Geometry - Theorems, Properties & Postulates

Ruler Postulate

Every point on a line can be paired with a unique real number. This number is called the coordinate of the point

Segment Addition Postulate

If points A, B, and C are on the same line with B in between A and C, then AB + BC = AC

Protractor Postulate

Given BA (→) and a point C not on BA (→), a unique real number from 0 to 180 can be assigned to BC (→)

0 is assigned to BA (→)

180 is assigned to BD (→)

Angle Addition Postulate

If point D is in the interior of <ABC, then m<ABD + m<DBC = m<ABC

Addition Property of Equality

If a = b, then a + c = b + c

Subtraction Property of Equality

If a = b, then a - c = b - c

Multiplication Property of Equality

If a = b, then a * c = b * c

Division Property of Equality

If a = b, and c is not equal to 0, then a/c = b/c

Reflexive Property of Equality

a = a

Symmetric Property of Equality

If a = b, then b = c

Transitive Property of Equality

If a = b, then b = c, then a = c

Substitution Property of Equality

If a = b, then b can replace a in any expression

Reflexive Property of Congruence

If segment AB is congruent to segment AB, then <A is congruent to <A

Symmetric Property of Congruence

If segment AB is congruent to segment CD, then segment CD is congruent to segment AB.

If <A is congruent to <B, then <B is congruent to <A

Transitive Property of Congruence

If segment AB is congruent to segment CD and segment CD is congruent to segment EF, then segment AB is congruent to segment EF.

If <A is congruent to <B and <B is congruent to <C, then <A is congruent to <C

Vertical Angles Theorem

Vertical angles are congruent

Congruent Complements Theorem

If two angles are complementary (two angles that adds up to 90 degrees) or to the same angle, then they are congruent

Congruent Supplements Theorem

If two angles are supplementary (two angles that adds up to 180 degrees) or to the same angle, then they are congruent

Linear Pairs Theorem

The sum of the measures of a linear pair is 180 degrees

Same-Side Interior Angles Postulate

m<1 + m<2 = 180 degrees

Alternate Interior Angles Theorem

<1 is congruent to <2

Corresponding Angles Theorem

<1 is congruent <2

Alternate Exterior Angles Theorem

<1 is congruent to <2

Converse of the Same-Side Interior Angles Postulate

If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel

Converse of the Alternate Exterior Angles Theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel

Triangle Angle Sum Theorem

The sum of the measures of all the triangles of a triangle is 180 degrees

Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles

Isosceles Triangle Theorem and the Converse

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Angle-Side-Angle (ASA) Congruence Criterion

If two angles of one triangle and the included side are congruent to two angles and the included side of another triangle, then the two triangles are congruent

Angle-Angle-Side (AAS) Congruence Criterion

If two angles and a non included side of one triangle are congruent to two angle sides and a non included side of another triangle, then the two triangles are congruent

Hypotenuse-Leg (HL) Theorem

If the hypotenuse and one leg of a right angle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

If two triangles are congruent, then each pair of corresponding sides is congruent and each pair of corresponding angles is congruent