Defined Terms and Postulates

April 3, 2008

Defined terms

• Yesterday, we talked about undefined terms. Today, we will focus on defined terms (which are based on the undefined terms from yesterday).

More about space and points

• Space is the set of all points (objects in space have length, width, and height).

• Collinear points are points all in one line.• Coplanar points are points all in one plane.

Intersections

• The intersection of two figures is the set of points that are in both figures.

• You have intersections between lines, between planes, and between a line and a plane.

Line segments and betweenness

• A line segment is a part of a line. It has a fixed length and is named for its two endpoints. In the figure below segment AB consists of points A and B and all points that are between A and B.

• C is between A and B.

Rays

• A ray has one end point and extends in the other direction upto infinity. It is represented by naming the end point and any other point on the ray.

• Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line.

More about segments

• The length a segment is always measured positively, like a distance.

• Congruent segments are segments that have equal lengths.

• The midpoint of a segment is the point that divides the segment into two congruent segments.

Even more about segments

• A bisector of a segment is a line, segment, ray or plane that intersects the segment at its midpoint.

About angles

• An angle is the figure formed by two rays with the same endpoint.

• This shared endpoint is called the vertex.• The two rays are called the sides of the angle.• We can name the angle B, angle ABC, angle

CBA, or angle 1.

Naming angles

• Find several ways to name the two angles.

Measuring angles

• Angles are measured in degrees. Like distance, degrees are positive.

• In geometry we measure angles in degrees and angles can have degree measures from 0° to 180°.

• Angles are classified according to their measures.– Acute angle: Measure between 0 and 90– Right angle: Measure 90– Obtuse angle: Measure between 90 and 180– Straight angle: Measure 180

Congruent Angles and Adjacent Angles

• Congruent angles are angles that have equal measures.

• Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.

• In essence it tells us that if the angles "overlap" then we cannot call them adjacent.

Making conclusions

• Do not assume anything that a diagram does not tell you!

• Notice the marks for congruence in line segments and angles, as well as the mark for a 90 degree angle.

What is a postulate?

• Postulates or axioms are known as basic assumptions. Assumptions are statements accepted without proof.

• Some will seem very basic.

Postulate 1 Ruler Postulate

1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1.

2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

Postulate 2 Segment Addition Postulate

• If B is between A and C, then AB + BC = AC

Postulate 3 Protractor Postulate

• On line AB in a given plane , choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with the real numbers from zero to 180 in such a way that:a. OA is paired with 0, and OB with 180.b. If OP is paired with x, and OQ with y, then the measure of POQ = | x - y |.

Postulate 4 Angle Addition Postulate

• If point B lies in the interior of angle AOC, then the measure of angle AOB + the measure of angle BOC = the measure of angle AOC.

• If angle AOC is a straight angle and B is any point not on AC, then the measure of angle AOB + the measure of angle BOC = 180.

Postulate 5

• A line contains at least two points; a plane contains at least 3 points not all in one line; space contains at least four points not all in one plane.

Postulate 6

• Through any two points there is exactly one line.

• True for plane geometry.

Postulate 7

• Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.

Postulate 8

• If two points are in a plane, then the line that contains the points is in that plane.

Postulate 9

• If two planes intersect, then their intersection is a line.