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TOPIC 1
DEFINITIONS OF GEOMETRY ITEMS
For most of our study of geometry we will use deductive reasoning and study geometry as an
“axiomatic system”. This means that we start with the simplest of ideas, and we use what we
have learned as the basis for new ideas.
We begin with the simplest of the geometrical objects, trying to define our terms, establish
relationships between objects and terminology, and decide whether statements are true or false.
During this process, we will write definitions, postulates, and theorems. We will need to learn
some of the basics of symbolic logic to build up this body of information which we call
“geometry”.
Try your hand at writing a geometric definition. A good definition does not contain any words
that have not been previously defined.
A point is ….
A line is ….
It is impossible to write definitions for these geometry items, if we assume that we know no
geometry in advance. We have to list these two as being undefined terms. Why are undefined
terms necessary in an axiomatic system? Because we have nothing to fall back on, and the
method of an axiomatic system is to use only what we know to move forward. When we begin,
we have “nothing to fall back on”.
We need to begin by leaving these three objects as undefined terms: point, line, and plane.
These will be the only undefined terms we will ever encounter in our study of geometry.
We know what they look like, and we can agree on a way to name them:

Points use a single capital letter to name them.

Lines are named either with
o A single lower case letter
o Two capital letters for points which lie along the line
o (occasionally, we will use three capital letters, using three points along
the line, as long as the three points are listed in order as they appear on
the line)
E
D
a
B
C
Line a

BC
DEF
Planes are named by
o One capital letter, written near the “corner” of the drawing of the plane
o Several capital letters which specify the surface on which those points
lie
T
R
S
P
Plane P or Plane RST
Beginning Definitions
Defn: Space is the set of all points.
Note: this means that space is too “big” to fit on a line or even on a plane. It is a
three dimensional object, and it is the largest geometrical object.
F
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Defn: Collinear points are points that all lie on one line.
Defn: Coplanar points are _________________________________________.
Defn: The intersection of two figures is the set of points which is __________.
Defn: A line segment is a part of a line with two endpoints.
The label for a segment is AB
Compare this to the label for a line: AB
Defn: A ray is a part of line with one endpoint.
The label for a ray is AB
A ray can have different names and yet be the same ray.
Example:
AB = AC = AD
Defn: Opposite rays are two collinear rays with the same endpoint but no other points
in common.
Example:
AB and AC are opposite rays.
Defn: Parallel lines are coplanar lines which have no points in common. (This means
that they do not intersect.)
The name for parallel lines is “||”
So AB || CD
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Find three pairs of parallel lines in this drawing.
Defn: Skew lines are noncoplanar lines. (This implies that they do not intersect as well,
but the lack of intersection is not listed in the definition.)
Find two pairs of skew lines in this drawing.
Pause for a minute: There are facts about these geometry items that are true but are not
included in the definitions. A definition is designed to identify an object and
make it clear how to distinguish it from other objects. A definition is not designed
to contain all of the true facts about an object.
If you try to state a definition of an apple, it is not good enough to call it a red
fruit, as there are other red fruits which are not called apples. But you do not have
to write pages and pages of descriptions of all of the biology truths that are known
about apples as part of the definition.
It is also important to notice that the definitions provided here are being given in a
particular order. When a previous word is needed in a definition, then that word
has to be listed earlier in the definition list.
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Defn: Parallel planes are planes which do not intersect (or share any points in
common).
Find two pairs of parallel planes. Remember how to label planes (one single
letter, if appropriate, or three noncollinear points otherwise. Four noncollinear
points are okay as well.)
Defn: The length of a line segment is a positive number which gives a relative size of
the segment, so it can be compared with other line segments.
The symbol for length is the endpoints of the segment without a bar above them.
Above:
and Z)
LZ is a line segment, and LZ is its length (or the distance between L
Defn: Line segments are congruent if they have the same length.
Note: Congruent has the symbol
same size and the same shape.
Example:
which means that the two items have the
AB = CD, so AB = CD
Defn: B is the midpoint of a line segment AC if A, B, and C are collinear and
AB = BC
(This also means that AB = BC)
An alternate version of the definition of midpoint is that B is the midpoint of AC
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1
if AB = AC, or if BC = AC .
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2
Defn: The bisector of a segment is a point, line, line segment, ray, or plane which
intersects the line segment at its midpoint.
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Defn: An angle is the union of two rays with the same endpoint.
An angle is named with three points, and the middle letter is the common
endpoint. If there is not more than one angle shown with that endpoint, the angle
can be named with a single letter, using the endpoint’s name. An angle is also
named with a single number.
