# Geometry Postulate

Written by Jerry Ratzlaff on . Posted in Geometry

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 of equality  -  If  $$\large{ a = b }$$ , then  $$\large{ b = a }$$
• Transitive property of equality  -  If  $$\large{ a = b }$$  and  $$\large{ b = c }$$ , then  $$\large{ a = c }$$
• Addition property of equality  -  If  $$\large{ a = b }$$ , then  $$\large{ a+c=b+c }$$
• Division property of equality  -  $$\large{ \frac{a}{b} = a \; \frac{1}{b} }$$
• Multiplication property of equality  -  If  $$\large{ a = b }$$ , then  $$\large{ a \; c=b \; c }$$
• Subtraction property of equality  -  If  $$\large{ a = b }$$ , then  $$\large{ a-c=b-c }$$
• Distributative property of equality  -  $$\large{ a \; \left(b+c\right) = a\;b + a\;c }$$
• Substitution property of equality  -  If  $$\large{ a = b }$$ , then  $$\large{ a }$$  can be replaced by  $$\large{ b }$$
• Reflexive property of angle measure  -  For any angle  $$\large{ A }$$ , then  $$\large{ m\angle A= m\angle A }$$
• Symmetric property of angle measure  -  If  $$\large{ m\angle A= m\angle B }$$ , then  $$\large{ m\angle B= m\angle A }$$
• Transitive property of angle measure  -  If  $$\large{ m\angle A= m\angle B }$$  and  $$\large{ m\angle B= m\angle C }$$ , then  $$\large{ m\angle A= m\angle C }$$
• Reflexive property of segment length  -  For any segment  $$\large{ AB }$$ , then  $$\large{ AB=BA }$$
• Symmetric property of segment length  -  If  $$\large{ AB = CD }$$ , then  $$\large{ CD = AB }$$
• Transitive property of segment length  -  If  $$\large{ AB = CD }$$  and  $$\large{ CD = EF }$$ , then  $$\large{ AB = EF }$$
• 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.