Postulates and Theorems
Connecting Reasoning and Proof

1.Postulate Through any two points there is exactly one line. 2.Postulate Through any three points not on the same line (noncollinear points) there is exactly one plane. 3.Postulate A line contains at least two points. 4.Postulate A plane contains at least three noncollinear points. 5.Postulate If two points lie in a plane, then the entire line containing those two points lies in that plane. 6.Postulate If two planes intersect then their intersection is a line. 7.Theorem Congruence of segments is reflexive, symmetric, and transitive. 8.Theorem Supplement Theorem If two angles form a linear pair, then they are supplementary angles. 9.Theorem Congruence of angles is reflexive, symmetric, and transitive. 10.Theorem Angles supplementary to the same angle or to congruent angles are congruent. 11.Theorem Angles complementary to the same angle or to congruent angles are congruent. 12.Theorem All right angles are congruent. (All rt. are ) 13.Theorem Vertical angles are congruent. (Vert. are ) 14.Theorem Perpendicular lines intersect to form four right angles. (lines form 4 rt. )

1.Postulate Corresponding Angles Postulate If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. 2.Theorem Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. 3.Theorem Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. 4.Theorem Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. 5.Theorem Perpendicular Transversal Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. 6.Postulate Two nonvertical lines have the same slope if and only if they are parallel. 7.Postulate Two nonvertical lines are perpendicular if and only if the product of their slopes is –1. 8.Postulate If two lines in a plane are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. 9.Postulate Parallel PostulateIf there is a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. 10.Theorem If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel. 11.Theorem If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles are supplementary, then the two lines are parallel. 12.Theorem If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel. 13.Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel.

1.Theorem Angle Sum Theorem The sum of the measures of the angles of a triangel is 180.

2.Theorem Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

3.Theorem Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

4.Corollary The acute angles of a right triangle are complementary

5.Corollary There can be at most one right or obtuse angle in a triangle.

6.Theorem Congruence of triangles is reflexive, symmetric, and transitive.

7.Postulate SSS Postulate If the sides of one triangle are congruent to the sides of a second triangle then the triangles are congruent.

8.Postulate SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.

9.Postulate ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

10.Theorem AAS If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle, the two triangles are congruent.

11.Theorem Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite those sides are congruent.

12.Theorem If two angles of a triangle are congruent then the sides opposite those angles are congruent. (2 of a are , the sides opp. the are .)

13.Corollary A triangle is equilateral if and only if it is equiangular. (An equilateral/equiangular is equiangular/equilateral.)

14.Corollary Each angle of an equilateral triangle measures 60°.

1.Theorem A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. (pt. on the bisector of seg. is equidistant from the endpt. of seg.)

2.Theorem A point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. (a pt. equidistant from the endpts of seg. lies on the bisector of seg.)

3.Theorem A point on the bisector of an angle is equidistant from the sides of the angle. (a pt. on bisector of is equidistant from the sides of )

4.Theorem A point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. (a pt. on or in the int. of an and equidistant from sides an is on bisector of )

5.Theorem Leg-Leg If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. (LL)

6.Theorem Hypotenuse-Angle If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent. (HA)

7.Theorem Leg-Angle If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. (LA)

8.Postulate Hypotenuse-Leg If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. (HL)

9.Theorem Exterior Angle Inequality Theorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles.

10.Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. (in if 2 & lg side opp lg )

11.Theorem If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angel is longer than the side opposite the lesser angle. (in if & lg opp lg side)

12.Theorem The perpendicular segment from a point to a line is the shortest segment from the point to the line. (the seg. From pt to line is shortest seg. From the pt. to the line).

13.Corollary The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. (the seg. From pt to plane is shortest seg. from the pt. to the plane).

14.Theorem Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

15.Theorem SAS Inequality Theorem (Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the included angle in one triangle is greater than the included angle in the other, then the third side of the first triangle is longer than the third side in the second triangle.

16.Theorem SSS Inequality If two sides of one triangle are congruent to two sides of another triangle, and third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle.