Notes

As early as 1350 B.C., Ramses II used the principles of geometry to divide the soil of Egypt among the people. But it was the early Greeks who developed the subject most fully.

The word geometry comes from two Greek words: ge (earth) and metrein (to measure).

Around 300 B.C. a Greek named Euclid wrote a book on geometry called The Elements.

equal or symbol =

congruent or symbol ≅

For example, one line is congruent to another line if the two lines have measurements that are equal.

The size and shape of the space are a geometric solid, or simply a solid.
All geometric solids have three dimensions—length, width, and height. Therefore, we call them three-dimensional figures.

A plane is a geometric term that's difficult to define clearly.
A plane is like these flat objects with two exceptions: it extends indefinitely in all directions, and it has no thickness. In Geometry, a plane is defined using at least three points.
Planes have only two dimensions—length and width. Therefore, they're two-dimensional figures.
A plane separates one portion of space from another.
the boundaries of a solid are planes.

The surface that forms the boundary of a sphere is a curved surface.

lines has no thickness, only length, and continues indefinitely in both directions.
A line is another geometric term that's difficult to define.
A line has only one dimension—length. Consequently, it's a one-dimensional figure.
a line divides a plane into two parts.

A point has no dimensions. It's simply used to indicate position. We indicate a point by a small dot with a capital letter near it.
a point separates parts of a line.
if two lines intersect, their intersection is a point.

A certain, definite part of a line, called a segment, has two end points.
A straight line is a continuous mark free from curves that extends on and on in both directions.

A curved line, or a curve, is a line in which no part is straight.
A curved line also forms at the intersection of a curved surface and a plane.

A broken line is a series of connected line segments that don't form a straight line.

----------------------------------------------------------------------------------------------------------------------------------------------
An angle is a figure formed by two lines that intersect.
The lines are the sides of the angle, and the point of intersection is the vertex.
The symbol ∠ represents the word angle.

Adjacent angles are two angles with the same vertex and a common side lying between them.
Vertical angles form when two lines intersect.
The vertical angles are the ones that are opposite each other.

Extending the sides of an angle doesn't change the size of that angle.

A circle divides into 360 equal parts, called degrees.
An angle of 1 degree is an angle that's 1⁄360 of a circle, or 1⁄360 of a complete revolution.
An angle of 1 degree is an angle that's 1⁄360 of a circle, or 1⁄360 of a complete revolution. Each degree divides into 60 minutes; therefore, each minute is 1⁄60 of a degree. Further, each minute divides into 60 seconds, so that a second is 1⁄60 of a minute. You can also say that there are 60 seconds in 1 minute and 60 minutes in 1 degree. You place the symbols for degrees, minutes, and seconds above and to the right of the number, like this: 45°31'18".

A 180° angle is a straight angle.
forming 90° (1⁄4 × 360⁄1 = 90) and is a right angle.
Any angle less than a right angle (less than 90°) is an acute angle.
Any angle greater than a right angle but less than a straight angle (less than 180°) is an obtuse angle.
Finally, any angle greater than a straight angle but less than a round angle (360°) is a reflex angle.

Complementary angles are two angles whose sum is 90°, or a right angle. Each angle is the complement of the other.
Supplementary angles are two angles whose sum is 180°, or a straight angle. Each angle is the supplement of the other.
A somewhat less common term is conjugate angles, whose sum is 360°.

Two lines that form a right angle are perpendicular lines.
The symbol ⊥ means perpendicular, perpendicular to, or is perpendicular to.
A vertical line is a line that's perfectly straight up and down. It's the direction of a plumb line. (A plumb line is a cord or string with a weight attached that hangs freely.)

If two straight lines intersect, the vertical angles are congruent.
If two straight lines intersect and one angle is 90°, the other three angles are 90°.

Lines drawn to show compass direction (north, south, east, and west) are perpendicular lines.

----------------------------------------------------------------------------------------------------------------------------------------------
A compass is a tool mainly used to draw circles
A protractor is a tool used to measure and draw angles.
A bisector is anything that divides a quantity into two equal parts.
If the bisector is perpendicular to the segment it bisects, it's said to be the perpendicular bisector of the segment.

----------------------------------------------------------------------------------------------------------------------------------------------
Two straight lines in the same plane that never meet, no matter how far you extend them, are parallel lines.
The symbol || means parallel, parallel to, or is parallel to. Parallel lines can be horizontal, vertical, or angled.

A transversal is a line that crosses a system of two or more lines.
When a transversal crosses a set of two parallel lines, eight angles form.

the angles formed on the inside of the lines are interior angles
Angles on opposite sides of the transversal are alternate angles.
corresponding angles (sometimes called exterior-interior angles) are those that are in corresponding positions in relation to the two traversed lines.

BRIEF SUMMARY OF PARALLEL LINES
A transversal crossing two parallel lines forms eight angles.
Four of these angles are acute and four are obtuse.
All acute angles are congruent and all obtuse angles are congruent.
Any acute angle is the supplement of any obtuse angle.