A postulate is a statement about a geometrical world that is accepted without justification and makes the frame
work for the geometrical world. A theorem is a conclusion that follows logically from the postulates using rational
argument and does require justification.
Theorem #1 – If two angles are right angles, then they are congruent.
Theorem #2 – If two angles are straight angles, then they are congruent.
Theorem #3 – If a conditional statement is true, then the contrapositive of the statement is also true. (If p, then q If ~q, then ~p.)
Theorem #4 -If angles are supplementary to the same angle then they are congruent.
Theorem #5 -If angles are supplementary to congruent angles, then they are congruent.
Theorem #6 -If angles are complementary to the same angle then they are congruent.
Theorem #7 -If angles are complementary to congruent angles, then they are congruent.
Theorem #8 -If a segment is added to two congruent segments, the sums are congruent.(Addition Property)
Theorem #9 -If an angle is added to two congruent angles, the sums are congruent.(Addition Property)
Theorem #10 -If congruent segments are added to congruent segments, the sums are congruent.(Addition Property)
Theorem #11 -If congruent angles are added to congruent angles, then the sums are congruent.(Addition Property)
Theorem #12 – If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent.(Subtraction Property)
Theorem #13 -If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.(Subtraction property)
Theorem #14 -If segments (or angles) are congruent, their like multiples are congruent. (Multiplication property)
Theorem #15 -If segments (or angles) are congruent, their like divisions are congruent. (Division property)
Theorem 16: If angles (or segments) are congruent to the same angle (or segment), then they are congruent to each other. (Transitive Property)
Theorem 17: If angles (or segments) are congruent to congruent angles (or segments), then they are congruent to each other. (Transitive Property)
Theorem 18: Vertical Angles are congruent.
Theorem 19: All radii of a circle are congruent.
Theorem 20: If two sides of a triangle are congruent, then the angles opposite the sides are congruent.
Theorem 21: If two angles of a triangle are congruent, then the sides opposite the angles are congruent.
Theorem 23: If two angles are both supplementary and congruent, then they are right angles.
Theorem 24: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of the segment.
Theorem 25: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
Theorem 26: If two nonvertical lines are parallel, then their slopes are equal
Theorem 27: If the slopes of two nonvertical lines are equal, then the lines are parallel.
Theorem 28: If two lines are perpendicular and neither is vertical, then each line’s slope is the opposite reciprocal of the others.
Theorem 29: If a line’s slope is the opposite reciprocal of another line’s slope, then the two lines are perpendicular.
Theorem 30: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
Theorem 31: If two lines are cut by a transversal such that two alternate interior angles are congruent, then the lines are parallel.
Theorem 32: If two lines are cut by a transversal such that two alternate exterior angles are congruent, then the lines are parallel.
Theorem 33: If two lines are cut by a transversal such that two corresponding angles are congruent, then the lines are parallel.
Theorem 34: If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, then the lines are parallel.
Theorem 35: If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, then the lines are parallel.
Theorem 36: If the two coplanar lines are perpendicular to the third line, then they are parallel.
Theorem 37: If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.
Theorem 38: If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.
Theorem 39: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles are congruent.
Theorem 40: If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.
Theorem 41: If two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are congruent.
Theorem 42: If two parallel lines are cut by a transversal, then each pair of exterior angles on the same side of the transversal are congruent.
Theorem 43: In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Theorem 44: If two lines are parallel to a third line, then, they are parallel to each other, (Transitive Property of Parallel Lines)
Theorem 45: A line and a point not on a line determine a plane.
Theorem 46: Two intersecting lines determine a plane.
Theorem 47: Two parallel lines determine a plane.
Theorem 48: If a line is perpendicular to two distinct lines, that lie in a plane and that pass through its foot, then it is perpendicular to the plane.
Theorem 49: If a plane intersects two parallel planes, then the lines of intersection are parallel.
Theorem 50: The sum of the measures of the three angles of a triangle is 180.
Theorem 51: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
Theorem 52: A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side.
Theorem 53: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No Choice Theorem)
Theorem 54: If there exists a correspondence between the vertices of two triangles such that two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. (AAS – Angle Angle Side)
Theorem 55: The sum S of the measures of the angles of a polygon with n sides is given by the formula S = (n-2)180
Theorem 56: If one exterior angle is taken at each vertex, the sum S of the measures of the exterior angles of a polygon, S = 360°.
