# Understanding the Mathematical Subject Plane Geometry

Plane geometry consists of straight lines, points, and curved lines in a plane.
A plane is a two-dimensional object that is an infinite square composed of
all the points within the square. A point has zero dimensions. One postulate of
geometry states that two points determine a “straight line”. A line is defined
as a “straight line” that goes in opposite directions from each of the
two points that determine it. A segment is a part of a line, which can be
determined by any two points that also determine a line. The segment will be
all points between and including the two points. A plane can be determined
by three points, a point and a line, or two lines. The first two can be proven to
form two lines as follows: Three points not on the same line determine two lines by using two of the points to determine one line, then the remaining point will determine the second line. A point and a line will determine the second line by using one of the points
on the line and the point to determine two lines. Two lines form an infinite number of
lines using one point on each line for each line. These lines form a plane.

Using the concept of x, y axes, we can plot curved lines on the plane. We can also
plot squares, rectangles, parallelograms, trapezoids, circles, triangles, etc.
Two lines are parallel if they do not intersect. Two lines in the same
plane will either intersect or be parallel. The formula for a straight line
is linear. This means it has an exponent of 1 for the variable. Examples are
x=1, y=x+3, y=x+39. The line x=1 intersects the y-axis at the point
(1,0). All the points have x=1 with all other real numbers as the
y coordinates. Examples are (1,9), (1,10), (1,-17), (1,-333).
The line y=x+3 has the points (0,3), (1,4), (9,12), (-1,2), (-15,-12).
The line y=x+39 includes the points (0,39), (1,40), (13,52), (-1,38).

There are several theorems in plane geometry. Some have to do with triangles.
Some of the most important are the congruence theorems for triangles. Two
triangles are congruent if their sides and angles are equal. The SSS proves
that two triangles are congruent if their sides are equal. The SAS proves two
triangles are congruent if two of their sides and one included angle are equal.
Euclidean Geometry is another name for plane geometry. Hyperbolic Geometry is
one of the most popular of the Non-Euclidean Geometry, which has to do with
curved lines. Hyperbolic Geometry allows all lines to be parallel if they
curve without intersecting. It also allows for triangles to have greater than
or less than the 180 degrees that is required in Euclidean Geometry.

Hilbert, one of the most famous mathematicians in geometry, gave ten unsolved
problems in mathematics, some of which have not been solved as of today.
The Riemann Hypothesis is one of them. It claims that all solutions to
the Zeta function lie on a “critical line” that has nontrivial real values equal to 1/2.