Examples
Defn: The vertex of an angle is the common endpoint of its two rays.
Angles have points which are in their interior and points which are on their exterior.
They also have points which are on the angle itself. Discuss a way to determine
whether a point is on the interior, on the angle, or in the exterior. Try to use only words
which have been defined at this time.
Single angles can be classified by their size:
Defn: An acute angle is an angle whose measure is less than 900.
Defn: A right angle is an angle whose measure is equal to 900.
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Defn: An obtuse angle is an angle whose measure is more than 900.
Note: In geometry, angles cannot be larger than 1800 or less than 00. This will be
defined differently when you study trigonometry.
Defn: A straight angle is an angle whose measure is equal to 1800.
Examples:
Give classifications to each of the angles shown.
Defn: Perpendicular lines are liens which meet at right angles.
Pairs of angles can be classified as well.
Defn: Congruent angles are two angles whose measures are equal.
The name / symbol is <ABC = <MJK
[This means the same thing as m<ABC = m<MJK]
Defn: Adjacent angles are two angles which share a common ray but no interior points.
Defn: Complementary angles are two angles whose measures add up to 900.
Defn: Supplementary angles are two angles whose measures add up to 1800.
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Note: “Complementary” comes before “supplementary” in the alphabet, and 90
comes before 180 in the order of numbers. This is a way to remember which
word matches with which sum.
Note: “Complementary” and “supplementary” angles can only be used with
pairs of angles, not with three or four angles which happen to add up to 90 or 180.
Note: The definitions of complementary and supplementary do not have the
word “adjacent” in them. Draw a pair of complementary angles which are
adjacent and another pair which are NOT adjacent.
Defn: Vertical angles are two angles which are made from two intersecting lines, share
a common vertex, but which are not adjacent.
From this definition, draw a pair of vertical angles. [Note that the word “vertical”
is not the same as you are used to thinking of it. It has a peculiar meaning here,
but the successful definition describes it carefully. “Vertical” here refers to the
fact that the two angles have a common “vertex”.]
Defn: A linear pair are two adjacent angles whose non-common sides form a line.
Draw a linear pair.
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Defn: A perpendicular pair are two adjacent angles whose non-common sides are
perpendicular.
Draw a perpendicular pair.
Defn: The bisector of an angle is a ray, a line segment, a line, or a plane which cuts the
angle its congruent parts. Alternately, a bisector of an angle is a ray, a line
segment, a line, or a plane which cuts an angle so that each of the two pieces is
one-half of the measure of the original angle.
Angles associated with parallel lines (and other lines)
Defn: Interior angles are angles which are between two lines
Defn: Exterior angles are angles which are not between two lines
Defn: A transversal is a line or line segment which cuts across two other lines.
t
5
6
1
3
7
2
a
4
8
b
Defn: Corresponding angles (in this situation) are two non-adjacent angles which are
on the same side of the transversal, but one is an exterior angle and the other is an
interior angle.
Defn: Alternate interior angles are two interior angles which are not on the same side
of the transversal.
Defn: Alternate exterior angles are two exterior angles which are not on the same side
of the transversal.
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Defn: Same-side interior angles are two interior angles which are on the same side of
the transversal.
Defn: Same-side exterior angles are two exterior angles which are on the same side of
the transversal.
Defn: A polygon is a figure which encloses a part of a plane. It has sides which are line
segments. Only two line segments intersect at a time, and each segment is
connected to two other segments.
Defn: An equilateral polygon has all of its sides congruent.
Defn: An equiangular polygon has all angles congruent.
Defn: A regular polygon is a polygon which is equilateral and equiangular.
Defns: A triangle is a polygon with three sides.
A quadrilateral is a polygon with four sides.
A pentagon is a polygon with five sides.
A hexagon (or sexagon) is a polygon with six sides.
A heptagon (or septagon) is a polygon with seven sides.
An octagon is a polygon with eight sides.
A nonagon is a polygon with nine sides.
A decagon is a polygon with ten sides.
A dodecagon is a polygon with twelve sides.
An icosagon is a polygon with twenty sides.
Defns: Triangles are defined by their sides as follows:
A scalene triangle is a triangle whose sides all have different lengths.
An isosceles triangle is a triangle with exactly two congruent sides.
An equilateral triangle is a triangle with all three sides of the same length.
Defns: Triangles are defined by their angle sizes, too.
An acute triangle contains three acute angles.
An obtuse triangle contains one obtuse angle and two acute angles.
A right triangle contains one right angle and two acute angles.
Defn: In a triangle, the side opposite a vertex (or an angle) is the line segment which
does not intersect one of the rays which makes the angle or meets at the
designated vertex.
