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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.   (or  If an angle's measure is greater than 0 and less than 90, then it is an Acute Angle.)

 

If an angle is a Right Angle, then its measure is 90.   (or   If an angle measures 90, then it is a Right Angle.)

 

If an angle is an Obtuse Angle, then its measure is greater than 90 and less than 180.   (or   If an angle's measure is greater than 90 and less than 180, then it is an Obtuse Angle.)

 

If an angle is a Straight Angle, then its measure is 180.   (or   If an angle measures  180, then it is a Straight Angle.)

 

If two angles are Congruent Angles, then they have the same measure.   (or  If two angles have the same measure, then they are Congruent Angles.)

 

If two segments are Congruent Segments, then they have the same length.   (or  If two segments have the same measure, then they are Congruent Segments.)

 

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.   (of "If a ray divides an angle into two congruent angles, then it bisects 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.   (of "If two rays divide an angle into three congruent angles, then they trisect 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 ( a straight angle), then they are supplementary.") 

 

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