Below is a list of the Definitions, Postulates, Theorems and Mathematical Properties that we have covered so far in Advanced Geometry. These are the justifications (reasons) that can be used in proofs.
POSTULATES:
THEOREMS: If
two angles are right angles, then they are congruent. If
two angles are straight angles, then they are congruent.
If
a conditional statement is true, then the contrapositive of the statement
is also true. (If p, then q If ~q, then ~p.) If
angles are supplementary to the same angle or to congruent angles, then
they are congruent. (Congruent Supplement Theorem) If
angles are complementary to the same angle or to congruent angles, then
they are congruent. (Congruent Complement Theorem)
DEFINITIONS: If
an angle is an Acute Angle, then its measure is greater than 0°
and less than 90°.
If
an angle is a Right Angle, then its measure is 90°.
If
an angle is an Obtuse Angle, then its measure is greater than 90°
and less than 180°. If
an angle is a Straight Angle, then its measure is 180°. If
two angles are Congruent Angles, then they have the same measure. If
two segments are Congruent Segments, then they have the same
length.
If
a point (segment, ray, or line) Bisects a segment, then it divides
the segment into two congruent segments.
The dividing point is called the Midpoint. (or
"If a point divides a segment it into two congruent segments
then it is the Midpoint of the segment." or
"If a point, segment, ray or line divides a segment into congruent
segments then it bisects the segment.") If
two points (segments, rays, or lines) Trisect a segment, then they
divide the segment into three congruent segments.
The points are called Trisection points.
(or "If two points divides a segment it into three
congruent segments then they are "Trisection points."
or "If two points, segments, rays or lines divide a segment
into three congruent segments then they trisect the segment.") If
a ray Bisects an angle, then it divides the angle into two
congruent angles. The dividing ray is called the Bisector of the angle. If
two rays Trisect an angle, then they divide the angle into three
congruent angles. The two
dividing rays are called Trisectors of the angle.
If
lines (rays, or segments) are Perpendicular then they intersect at
right angles. (or "If lines (rays, or segments) intersect at
right angles, then they are perpendicular.") If
two angles are Complementary Angles, then their sum is 90°
(a right angle). Each of the
two angles is called the Complement of the other. (or
"If two angles sum to 90°
(a right angle), then they are complementary.") If
two angles are Supplementary Angles, then their sum is 180°
(a straight angle). Each of
the two angles is called the Supplement of the other. (or
"If two angles sum to 180°
(
ASSUMPTIONS: Givens Straight
Lines and Straight Angles Collinearity
of points Betweenness
of points Relative
positions of points
Do not assume:
Right Angles
Congruent Segments
Congruent Angles
Relative sizes of segments and angles Mathematical
Operations / Properties: Addition Subtraction Multiplication Division Substitution Transitive Property
