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Points Lines and Angles
Points
A point indicates position, it is represented by a dot and named by a capital letter. The Figure below illustrates point A, point B, and point C.
Lines
A line (straight line) is a set of continuous points that extend indefinitely in either direction. It is named by any two points on the line. The symbol ↔ written on top of two letters is used to denote the line. A line may also be named by one small letter. The Figure below illustrates one line can be written as, line AB, or line m.
Postulate 1.1: A line contains at least two points.
Planes
A plane is a set of points that forms a flat surface which extends indefinitely in all directions. It is usually represented as a closed four sides figure. A capital letter is placed at one of the vertex. The Figure below illustrates the plane P and plane Q.
Line Segments
A line segment is a part of a line. It has two endpoints and is named by its endpoints. Sometimes, the symbol ̅ written on top of two letters is used to denote the segment. The Figure below showed the line segment AB.
Rays
A ray is a part of a line, except that it has only one endpoint and continues forever in one direction. It is named by the letter of its endpoint and any other point on the ray. The symbol → written on top of the two letters is used to denote the ray. Figure below shows ray AB.
Midpoint
A midpoint of a line is the point that has equidistant from the two endpoints. See Figure below, C is the midpoint of line segment AB because the length of AC is the same as the length of CB, that is, AC = CB or because AC is equal to one-half AB and CB is equal to one-half AB.
Collinear Points
Collinear points are points which lie on the same line. See Figure below, points A, C, and B are collinear because they lie on the same line.
See Figure below, points E, G, F are not collinear because point G do not lie on the line EF.
Two Points Determine One Line
Through any two points, there is exactly one line. See Figure below, these is only one line l that pass through point A and point B.
Two Lines Intersect
If two lines intersect, then they intersects in exactly one point. See Figure below, both line m and line n intersect at point P.
The Shortest Distance between Two Points
Of all connections between two endpoints, the straight line has the shortest distance. See Figure below, both the line segment l and the curve m have the same endpoints, but the length of the straight line l is less than the length of the curve m.
The straight line segment AB is the shortest distance from point A to point B.
Distance of a Line Segment
The distance between two points is the length of the line segment that join the two endpoints. See Figure below, the letter d denote the distance from A to B. The length of the segment AB can be written as d = AB.
Note: the term “line” will always be understood to represent a straight line. The term “segment” represents a part of a line.
Point Lies between Two Endpoints
If point P lies between points A and B on a line, then AP + PB = AB. See Figure below, since points A, P, and B are collinear, and point P lies between points A and B, then AP + PB = AB.
More Than One Plane Contain Two Points
A plane contains at least three not collinear points. See Figure below, both plane P and plane Q contain point A and point B. More than one plane contain two points.
Three Not CollinearPpoints Determine a Plane
Through any three not collinear points, there is exactly one plane. See Figure below, point A, point B, and point C lie on plane P.
Two Planes Intersect
If two plane intersect, then their intersection is a line. See Figure below, plane P and plane Q intersect along line l.
Line and a Point Out of the Line Determine a Plane
If a point lies outside a line, then exactly one plane contains both the line and the point. See Figure below, the plane P contains line l and point A.
Two Intersect Lines Form A Plane
If two lines intersect, then exactly one plane contains both lines. See Figure below, the plane Q contains line m and line n, which intersect at point A.
Coordinate of Points
Each point on a line can be paired with exactly one real number called its coordinate. The distance between two points is the positive difference of their coordinates.
See Figure above, the coordinate of point B is b, the coordinate of point A is a. Since the line has only one direction, then b > a, then the distance of the segment AB is equal to the coordinate of B subtract the coordinate of A. The distance AB = b – a.
Example 1:
Example 2:
Example 3: