Geometry 3

Property

Statements often proved in algebra

The Reflexive Property

a=a (Any number is equal to itself)

The Substitution Property

If a=b, then a can be substituted for b in any expression

The Addition Property

If a=b, the a+c = b+c

The Subtraction Property

If a=b, then a - c = b - c

The Multiplication Property

If a = b, the ac = bc

The Division Property

If a=b and c is not equal to 0, then a/c = b/c

coordinate and distance

points on a line are numbered so that to every point there corresponds exactly one real number called its coordinate and to every real number there corresponds exactly one point.

to every pair of points on the line there corresponds a real number called the distance between them

Postulate 3 - The Ruler Postulate

The points on a line can be numbered so that positive number differences measure distances

Definition- Betweenness of Points

A point is between two other points on the same line iff its coordinate is between their coordinates.

Theorem 1. The Betweenness of Points Theorem

If A-B-C, then AB + BC = AC

degree

unit chosen to measure angles

rotation of rays and half rotation of rays

all the position through which a ray with its endpoint at the center of the protracter turning exactly once around the scale

All the rays that correspone to a protracter with the meausre of 180 degrees

measure of the angle

To every pair of rays there corresponds a real number called the measure of the angle

Postulate 4. The Protractor Postulate

The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles.

Definitions of angles- acute, right, obtuse, and straight

acute iff it is less than 90 degrees

right iff it is 90 degrees

obtuse iff it is more than 90 degrees but less than 180 degrees

straight iff it is 180 degrees

Definition- Betweenness of Rays

A ray is between two others in the same half-rotation iff its coordinate is between their coordinates.

Thereom 2. The Betweeness of Rays Thereom

If OA-OB-OC, then angle AOB + angle BOC = angle AOC

Definition- midpoint of line segment

A point is the midpoint of a line segment iff it divides the line segment into two equal segments.

Definition- line that bisects an angle

A line bisects an angle iff it divides the angle into two equal angles

Corollary

Thereom that can be easily proved as a consequence of a postulate or another thereom

Corollary to the Ruler Postulate

A line segment has exactly one midpoint

Corollary to the Protractor Postulate

An angle has exactly one ray that bisects it

Definition of complementary and supplementary angles

Two angles are complemetary if their sum is 90 degrees ( each angle is called the complement of the other)

Two angles are supplementary iff their sum is 180 degrees ( each angle is called the supplement of the other)

Theorem 3. (complements)

Complements of the same angle are equal

Thereom 4 ( Supplements)

Supplements of the same angle are equal

opposite rays

Opposite rays are two rays that share a common endpoint and extend in exactly opposite directions, forming a straight line (a 180° angle).

Definition - linear pair and vertical angles

Two angles are a linear pair iff they have a common side, and their other sides are opposite rays

Two angles are vertical angles iff the sides of one angle are opposite rays to the sides of the other

Theorem 5

The angles in a linear pair are supplementary

Theorem 6 (vertical angles)

Vertical angles are equal

Definition of perpendicular lines

Two lines are perpendicular if they form a right angle

Theorem 7 (perpendicular lines)

Perpendicular lines form 4 right angles

Collorary to the definition of a right angle

All right angles are equal

Thereom 8 (linear pair and perpendicular lines)

If the angles in a linear pair are equal, then their sides are perpendicular.

Definition of parallel lines

Two lines are parallel iff they lie in the same place and do not intersect

Axioms

Postulates