Review #0

Geometry Definitions

Final Exam Review

Points

• Three Points C, M, Q• A point is the most fundamental object in

geometry. It is represented by a dot and named by a capital letter. A point represents position only; it has zero size (that is, zero length, zero width, and zero height).

Lines

• A line (straight line) can be thought of as a connected set of infinitely many points. It extends infinitely far in two opposite directions. A line has infinite length, zero width, and zero height. Any two points on the line can be used to name it. The symbol ↔ written on top of two letters is used to denote that line. A line may also be named by one small letter l.

• Diagram of two lines: Remember lines can be labeled in two ways

Types of Points

• Points that lie on the same line are called collinear points. If there is no line on which all of the points lie, then they are non-collinear points.

Collinear>

Non-collinear>

Types of Lines

• PARALLEL LINES- two lines that are always the same distance apart, and will never intersect. Parallel can be abbreviated as ||. An example of parallel lines is on the Italian flag. Lines a and b on the flag are parallel.

Types of Lines• PERPENDICULAR LINES -

two coplanar lines that intersect and form angles measuring exactly 90 degrees, like the edges of a building. If an angle measures 90 degrees, a square is place where the lines intersect to show that it is a right angle. Perpendicular is often abbreviated as _|_.

• Line a _|_ b reads as line a is perpendicular to line b.

Types of Lines

• SKEW LINES - two lines that do not intersect, and are not parallel. Skew lines are always non-coplanar. An overpass on a highway is an excellent example of skew lines.

• This only occurs when you consider lines in 3 dimensional space.

Plane• A plane may be considered as an infinite set of points

forming a connected flat surface extending infinitely far in all directions.

• A plane has infinite length, infinite width, and zero height (or thickness). It is usually represented in drawings by a four-sided figure.

• A single capital letter is used to denote a plane. The word plane is written with the capital letter so as not to be confused with a point.

Relationships to Planes

• COPLANAR - on the same plane. Points or objects may not be collinear, but if they lie in the same plane they are coplanar. These bales of hay lie on the same field, so they are like points lying on the same plane.

• NONCOPLANAR - any number of points not lying in the same plane.

Line Segment

• We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper. A line segment does not extend forever, but has two distinct endpoints. We write the name of a line segment with endpoints A and B as . Note how there are no arrow heads on the line over AB such as when we denote a line or a ray.

AB

A

B

Midpoint and Bisector of a Segment

• MIDPOINT-A point on the line segment that cuts the segment into two congruent pieces.

• BISECTOR OF A SEGMENT-A line, segment, ray, or plane that intersects the segment at its midpoint.

Ray

• We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction.

• The point where the ray begins is known as its endpoint.

• We write the name of a ray with endpoint A and passing through a point B as .

Angles

• Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.

Angles, Cont.

• We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as or as ABC .CBA

Degrees: Measuring Angles

• We measure the size of an angle using degrees.

• The Protractor Postulate

Types of Angles• Acute Angle:

Measures between 0° and 90°

• Right Angle: Measure of 90°

• Obtuse Angle: Measure between 90° and 180°

• Straight Angle: Measure of 180°

Angle Relationships

• Complementary Angles: Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees.

• Supplementary Angles: Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees.

Angle Relationships• Congruent Angles: Angles

with equal measures.• Adjacent Angles: Share a

vertex and a common side but no interior points.

• Bisector of an angle: a ray that divides the angle into two congruent angles. In this picture is the angle bisector.

OY

12 3

4

5

67 8

9

1011

12

13 14

Postulates

• A statement that is accepted without proof.• Usually these have been observed to be true

but cannot be proven using a logic argument.

Postulates Relating Points, Lines, and Planes

• Postulate 5: A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.

Postulates Relating Points, Lines, and Planes

• Postulate 6: Through any two points there is exactly one line.

Postulates Relating Points, Lines, and Planes

• Postulate 7: Through any three points there is at least one plane (if collinear), and through any three non-collinear points there is exactly one plane.

Postulates Relating Points, Lines, and Planes

• Postulate 8: If two points are in a plane, then the line that contains the points is in that plane.

A .B .

Postulates Relating Points, Lines, and Planes

• Postulate 9: If two planes intersect, then their intersection is a line.

Theorems

• Theorems are statements that have been proven using a logic argument.

• Many theorems follow directly from the postulates.

Theorems Relating Points, Lines, and Planes

• Theorem 1-1: If two lines intersect, then they intersect in exactly one point.

• Theorem 1-2: Through a line and a point not in the line there is exactly one plane.

• Theorem 1-3: If two lines intersect, the exactly one plane contains the lines.