# Geometry Postulate

A postulate is a statement that is assumed true without proof.

## Properties

**Reflexive property of equality**- A quantity is congruent to itself. \(\large{ a = a }\)

**Symmetric property**- If \(\large{ a = b }\) , then \(\large{ b = a }\)**of equality**

**Transitive property**- If \(\large{ a = b }\) and \(\large{ b = c }\) , then \(\large{ a = c }\)**of equality**

**Addition property**- If \(\large{ a = b }\) , then \(\large{ a+c=b+c }\)**of equality**

**Division property**- \(\large{ \frac{a}{b} = a \; \frac{1}{b} }\)**of equality**

**Multiplication property**- If \(\large{ a = b }\) , then \(\large{ a \; c=b \; c }\)**of equality**

**Subtraction property**- If \(\large{ a = b }\) , then \(\large{ a-c=b-c }\)**of equality**

**Distributative property**- \(\large{ a \; \left(b+c\right) = a\;b + a\;c }\)**of equality**

**Substitution property**- If \(\large{ a = b }\) , then \(\large{ a }\) can be replaced by \(\large{ b }\)**of equality**

**Reflexive property of angle measure**- For any angle \(\large{ A }\) , then \(\large{ m\angle A= m\angle A }\)

**Symmetric property**- If \(\large{ m\angle A= m\angle B }\) , then \(\large{ m\angle B= m\angle A }\)**of****angle measure**

**Transitive property**- If \(\large{ m\angle A= m\angle B }\) and \(\large{ m\angle B= m\angle C }\) , then \(\large{ m\angle A= m\angle C }\)**of****angle measure**

**Reflexive property of segment length**- For any segment \(\large{ AB }\) , then \(\large{ AB=BA }\)

**Symmetric property**- If \(\large{ AB = CD }\) , then \(\large{ CD = AB }\)**of****segment length**

**Transitive property**- If \(\large{ AB = CD }\) and \(\large{ CD = EF }\) , then \(\large{ AB = EF }\)**of****segment length**

**Associative law of addition**- \(\large{ \left(a+b\right)+c = a+\left(b+c\right) }\)

**Associative law of multiplication**- \(\large{ \left(a\;b\right)\;c = a\; \left(b\;c\right) }\)

**Commutative law of addition**- \(\large{ a + b = b + a }\)

**Commutative law of multiplication**- \(\large{ a \; b = b \; a }\)

**Zero property of multiplication**- \(\large{ a \; 0 = 0 }\)

**Additive identity**- \(\large{ a+0 = a }\)

**Additive inverse**- \(\large{ a+ \left(-a\right) = 0 }\)

**Multiplicative inverse**- \(\large{ a = \frac{1}{a} }\)

**Multiplicative identity**- \(\large{ a \; 1 = a }\)

**Multiplicative identity**- \(\large{ a \; \frac{1}{a} = 1 }\)

**Definition of subtraction**- \(\large{ a-b = a+\left(-b\right) }\)

- If \(\large{ a = b }\) , then \(\large{ \frac{a}{c}=\frac{b}{c} }\)

- If \(\large{ a = b }\) and \(\large{ c \ne 0 }\) , then \(\large{ \frac{a}{c}=\frac{b}{c} }\)

- If \(\large{ a + b = a + b' }\) , then \(\large{ b = b' }\)

- If \(\large{ a \; b = a b' }\) and \(\large{ a + a \ne a }\) , then \(\large{ b = b' }\)

- If \(\large{ a, b }\) are real numbers, then \(\large{ a+b }\) is a real number and \(\large{ a \times b }\) is a real number.

## Angle

**Angle addition postulate**- From any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts.

## Circle

**Arc addition postulate**- The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

## Line

**Converse of Corresponding angles postulate**- If two lines are intersected by a transversal and corresponding angles are equal in measure, then the lines are parallel.

**Corresponding angles postulate**- If two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure.

**Postulate**- Through any two points there is exactly one line.

**Postulate**- If two lines intersect, then they intersect at exactly one point.

**Postulate**- Through a point not on a given line, there is one and only one line parallel to the given line.

**Segment addition postulate**- For any segment, the measure of the whole is equal to the sum of the measures of its non-overlapping parts.

## Plane

**Postulate**- Through any three noncollinear points there is exactly one plane containing them.

**Postulate**- If two points lie in a plane, then the line containing those points lies in the plane.

## Polygon

**Area addition postulate**- The area of a region is equal th the sum of the areas of its nonnoverlapping parts.

## Triangle

**Angle-angle similarity postulate (AA)**- If two angles of one triangle are equal in measure to two angles of another triangle, then the two triangles are similar.

**Angle-side-angle congruence postulate (ASA)**- If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

**Side-angle-side congruence postulate (SAS)**- If two sides and the included angle of one triangle are equal in measure to the correrponding sides and angle of another triangle, then the triangles are congruent.

**Side-side-side congruence postulate (SSS)**- If three sides of one triangle are equal in measure to the correrponding sides of another triangle, then the triangles are congruent.