# 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°

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

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

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

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

Example: