Undefined Terms, Postulates, Properties, Definitions and Theorems
Undefined Terms
Point
Line
Plane
A geometric point has no length, width, or thickness; it
merely represents a place or position, usually named with a
capital letter.
A line is an infinite set of points that extends endlessly in
both directions. To name a line we usually use two capital
letters that name two points on the line or we can name
the line using a single lower case letter
A plane is a set of points that forms a completely flat
surface extending indefinitely in all directions. A plane may
be named by using letters that name 3 points in the plane,
provided that they are not on the same line.
Point P
Line AB
AB
Plane ABC
Properties 1) Reflexive Property of Equality – A quantity is equal to itself. a  a
2) Symmetric Property of Equality – An equality may be expressed in either order. If a  b , then b  a
3) Transitive Property of Equality – If quantities are equal to the same quantity, they are equal to each other. If a  b and b  c , then a  c
4) Distributive Property - a(b  c)  ab  ac
Note: These properties also apply to congruency of line segments and angles.
Postulates – A postulate is a statement whose truth is accepted without proof.
1) Segment Addition Postulate, Angle Addition Postulate, Partition Postulate –
The whole is equal to the sum of its parts.
2) Substitution Postulate or property– A quantity may be substituted for its equal in any expression.
3) Addition Postulate or property – If equal quantities are added to equal quantities. If a  b, and c  d , then a  c  b  d .
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If congruent segments are added to congruent segments, the sums are congruent.
If congruent angles are added to congruent angles, the sums are congruent.
4) Subtraction Postulate or property – If equal quantities are subtracted from equal quantities, the differences are equal.
If a  b, and c  d , then a  c  b  d .
5) Multiplication Postulate or property - If equal quantities are multiplied by equal quantities, the products are equal.
If a  b, and then ca  cd .
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Doubles of equal quantities are equal.
6) Division Postulate or Property - If equal quantities are divided by equal quantities, the quotients are equal.
7) A line segment can be extended to any length in either direction.
8) Through two given points, one and only one line can be drawn or two points determine a line.
9) A line segment has one and only one midpoint.
10) An angle has one and only one angle bisector.
11) The sum of the degree measures of all the angles on one side of a given line whose common vertex is a given point on the line is 180.
12) The sum of the degree measures of all the angles about a given point is 360.
12) SAS Postulate – Two triangles are congruent if two sides and the included angle in one triangle are congruent to the two corresponding sides and
included angle in the other triangle.
13) ASA Postulate – Two triangles are congruent if two angles and the included side in one triangle are congruent to two corresponding angles and the
included side in the other triangle.
14) SSS Postulate – Two triangles are congruent if three sides in one triangle are congruent to three sides in the other triangle.
15) Two lines each parallel to same line are parallel to each other.
Definitions – A definition is a statement of the precise meaning of a term. A definition must be reversible, in other words, if expressed as a
conditional, the converse of the condition is also true in a definition. For example, the definition of congruent line segments can be
expressed as “If two line segments are congruent, then they are equal in measure” or “If two lines segments are equal in measure, then they
are congruent”. Every definition may be expressed as a biconditional statement.
Acute Angle
An acute angle is an angle whose degree measure is greater than 0 and less than
90.
Acute Triangle
Adjacent Angles
An acute triangle is a triangle that has three acute angles.
Adjacent angles are two angles in the same plane that have a common vertex and
a common side but do not have any interior points in common.
Altitude of a triangle
An altitude of a triangle is a line segment drawn from any vertex of the triangle,
perpendicular to and ending in the line that contains the opposite side.
Angle
An angle is the set of points that is the union of two rays having the same
endpoint.
1
Angle Bisector
A bisector of an angle is a ray whose endpoint is the vertex of the angle, and that
divides the angle into two congruent angles.
Bisector of a line
segment.
The bisector of a line segment is any line or subset of a line that intersects the
segment at its midpoint
Collinear Set of Points
A collinear set of points is a set of points all of which lie on the same
straight line.
Complementary
angles
Complementary angles are two angles the sum of whose degree measures is 90.
Do not have to be adjacent angles.
Congruent Angles
Congruent angles are angles that have the same measure.
