Postulates
Section 1.3
1. Through two distinct points, there is exactly one line.
2. (Ruler Postulate) The measure of any line segment is a
unique positive number.
3. (Segment-Addition Postulate) If X is a point on and
A-X-B,then .
4. If two lines intersect, they intersect at a point.
5. Through three noncollinear points, there is exactly one
plane.
6. If two distinct planes intersect, then their intersection is
a line.
7. Given two distinct points in a plane, the line containing
these points also lies in the plane.
Section 1.4
8. (Protractor Postulate) The measure of an angle is a unique
positive number.
9. (Angle-Addition Postulate) If a point D lies in the interior
of angle ABC, then m∠ABD + m∠DBC = m∠ABC.
Section 2.1
10. (Parallel Postulate) Through a point not on a line, exactly
one line is parallel to the given line.
11. If two parallel lines are cut by a transversal, then the pairs
of corresponding angles are congruent.
Section 3.1
12. If the three sides of one triangle are congruent to the three
sides of a second triangle, then the triangles are congruent
(SSS).
13. If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a second
triangle, then the triangles are congruent (SAS).
14. If two angles and the included side of one triangle are
congruent to two angles and the included side of a second
triangle, then the triangles are congruent (ASA).
Section 5.2
15. If the three angles of one triangle are congruent to the
three angles of a second triangle, then the triangles are
similar (AAA).
Section 6.1
16. (Central Angle Postulate) In a circle, the degree measure
of a central angle is equal to the degree measure of its
intercepted arc.
17. (Arc-Addition Postulate) If and intersect only at
point B, then .
Section 8.1
18. (Area Postulate) Corresponding to every bounded region is a
unique positive number A, known as the area of that region.
19. If two closed plane figures are congruent, then their areas are
equal.
20. (Area-Addition Postulate) Let R and S be two enclosed
regions that do not overlap. Then .
21. The area A of a rectangle whose base has length b and whose
altitude has length h is given by .
Section 8.4
22. The ratio of the circumference of a circle to the length of its
diameter is a unique positive constant.
Section 8.5
23. The ratio of the degree measure m of the arc (or central angle) of
a sector to 360° is the same as the ratio of the area of the sector
to the area of the circle.
Section 9.1
24. (Volume Postulate) Corresponding to every solid is a unique
positive number V known as the volume of that solid.
25. The volume of a right rectangular prism is given by
where measures the length, w the width, and h the altitude
of the prism.
26. The volume of a right prism is given by
where B is the area of a base and h is the length of the
altitude of the prism
Definitions
acute angle an angle whose measure is between 0° and 90°
acute triangle a triangle whose three interior angles are all
acute
adjacent angles two angles that have a common vertex and a
common side between them
altitude of cone (pyramid) the line segment from the vertex
(apex) of the cone perpendicular to the plane of the base
altitude of cylinder (prism) a line segment between and
perpendicular to each of the two bases
altitude of parallelogram a line segment drawn perpendicularly
from one side to a nonadjacent side (or extension of
that side)
altitude of trapezoid a line segment drawn perpendicularly
from a vertex to the remaining parallel side
altitude of triangle a line segment drawn perpendicularly from
a vertex of the triangle to the opposite side of the triangle; the
length of the altitude is the height of the triangle
angle the plane figure formed by two rays that share a common
endpoint
angle bisector see bisector of angle
angle of depression (elevation) acute angle formed by a
horizontal ray and a ray determined by a downward (an
upward) rotation
apex of pyramid (cone) same as vertex of pyramid (cone)
apothem of regular polygon any line segment drawn from
the center of the regular polygon perpendicular to one of
its sides
arc the segment (part) of a circle determined by two points on
the circle and all points between them
area the measurement, in square units, of the amount of region
within an enclosed plane figure
auxiliary line a line (or part of a line) added to a drawing to
help complete a proof or solve a problem
base a side (of a plane figure) or face (of a solid figure) to
which an altitude is drawn
base angles of isosceles triangle the two congruent angles
of the isosceles triangle
base of isosceles triangle the side of the triangle whose
length is unique; the side opposite the vertex
bases of trapezoid the two parallel sides of the trapezoid
bisector of angle a ray that separates the given angle into two
smaller, congruent angles
Cartesian plane the two-dimensional coordinate system
determined by x and y axes
Cartesion space the three-dimensional coordinate system
