent. (p. 468)
11-5 In a circle, a diameter bisects a chord and its
arc if and only if it is perpendicular to the chord.
(p. 469)
11-6 In a circle or in congruent circles, two chords
are congruent if and only if they are equidistant
from the center. (p. 475)
11-7 Circumference of a Circle If a circle has a
circumference of C units and a radius of r units,
then C = 2πr or C = πd (p. 479)
11-8 Area of a Circle If a circle has an area of A
square units and a radius of r units, then A = πr^2
(p. 483)
11-9 Area of a Sector of a Circle If a sector of a
circle has an area of A square units, a central angle
measurement of N degrees, and a radius of r units,
then A = N/360 (πr^2)
Chapter 12 Surface Area and Volume
12-1 Lateral Area of a Prism If a prism has a
lateral area of L square units and a height of h units
and each base has a perimeter of P units, then
L = Ph (p. 505)
12-2 Surface Area of a Prism If a prism has a
surface area of S square units and a height of h
units and each base has a perimeter of P units and
an area of B square units, then S = Ph + 2B (p. 505)
12-3 Lateral Area of a Cylinder If a cylinder has
a lateral area of L square units and a height of h
units and the bases have radii of r units, then
L = 2πrh. (p. 507)
12-4 Surface Area of a Cylinder If a cylinder has
a surface area of S square units and a height of h
units and the bases have radii of r units, then
S = 2πrh +2πr^2. (p. 507)
12-5 Volume of a Prism If a 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. (p. 511)
12-6 Volume of a Cylinder If a cylinder has a
volume of V cubic units, a radius of r units, and a
height of h units, then V = πr^2h. (p. 512)
12-7 Lateral Area of a Regular Pyramid If a
regular pyramid has a lateral area of L square units,
a base with a perimeter of P units, and a slant
height of l units, then L = ½Pl
12-8 Surface Area of a Regular Pyramid If a
regular pyramid has a total surface area of S square
units, a slant height of l units, and a base with
perimeter of P units and an area of B square units,
then S = ½Pl + B. (p. 517)
12-9 Lateral Area of a Cone If a cone has a lateral
area of L square units, a slant height of l units, and
a base with a radius of r units, then L = πrl (p. 519(
12-10 Surface Area of a Cone If a cone has a
surface area of S square units, a slant height of l
units, and a base with a radius of r units, then
S = πrl + πr^2. (p. 519)
12-11 Volume of a Pyramid If a pyramid has a
volume of V cubic units, and a height of h units,
and the area of the base if B square units, then
V = 1/3Bh. (p. 523)
12-12 Volume of a Cone If a cone has a volume of
V cubic units, a radius of r units, and a height of h
units, then V = 1/3πr^2h. (p. 523)
12-13 Surface Area of a Sphere If a sphere has a
surface area of S square units and a radius of r
units, then S = 4πr^2. (p. 529)
12-14 Volume of a Sphere If a sphere has a
volume of V cubic units and a radius of r units, then
V = 4/3πr^3. (p. 529)
12-15 If two solids and similar with a scale factor
of a:b, then the surface areas have a ratio of
a^2:b^2 and the volumes have a ratio of a^3:b^3.
(p. 536)
Chapter 13 Right Triangle and Trigonometry
13-1 45°-45°-90° Triangle Theorem In a 45-45-90
triangle, the hypotenuse is √2 times as long as a leg.
(p. 555)
13-2 30°-60°-90° Triangle Theorem In a 30-60-90
triangle, the hypotenuse is twice as long as the
shorter leg, and the longer leg is √3 as long as the
shorter leg. (p. 560)
13-3 If x is the measure of an acute angle of a right
triangle, then sin x/cos x = tan x. (p. 574)
13-4 If x is the measure of an acute angle of a right
triangle, then sin^2 + cos^2x = 1. (p. 577)
Chapter 14 Circle Relationships
14-1 The degree measure of an inscribed angle
equals one-half the degree measure of its
intercepted arc. (p. 587)
14-2 If inscribed angles intercept the same arc or
congruent arcs, then the angles are congruent.
(p. 588)
14-3 If an inscribed angle of a circle intercepts a
semicircle, then the angle is a right angle. (p. 589)
14-4 In a plane, if a line is tangent to a circle, then
it is perpendicular to the radius drawn to the point
of tangency. (p. 592)
14-5 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. (p. 592)
14-6 If two segments from the same exterior point
are tangent to a circle, then they are congruent.
(p. 594)
14-7 A line or line segment is a secant to a circle if
and only if it intersects the circle in two points.
(p. 600)
14-8 If a secant angle has its vertex inside a circle,
then its degree measure is one-half the sum of the
degree measures of the arcs intercepted by the
angle and its vertical angle. (p. 601)
14-9 If a secant angle has its vertex outside a circle,
then its degree measure is one-half the difference of
the degree measures of the intercepted arcs.(p. 601)
14-10 If a secant-tangent angle has its vertex
outside the circle, then its degree measure is onehalf the difference of the degree measures of the
intercepted arcs. (p. 606)
14-11 If a secant-tangent angle has its vertex on the
circle, then its degree measure is one-half the
degree measure of the intercepted arc. (p. 606)
14-12 The degree measure of a tangent-tangent
angle is one-half the difference of the degree
measures of the intercepted arcs. (p. 607)
14-13 If two chords of a circle intersect, 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. (p. 612)
14-14 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 segment and its external secant
segment. (p. 613)
14-15 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. (p. 614)
14-16 General Equation of a Circle The equation
of a circle with center at (h, k) and a radius of r
units is (x – h)^2 + (y – k)^2 = r^2. (p. 618)