Theorems and Properties List


	Theorem/Properties Sheet for Proofs

This is a partial listing of the more popular theorems, postulates and properties
needed when working with Euclidean proofs. You need to have a thorough understanding of these items.

Your textbook (and your teacher) may want you to remember these theorems with slightly different wording.
Be sure to follow the directions from your teacher.

The "I need to know, now!" entries are highlighted in blue.

Properties:

Reflexive Property	A quantity is congruent (equal) to itself. a = a
Symmetric Property	If a = b, then b = a.
Transitive Property	If a = b and b = c, then a = c.

Postulates: (assumed to be true)

Addition Postulate	If equal quantities are added to equal quantities, the sums are equal.
Subtraction Postulate	If equal quantities are subtracted from equal quantities, the differences are equal.
Multiplication Postulate	If equal quantities are multiplied by equal quantities, the products are equal. (also Doubles of equal quantities are equal.)
Division Postulate	If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)
Substitution Postulate	A quantity may be substituted for its equal in any expression.
Parallel Postulate	If there is a line and a point not on the line, then there exists one line through the point parallel to the given line.
Corresponding Angles Postulate	If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding Angles Converse Postulate	If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
Side-Side-Side (SSS) Congruence Postulate	If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS) Congruence Postulate	If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA) Congruence Postulate	If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Angle (AA) Similarity Postulate	If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Theorems: (can be proven true)

Right Angles	All right angles are congruent.
Congruent Supplements	If two angles are supplementary to the same angle (or to congruent angles) then the two angles are congruent.
Congruent Complements	If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.
Vertical Angles	Vertical angles are congruent.
Alternate Interior Angles	If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
Alternate Exterior Angles	If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
Interiors on Same Side	If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.
Alternate Interior Angles Converse	If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
Alternate Exterior Angles Converse	If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.
Interiors on Same Side Converse	If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.
Triangle Sum	The sum of the interior angles of a triangle is 180º.
Exterior Angle	The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
Angle-Angle-Side (AAS) Congruence	If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Base Angle Theorem (Isosceles Triangle)	If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse (Isosceles Triangle)	If two angles of a triangle are congruent, the sides opposite these angles are congruent.
Hypotenuse-Leg (HL) Congruence (right triangle)	If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
Mid-segment Theorem (also called mid-line)	The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Parallelograms	About Sides	* If a quadrilateral is a parallelogram, the opposite sides are parallel. * If a quadrilateral is a parallelogram, the opposite sides are congruent.
	About Angles	* If a quadrilateral is a parallelogram, the opposite angles are congruent. * If a quadrilateral is a parallelogram, the consecutive angles are supplementary.
	About Diagonals	* If a quadrilateral is a parallelogram, the diagonals bisect each other. * If a quadrilateral is a parallelogram, the diagonals form two congruent triangles.
Parallelogram Converses	About Sides	* If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. * If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
	About Angles	* If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram. * If the consecutive angles of a quadrilateral are supplementary, the quadrilateral is a parallelogram.
	About Diagonals	* If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. * If the diagonals of a quadrilateral form two congruent triangles, the quadrilateral is a parallelogram.
Side Proportionality	If two triangles are similar, the corresponding sides are in proportion.

	Roberts