determined by x, y, and z axescenter of circle the interior point of the circle whose distance
from all points on the circle is the same
center of regular polygon the common center of the
inscribed and circumscribed circles of the regular polygon
center of sphere the interior point of the sphere whose
distance from all points on the sphere is the same
central angle of circle an angle whose vertex is at the center
of the circle and whose sides are radii of the circle
central angle of regular polygon an angle whose vertex is at
the center of the regular polygon and whose sides are two
consecutive radii of the polygon
centroid of triangle the point of concurrence for the three
medians of the triangle
chord of circle any line segment that joins two points on
the circle
circle the set of points in a plane that are at a fixed distance
from a given point (the center of the circle) in the plane
circumcenter of triangle the center of the circumscribed circle
of a triangle; the point of concurrence for the perpendicular
bisectors of the three sides of the triangle
circumference the linear measure of the distance around a circle
circumscribed circle a circle that contains all vertices of a
polygon whose sides are chords of the circle
circumscribed polygon a polygon whose sides are all tangent
to a circle in the interior of the polygon
collinear points points that lie on the same line
common tangent a line (or segment) that is tangent to more
than one circle; can be a common external tangent or a
common internal tangent
complementary angles two angles whose sum of measures
is 90°
concave polygon a polygon in which at least one diagonal lies
in the exterior of the polygon
concentric circles (spheres) two or more circles (spheres)
that have the same center
conclusion the “then” clause of an “If, then” statement;
the part of a theorem indicating the claim to be proved
concurrent lines (planes) three or more lines (planes) that
contain the same point
congruent refers to figures (such as angles) that can be made
to coincide
converse relative to the statement “If P, then Q,” this
statement has the form “If Q, then P”
convex polygon a polygon in which all diagonals lie in the
interior of the polygon
coplanar points points that lie in the same plane
corollary a theorem that follows from another theorem as a
“by-product”; a theorem that is easily proved as the
consequence of another theorem
cube a right square prism whose edges are congruent
cyclic polygon a polygon that can be inscribed in a circle
cylinder (circular) the solid generated by all line segments
parallel to the axis of the cylinder and that contain corresponding
endpoints on the two congruent circular bases
decagon a polygon with exactly 10 sides
deduction a form of reasoning in which specific conclusions
are reached through the use of established principles
degree the unit of measure that corresponds to of a
complete revolution; used with angles and arcs
diagonal of polygon a line segment that joins two nonconsecutive
vertices of a polygon
diameter any line segment that joins two points on a circle
(or sphere) and that also contains the center of the circle (or
sphere)
dodecagon a polygon that has exactly 12 sides
dodecahedron a polyhedron that has exactly 12 faces
dodecahedron (regular) a polyhedron that has exactly
12 faces that are congruent regular pentagons
edge of polyhedron any line segment determined by the
intersection of two faces of the polyhedron (includes prisms
and pyramids)
equiangular polygon a type of polygon whose angles are
congruent (equal)
equilateral polygon a type of polygon whose sides are
congruent (equal)
equivalent equations equations for which the solutions are
the same
extended proportion a proportion that has three or more
members, such as
extended ratio a ratio that compares three or more numbers,
such as a:b:c
exterior refers to all points that lie outside an enclosed (bounded)
plane or solid figure
exterior angle of polygon an angle formed by one side of the
polygon and an extension of a second side that has a common
endpoint with the first side
extremes of a proportion the first and last terms of a
proportion; in , a and d are the extremes
face of polyhedron any one of the polygons that lies in a
plane determined by the vertices of the polyhedron; includes
base(s) and lateral faces of prisms and pyramids
geometric mean the repeated second and third terms of
certain proportions; in , b is the geometric mean of a
and c
height the length of the altitude of a geometric figure
heptagon a polygon that has exactly seven sides
heptahedron a polyhedron that has exactly seven faces
hexagon a polygon that has exactly six sides
hexahedron a polyhedron that has exactly six faces
hexahedron (regular) a polyhedron that has six congruent
square faces; also called a cube
hypotenuse of right triangle the side of a right triangle that
lies opposite the right angle
hypothesis the “if” clause of an “If, then” statement; the part
of a theorem providing the given information
icosahedron (regular) a polyhedron with 20 congruent faces
that are equilateral triangles
incenter of triangle the center of the inscribed circle of a
triangle; the point of concurrence for the three bisectors of
the angles of the triangle
induction a form of reasoning in which a number of specific
observations are used to draw a general conclusion
inscribed angle of circle an angle whose vertex is on a circle
and whose sides are chords of the circle
inscribed circle a circle that lies inside a polygon in such a
way that the sides of the polygon are tangents of the circle
inscribed polygon a polygon whose vertices all lie on a circle
in such a way that the sides of the polygon are chords of
the circle
intercepted arc the arc (an arc) of a circle that is cut off in the
interior of an angle (or related angle)
intercepts the points at which the graph of an equation intersects
the axes
interior refers to all points that lie inside an enclosed (bounded)
plane or solid figure
interior angle of polygon any angle formed by two consecutive
sides of the polygon in such a way that the angle lies in
the interior of the polygon
intersection the elements that two sets have in common; the
points that two geometric figures share
intuition drawing a conclusion through insight
inverse relative to the statement “If P, then Q,” this statement
has the form “If not P, then not Q”
isosceles trapezoid a trapezoid that has two congruent legs
(its nonparallel sides)
isosceles triangle a triangle that has two congruent sides
kite a quadrilateral that has two distinct pairs of congruent
adjacent sides
lateral area the sum of areas of the lateral faces of a solid or
the area of the curved lateral surface, excluding the base
area(s) (as in prisms, pyramids, cylinders, and cones)
legs of isosceles triangle the two congruent sides of the
triangle
legs of right triangle the two sides that form the right angle
of the triangle
legs of trapezoid the two nonparallel sides of the trapezoid
lemma a theorem that is introduced and proved so that a later
theorem can be proved
line of centers the line (or line segment) that joins the centers
of two coplanar circles
line segment the part of a line determined by two points and
all points on the line that lie between those two points
locus the set of all points that satisfy a given condition or
conditions
major arc an arc whose measure is between 180° and 360°
mean proportional see geometric mean
means of a proportion the second and third terms of a
proportion; in , b and c are the means
median of trapezoid the line segment that joins the
midpoints of the two legs (nonparallel sides) of the
trapezoid
median of triangle the line segment joining a vertex of the
triangle to the midpoint of the opposite side
midpoint the point on a line segment (or arc) that separates the
line segment (arc) into two congruent parts
minor arc an arc whose measure is between 0° and 180°
nonagon a polygon that has exactly nine sides
noncollinear points three or more points that do not lie on the
same line
noncoplanar points four or more points that do not lie in the
same plane
obtuse angle an angle whose measure is between 90°
and 180°
obtuse triangle a triangle that has exactly one interior angle
that is obtuse
octagon a polygon that has exactly eight sides
octahedron a polyhedron that has exactly eight faces
octahedron (regular) a polyhedron with eight congruent faces
that are equilateral triangles
opposite rays two rays having a common endpoint and that
together form a line
orthocenter of triangle the point of concurrence for the three
altitudes of the triangle
parallel lines (planes) two lines in a plane (or two planes) that
do not intersect
parallelogram a quadrilateral that has two pairs of parallel
sides
parallelepiped a right rectangular prism; a box
pentagon a polygon that has exactly five sides
pentahedron a polyhedron that has exactly five faces
perimeter of polygon the sum of the lengths of the sides of
the polygon
perpendicular bisector of line segment a line (or part of a
line) that is both perpendicular to a given line segment and
bisects that line segment
perpendicular lines two lines that intersect to form congruent
adjacent angles
point of tangency (contact) the point at which a tangent to a
circle touches the circle
polygon a plane figure whose sides are line segments that
intersect only at their endpoints
polyhedron a solid figure whose faces are polygons that intersect
other faces along common sides of the polygons
postulate a statement that is assumed to be true
quadrilateral a polygon that has exactly four sides
radian the measure of a central angle of a circle whose
intercepted arc has a length equal to the radius of the circle
radius any line segment that joins the center of a circle (or
sphere) to a point on the circle (or sphere)
radius of regular polygon any line segment that joins the
center of the polygon to one of its vertices.
