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.