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

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