Coplanar Set of points
A coplanar set of points is a set of points all of which lie on the same plane.
Corresponding Angles
Corresponding angles are a pair of angles on the same side of the transversal, not
sharing a common vertex, and one is interior and one is exterior.
Deductive reasoning
Distance from a point
to a line
Deductive reasoning is the process by which a person makes conclusions based on
previously known facts
The distance from a point to a line is the length of the perpendicular from the
point to the line.
Equiangular triangle
An equiangular triangle is a triangle that has 3 congruent angles.
Equilateral Triangle
Inductive reasoning
Isosceles Triangle
Line Segment
Linear Pair
An equilateral triangle is a triangle that has three congruent sides.
Inductive reasoning is the process of arriving at a conclusion based on a set of
observations
An isosceles triangle is a triangle that has two congruent sides.
A line segment is a set of points consisting of two points on a line called endpoints
and all the points on the line between the endpoints
A linear pair of angles are two adjacent angles whose sum is a straight angle.
Median of a triangle
A median of a triangle is a line segment that joins any vertex of the triangle to the
midpoint of the opposite side.
Midpoint
The midpoint of a segment is the point of that line segment that divides the
segment into two congruent segments.
Non Collinear Set of
Points
A non-collinear set of points is a set of 3 or more points that do not all lie on the
same straight line.
Obtuse Angle
An obtuse angle is an angle whose degree measure is greater than 90 and less
than 180.
AB
Obtuse triangle
Opposite Rays
An obtuse triangle is a triangle that has an obtuse angle.
Opposite rays are two rays of the same line with a common endpoint and no other
point in common.
Orthocenter of a
Triangle
The orthocenter of a triangle is the intersection of the 3 altitudes of a triangle.
Parallel Lines
Perpendicular
bisector
Parallel lines are coplanar lines that do not intersect or coplanar lines are a parallel
if and only if they have no points in common or if the lines coincide and, therefore,
have all points in common.
The perpendicular bisector of a line segment is a line, a line segment or ray that is
perpendicular to the line segment and bisects the line segment.
Perpendicular Lines
Perpendicular lines are two lines that intersect to form right angles.
Polygon
A polygon is a closed plane figure made up of several line segments that are joined
together.
Ray
A ray is part of a line that consists of a point on the line, called an endpoint, and all
the points on one side of the endpoint.
Right Angle
A right angle is an angle whose degree measure is 90o.
Right triangle
A right triangle is a triangle that has a right angle.
Scalene Triangle
Skew Lines
A scalene triangle is a triangle that has no congruent sides.
Skew lines are non-coplanar lines that are neither parallel nor intersecting
Straight Angle
A straight angle is an angle whose degree measure is 180o.
Supplementary angles
Supplementary angles are two angles the sum of whose degree measure is 180.
Do not have to be adjacent angles.
Transversal
A transversal is a line that intersects two or more coplanar lines in different points
Triangle
Vertical angles
A triangle is a polygon that has exactly 3 sides
Vertical angles are the opposite angles formed when two lines intersect.
Theorems – A theorem is a statement that is proved by deductive reasoning. Theorems can be stated as conditional statements, where the
converse of the conditionally stated theorem is not necessarily true.
1) If two angles are right angles, then they are congruent or all right angles are congruent.
2) If two angles are straight angles, then they are congruent.
3) If two angles are vertical angles, then they are congruent.
4) If two angles are complements of the same angle then they are congruent.
5) If two angles are congruent, then their complements are congruent.
6) If two angles are supplements of the same angle, then they are congruent.
7) If two angles are congruent, then their supplements are congruent.
8) If two angles form a linear pair then they are supplementary.
9) If two sides of a triangle are congruent, then the angles opposite those sides are congruent or the base angles of an isosceles triangle are
congruent.
a. The bisector of the vertex angle of an isosceles triangle bisects the base.
b. The bisector of the vertex angle of an isosceles triangle is perpendicular to the base.
c. Every equilateral triangle is equiangular.
10) The sum of the lengths of two sides of a triangle is greater than the third side.
11) The measure of an exterior angle of a triangle is greater than the measure of either non-adjacent interior angle.
