Geometry Definitions POINTS, LINES, and PLANES

Points 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. Point C A point represents position only; it has zero size (that is, zero length, zero width, and zero height).  

Lines l l  

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 points M,A, and N Non-collinear points T, I, and C

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. Line a is parallel to line b: a // b

Types of Lines PERPENDICULAR LINES - two coplanar lines that intersect and form angles measuring exactly 90°. If an angle measures 90°, a square is place where the lines intersect to show that it is a right angle. Perpendicular is often abbreviated as |. line a is perpendicular to line b: a | 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 3D space.

Plane A plane is 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 thickness. It is usually represented in drawings by a four-sided figure. A single capital cursive letter is used to denote a plane. T R plane T plane R

Relationships to Planes COPLANAR – points or objects on the same plane. Points or objects may not be collinear, but if they lie in the same plane they are coplanar. NONCOPLANAR - any number of points not lying in the same plane.

Line Segment A line segment is a "straight" arrangement of points between two distinct points– 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. B A

Midpoint and Bisector of a Segment MIDPOINT - A point on the line segment that cuts the segment into two congruent pieces. Slash marks are used to indicate the congruent pieces. BISECTOR OF A SEGMENT - A line, segment, ray, or plane that intersects the segment at its midpoint.

Ray A ray is 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 . The endpoint must be listed first. A G B   H

Angles Two rays that share a common 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 or as This angle may also be this written as , because B is the vertex point to only ONE angle.

Multiple angles with a shared vertex   C B     D   E There are 4 angles with vertex B, therefore each angle MUST be named using 3 points specific to that angle.

Degrees: Measuring Angles We indicate the size of an angle using degrees. We use a protractor to measure an angle’s degrees.

Types of Angles Acute Angle: Right Angle: Obtuse Angle: Measures between 0° and 90° Right Angle: Measure of 90° Obtuse Angle: Measure between 90° and 180°

Angle Relationships Complementary Angles: Two angles whose sum of their degree measurements equals 90°. Supplementary Angles: Two angles whose sum of their degree measurements equals 180°.

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.

Congruent vs Equal Congruent: Equal:   Equal: Refers to two numbers (the measures of the segments) that are equal. AB without the bar on top, means the length of the segment.

Conjectures: Postulates and Theorems Postulate: A statement that is accepted without proof. Usually these have been observed to be true but cannot be proven using a logic argument. Theorem: A statement that has been proven using a logic argument. Many theorems follow directly from postulates. Throughout this textbook Postulates and Theorems are referred to as Conjectures.

Conjectures Relating Points, Lines, and Planes 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.

Conjectures Relating Points, Lines, and Planes Through any two points there is exactly one line.

Conjectures Relating Points, Lines, and Planes Through any three points there is at least one plane (if collinear), and through any three non-collinear points there is exactly one plane.

Conjectures Relating Points, Lines, and Planes If two points are in a plane, then the line that contains the points is in that plane.  

Conjectures Relating Points, Lines, and Planes If two planes intersect, then their intersection is a line.

Major Conjectures Relating Points, Lines, and Planes If two lines intersect, then they intersect in exactly one point. Through a line and a point not in the line there is exactly one plane. If two lines intersect, then exactly one plane contains the lines.

The Measure of…notation

Time to ponder….. Coplanar lines that do not intersect vertical parallel ray plane A an infinite number of points that has two distinct end points bisect line segment intersect congruent

Time to ponder….. Any three or more points that lie in the same plane coplanar line plane collinear 4. Coplanar → Points on the same line True        False 5. Line → one endpoint and extends indefinitely in one direction

Time to ponder….. 6. Collinear → an infinite number of points that goes on forever in both directions True        False 7. Plane → an infinite number of points that goes on forever in both directions

  9. Give another names for plane S. 10. Name three collinear points. 11. Name the point of intersection of line AC and plane S.

You are Learning a new Language. The End Once you study all the fancy words/vocabulary, Geometry is very easy to understand…so STUDY! You are Learning a new Language.