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27 terms

Geometry chapter 3: Postulates, theorems and properties

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Unique measure assumption
Every angle has a unique measure from 0° to 180°
Unique angle assumption
Given any ray BC and a real number r between 0 and 180, there is a unique angle ABC on each side of BC such that m∠ABC = r
Straight angle assumption
If BA and BC are opposite rays, then m∠ABC = 180°
Zero angle assumption
If AB and AC are the same ray, then m∠ABC = 0°
Angle addition Assumption
If angles CBD and CBA are adjacent angles, then m∠CBD + m∠DBA = m∠ABC
Equal angle measures Theorem
If two angles have the same measure, their complements have the same measure. If two angles have the same measure, their supplements have the same measure.
Linear pair theorem
If two angles form a linear pair, then they are supplementary
Vertical angles theorem
If two angles are vertical angles, then their measures are equal
Reflexive property of equality
a = a
Symmetric property of equality
if a = b, then b = a
Transitive property of equality
if a = b, and b = c, then a = c
Addition property of equality
If a = b, then a + c = b + c
Multiplication property of equality
If a = b, then ac = bc
Transitive property of inequality
if a < b and b < c, then a < c
Addition property of inequality
If a < b, then a + c < b + c
Multiplication property of inequality
If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc
Equation to inequality property
If a and b are positive numbers and a + b = c, then c > a and c > b
Substitution property
If a = b, the a may be substituted for b in any expression
Corresponding angles postulate
If two corresponding angles have the same measure, then the lines are parallel. If the lines are parallel, then corresponding angles have the same measure.
Parallel lines and slopes theorem
Two nonvertical lines are parallel if and only if they have the same slope
Transitivity of parallelism theorem
If line l is parallel to line m and lin m is parallel to line n, then line l is parallel to line n
Sk Theorem 1: size property
Under a size change Sk, the line through any two preimage points is parallel to the line through their images
Sk Theorem 2: Collinearity is preserved
Under Sk, the images of collinear points are collinear
Sk Theorem 3: Angle measure is preserved
Under Sk, an angle and its image have the same measure
Two perpendiculars theorem
If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other
Perpendicular to parallels theorem
In a plane, if a line is perpendicular to one of the two parallel lines, then it is also perpendicular to the other
Perpendicular lines and slopes theorem
Two nonvertical lines are perpendicular if and only if the product of their slopes is -1