ratio a comparison between two quantities a and b, generally
written or a:b
ray the part of a line that begins at a point and extends infinitely
far in one direction
rectangle a parallelogram that contains a right angle
reflex angle an angle whose measure is between 180°
and 360°
regular polygon a polygon whose sides are congruent and
whose interior angles are congruent
regular polyhedron a polyhedron whose edges are congruent
and whose faces are congruent
regular prism a right prism whose bases are regular
polygons
regular pyramid a pyramid whose base is a regular polygon
and whose lateral faces are congruent isosceles triangles
rhombus a parallelogram with two congruent adjacent sides
right angle an angle whose measure is exactly 90°
right circular cone a cone in which the line segment joining
the vertex of the cone to the center of the circular base is
perpendicular to the base
right circular cylinder a cylinder in which the line segment
joining the centers of the circular bases is perpendicular to
the plane of each base
right prism a prism in which lateral edges are perpendicular to
the base edges that they intersect
right trapezoid a trapezoid that contains a right angle
right triangle a triangle in which exactly one interior angle is
a right angle
scalene triangle a triangle in which no two sides are
congruent
sector of circle the plane region bounded by two radii of the
circle and the arc that is intercepted by the central angle
formed by those radii
segment of circle the plane region bounded by a chord and a
minor arc (major arc) that has the same endpoints as that
chord
semicircle the arc of a circle determined by a diameter; an arc
of a circle whose measure is exactly 180°
set any collection of objects, numbers, or points
similar polygons polygons that have the same shape
sine in a right triangle, the ratio
skew quadrilateral a quadrilateral whose sides do not all lie
in one plane
slant height of cone any line segment joining the vertex
(apex) of the cone to a point on the circular base
slant height of regular pyramid a line segment joining the
vertex (apex) of the pyramid to the midpoint of a base edge
of the pyramid
slope a measure of the steepness of a line; in the rectangular
coordinate system, the slope m of the line through
and is where
sphere the set of points in space that are at a fixed distance
from a given point (the center of the sphere)
straight angle an angle whose measure is exactly 180°; an
angle whose sides are opposite rays
straightedge an idealized instrument used to construct parts
of lines
supplementary angles two angles whose sum of measures
is 180°
surface area the measure of the total area of a solid; the sum
of the lateral area and base area in many solid figures.
symmetry with respect to a line ( ) figure for which every
point A on the figure has a second point B on the figure for
which is the perpendicular bisector of
symmetry with respect to a plane (R) solid figure for which
every point A on the figure has a second point B on the figure
for which plane R is the perpendicular bisector of .
symmetry with respect to a point (P) figure for which every
point A on the figure has a second point C on the figure for
which P is the midpoint of .
tangent in a right triangle, the ratio
tangent circles two circles that have one point in common;
the circles may be externally tangent or internally tangent
tangent line of circle a line that touches a circle at only one
point
tetrahedron a polyhedron that has exactly four faces
tetrahedron (regular) a four-faced solid in which the faces
are congruent equilateral triangles
theorem a statement that follows logically from previous
definitions and principles; a statement that can be proved
torus a three-dimensional solid that has a “doughnut” shape
transversal a line that intersects two or more lines, intersecting
each at one point
trapezoid a quadrilateral having exactly two parallel sides
triangle a polygon that has exactly three sides
triangle inequality a statement that the sum of the lengths of
two sides of a triangle cannot be greater than the length of
the third side
union the joining together of any two sets, such as geometric
figures
valid argument an argument in which the conclusion follows
logically from previously stated (and accepted) premises or
assumptions
vertex angle of isosceles triangle the angle formed by the
two congruent sides of the triangle
vertex of angle the point at which the two sides of the angle
meet
vertex of isosceles triangle the point at which the two congruent
sides of the triangle meet
vertex of polygon any point at which two sides of the polygon
meet
vertex of polyhedron any point at which three (or more)
edges of the polyhedron meet
vertical angles a pair of angles that lie in opposite positions
when formed by two intersecting lines
volume the measurement, in cubic units, of the amount of
space within a bounded region of space