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Study Notes 008:
By William E. Steinman:
Geometry 1:
Geometry is derived from Greek and means literally earth measuring.
Although there were many contributions from Greek scientists
such as Thales and Pythagoras, the definitive plane geometry text
“Elements” was completed by
Euclidean geometry, can be derived by rigorous logical steps
from the five postulates stated by
Postulate one states that it is possible to draw a straight line from any point to any other point.
Postulate two states that it is possible to produce a finite straight line continuously in a straight line.
Postulate three states that it is possible to describe a circle with any center and radius.
Postulate four states that all right angles are equal to one another.
Postulate five states that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. Later, I will get to non-Euclidean geometry, which replaces the fifth postulate with either of two alternative postulates.
To discuss geometry we must have a set of definitions.
There are three undefined terms of plane geometry.
These are the point, the line, and the plane.
The point is considered to have no length, width, or thickness. It has position only.
A line has length, but no width or thickness.
A plane has length and width, but no thickness. A plane is a flat surface.
Given the above, reveals that plane geometry is a geometry of things that can be drawn on a plane, I. E. things that have no thickness.
A line may be curved or straight or some combination of curved and straight.
A ray is part of a straight line, which begins at a point and extends without limit in one direction.
A straight line is unlimited in extent.
A straight line segment is the part of a straight line between two point on the straight line, including the points.
For example, A B
AB is a line segment in the above line.
Two line segments having the same length are called congruent.
A circle is defined as the set of all points in a plane that are the same distance from the center.
The circumference of a circle is the distance around it, IE. 360 degrees or 360o.
A radius is a straight line segment joining the center of the circle to a point on the circle.
A chord is a straight line segment joining any two points on the circle.
A diameter is a chord through the center of the circle. Therefore, a diameter is twice the radius.
An
arc is a continuous part of a circle. The symbol is .
Therefore A B represents the arc AB.
A semicircle is an arc that is one-half the circumference of a circle and is therefore 1800.
A central angle is and angle formed by two radii.
The arc between the two radii of a central angle is measured in degrees.
Congruent
circles are circles having congruent radii.
Arc B
Chord
Central
O Angle
A Diameter C
Semicircle
The size of an angle is the distance in degrees that one side must be rotated to meet the other side.
For example, if one side must be rotated 60o to meet the other side, we have an angle of 60o.
An acute angle is one that measures less than 90o.
A right angle is one that measures 90o.
An obtuse angle is one that measures more than 900 and less than 180o.
A straight angle is one that measures 180o.
A reflex angle is one that measures more than 180o and less than 360o.
Congruent angles are angles that have the same number of degrees.
Perpendiculars are rays that meet at right angles.
A perpendicular bisector is a ray that is at right angles to a given line segment and bisects it.
Yada, yada, yada!