“Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”
― Johannes Kepler
Fundamental Concept on Geometry
We may think of a point as a “dot” on a piece of paper or the pinpoint on a board. In geometry, we usually identify this point with a number or letter. A point has no length, width, or height – it just specifies an exact location. It is zero-dimensional.
Every point needs a name. To name a point, we can use a single capital letter. The following is a diagram of points A, B, and M:
We can use a line to connect two points on a sheet of paper. A line is one-dimensional. That is, a line has length, but no width or height. In geometry, a line is perfectly straight and extends forever in both directions. A line is uniquely determined by two points.
Lines need names just like points do, so that we can refer to them easily. To name a line, pick any two points on the line.
The line passing through the points A and B
is denoted by
A set of points that lie on the same line are said to be collinear.
Pairs of lines can form intersecting lines, parallel lines, perpendicular lines and skew lines.
Because the length of any line is infinite, we sometimes use parts of a line. A line segment connects two endpoints. A line segment with two endpoints A and B is denoted by
A line segment can also be drawn as part of a line.
The midpoint of a segment divides the segment into two segments of equal length. The diagram below shows the midpoint M of the line segment AB. Since M is the midpoint, we know that the lengths AM = MB.
A ray is part of a line that extends without end in one direction. It starts from one endpoint and extends forever in one direction.
A ray starting from point A and passing through B
is denoted by
Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides. Planes are thought of as flat surfaces, like a tabletop. A plane is made up of an infinite amount of lines. Two-dimensional figures are called plane figures.
All the points and lines that lie on the same plane are said to be coplanar.
Space is the set of all points in the three dimensions – length, width and height. It is made up of an infinite number of planes. Figures in space are called solids.
Figures in space
How to measure angles and types of angles
An angle consists of two rays with a common endpoint. The two rays are called the sides of the angle and the common endpoint is the vertex of the angle.
Each angle has a measure generated by the rotation about the vertex. The measure is determined by the rotation of the terminal side about the initial side. A counterclockwise rotation generates a positive angle measure. A clockwise rotation generates a negative angle measure. The units used to measure an angle are either in degrees or radians.
Angles can be classified base upon the measure: acute angle, right angle, obtuse angle, and straight angle.
If the sum of measures of two positive angles is 90°, the angles are called complementary.
If the sum of measures of two positive angles is 180°, the angles are called supplementary.
1) Two angles are complementary. One angle measures 5x degrees and the other angle measures 4x degrees. What is the measure of each angle?
2) Two angles are supplementary. One angle measures 7x degrees and the other measures (5x + 36) degrees. What is the measure of each angle?
Straight line intersecting two parallel lines:
We will examine now the important aspects of a straight line intersecting two parallel lines.
Angles are of three types:
Acute angle – which is less than 900
Obtuse angle – which is more than 900
Right angle – which is 900
In any triangle, an external angle is equal to the sum of two of its opposite angles.