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Defn: The exterior angle of a polygon is the linear pair for an interior angle.
Defn: The median of a triangle is a line segment which connects a vertex of the triangle
with the midpoint of the opposite side.
Defn: The midline of a triangle is a line segment which connects the midpoints of any
two sides of the triangle.
Defn: An altitude of a triangle is a line segment which connects a vertex with a point on
the opposite side such that the segment is perpendicular to the opposite side, or
the opposite side extended.
Defn: A diagonal of a polygon is a line segment which connects two non-consecutive
vertices.
Defn: Points of concurrence are points where three or more lines intersect.
The incenter of a triangle is the point of concurrence of the three angle bisectors.
The circumcenter of a triangle is the point of concurrence of the three
perpendicular bisectors of the sides of the triangle.
The orthocenter is the point of concurrence of the three altitudes of the triangle.
The centroid of a triangle is the point of concurrence of the three medians of a
triangle.
Defns: Quadrilaterals have special cases.
A parallelogram is a quadrilateral with two pairs of parallel sides.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
A rhombus is a quadrilateral with four congruent sides.
A rhombus is a parallelogram with congruent sides.
A rhombus is an equilateral quadrilateral
A rectangle with a quadrilateral with four right angles
A rectangle is a parallelogram with four right angles
A rectangle is a parallelogram with consecutive sides which are
perpendicular.
A rectangle is an equiangular quadrilateral
A square is a quadrilateral with congruent sides and right angles.
A square is a rectangle with congruent sides.
A square is a parallelogram with congruent sides and right angles.
A square is a regular quadrilateral.
A kite is a quadrilateral with two pairs of consecutive sides which are congruent.
An inscriptable quadrilateral is a quadrilateral whose four vertices lie on a
circle.
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The median of a trapezoid is a line segment which connects the midpoints of the
two non-parallel sides.
The altitude (synonym: height) of a quadrilateral a line segment which connects
a point on one side with a point on the opposite side such that the segment is
perpendicular to the opposite side, or the opposite side extended.
Defn: A circle is the set of all points in a place which are equidistant from one point,
called the center of the circle.
A radius of the circle is a line segment from the center to any point on the circle.
A diameter is a line segment from any point an a circle through the center to
another point on the circle.
The circumference is the length around the circle one time.
A chord is a line segment which has its endpoints on the circle.
A secant is a line which intersects a circle two times.
An arc is a part of the circle, defined by two points on the circle.
A tangent line is a line (or line segment) which hits a circle one time.
A central angle is an angle whose vertex is at the center of the circle.
An inscribed angle is an angle whose vertex is on the circle and whose sides are
chords.
In three dimensions, shapes are called solids. Some solids have both bases and lateral
faces.
Defn: A solid is a surface, or a collection of surfaces, which are not all coplanar and
which enclose a part of space.
Defn: A lateral face is a polygon which is located on the sides (not bases).
Defn: A prism is a solid with two congruent bases which are connected with either
parallelograms or rectangles
A right prism is a prism whose bases are perpendicular to the lateral
faces. In this case, the lateral faces are rectangles.
(Note: Prisms are described by the shape of their bases, such as a triangular prism
or a square prism.)
A cylinder is a prism with circular bases.
A pyramid is a solid with one base and triangular lateral faces.
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The apex of a pyramid is the point where the lateral faces concur.
A right pyramid is one in which the apex is located directly over the
middle of the base.
(Note: Pyramids are described by the shape of their bases, such as square
pyramids or trapezoidal pyramids)
A cone is a pyramid with a circular base.
A sphere is a solid such that all of the points on its surface are equidistant in
space from a single point, called its center.
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From http://www.helpingwithmath.com/by_subject/geometry/geo_shapes3D.htm
Defn: A polyhedron is a solid whose faces are all polygons. (In many cases, the faces
are regular polygons.) Plural form is polyhedra.
A tetrahedron is a polyhedron whose faces are all equilateral triangles. It has 4
such faces.
A cube (or hexahedron) is a polyhedron whose faces are all squares. It has 6
such faces.
An octahedron is a polyhedron whose faces are all equilateral triangles. It has 8
such faces.
An icosahedron is a polyhedron whose faces are all equilateral triangles. It has
20 such faces.
A dodecahedron is a polyhedron whose faces are all regular pentagons. It has 12
such faces.
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From http://images.search.yahoo.com/yhs/search?_adv_prop=image&fr=yhs-Babylon002&va=polyhedral+shapes&hspart=Babylon&hsimp=yhs-002