12) If the lengths of two sides of a triangle are unequal, the measures of the angles opposite these sides are unequal and the greater angle lies
opposite the greater side and the smallest angle lies opposite the shortest side.
13) If the measures of two angles of a triangle are unequal, the lengths of the sides opposite those angles are unequal and the greater side lies
opposite the greater angle, and the shortest side lies opposite the smallest angle.
14) If two intersecting lines form congruent adjacent angles, the lines are perpendicular.
15) If two points are each equidistant from the endpoints of a line segment, the points determine the perpendicular bisector of the line segment.
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Any point on the perpendicular bisector is equidistant from the endpoints of the line segment.
16) If coplanar lines are not parallel lines, then they are intersecting lines.
17) Through a given point not on a given line, there exists one and only one line parallel to the given line.
18) If a line intersects one of two parallel lines, it intersects the other.
Skills
1) Naming Points , Lines and Planes.
2) Determining Coplanar points
3) Determining the intersection of two figures.
4) Know symbols in geometry. http://www.mathsisfun.com/geometry/symbols.html
5) Construct a copy of a line segment. Construct line segments that are counting number multiples of the size of a given
line segment.
6) Solve algebraic examples involving midpoints and line segment bisectors.
7) Naming angles.
8) Measuring angles using a protractor.
9) Classify angles according to their measure.
10) Construct a copy of an angle and construct angles that are counting number multiples of the size of a given angle.
11) Students will be able to construct an angle bisector.
12) Solve algebraic examples involving angle bisectors.
13) Construct angles 1.5, 2.5, 3.5, etc, times bigger than a given angle.
14) Solve algebraic examples involving vertical, complementary and supplementary angles.
15) Solve algebraic examples involving perpendicular lines.
16) Construct a perpendicular bisector.
17) Divide a line segment into an even number of equal segments via construction.
18) Construct a 45 degree angle.
19) Classify triangles according to their sides, i.e. scalene, isosceles, and equilateral.
20) Construct an equilateral triangle, an isosceles triangle and a scalene triangle.
21) Determine if a triangle is equilateral via construction.
22) Construct 30 degree angles.
23) Solve algebraic examples involving equilateral and isosceles triangles.
24) Classify triangles according to their angles, i.e. acute, obtuse, equiangular, right, obtuse.
25) Construct a line perpendicular to a given line through a point not on the line.
26) Construct an altitude of a triangle.
27) Locate the orthocenter of a triangle via construction.
28) Know the different connectives in logic and their associated truth tables.
29) Determine the truth value of statements.
30) State the negation of a statement.
31) Know the various forms of conditional statements.
32) Determine if a statement is a valid argument (tautology) or an invalid argument.
33) Determine if statements are logically equivalent.
34) Know various laws of reasoning such as the Law of Detachment (Law of Modus Ponens), Law of Modus Tollens, Law
of Syllogism (Chain Rule), Law of Disjuntive Inference, DeMorgan’s Law, etc.
35) Proofs in logic.
36) State counterexample, or identify a counterexample.
37) Identify examples of inductive reasoning and examples of deductive reasoning.
38) Prove triangles congruent by SAS, ASA, SSS, AAS and HL.
39) Construct a copy of a triangle.
40) Solve algebraic examples involving the median of a triangle.
41) Construct a median of a triangle.
42) Locate the centroid of a triangle via construction.
43) Know the properties of the centroid of a triangle.
44) Locate the circumcenter of a triangle via construction.
45) Know the properties of the circumcenter of a triangle.
46) Know the properties of the circumcircle of a triangle.
47) Locate the incenter of a triangle via construction.
48) Know the properties of the incenter of a triangle.
49) Know the properties of the incircle of a triangle.
50) Prove corresponding parts of congruent triangles are congruent.
51) Solve algebraic examples involving congruent triangles.
52) Solve algebraic examples involving isosceles triangles and equilateral triangles.
53) Prove overlapping triangles congruent.
54) Use two pairs of congruent triangles.
55) Examples involving the possible lengths of the sides of a triangle.
56) Solve examples relating the sides of a triangle and the angles opposite those sides.
57) Prove lines are perpendicular.
58) Construct Parallel Lines
Common errors
1) Students confusing the definitions of perpendicular lines and perpendicular bisector.