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.
