Geometry: Understanding Geometric Planes

Geometric Concepts

• A plane is a flat surface that only occupies two dimensions and extends infinitely in those dimensions.
• There are no real-world examples of an actual geometric plane, but it can be modeled using a flat surface such as a chalkboard, table, or piece of paper.
• A plane has no thickness and only exists in two dimensions.
• There are different equations to describe a plane, including standard, normal, Hesse's normal, Cartesian, and parametric equations.

Plane Representation

• A plane can be drawn or modeled as a parallelogram with arrows pointing away from its sides to represent its infinite nature.
• A plane can be labeled or named using a single capital letter written in script or italics in one of its corners.
• A plane can also be named using any three or four points drawn on the edges of or within the parallelogram and labeled with letters.

Geometry Basics

• Geometry is a branch of mathematics that deals with shapes, measurements, and relationships of points, lines, and angles.
• A point is a location in space with no length or width, usually marked with a dot.
• A line is a straight series of points that extend in opposite directions infinitely, represented by an arrow at each end.
• A line segment is part of a line, has two endpoints, and does not extend infinitely.

Rays and Angles

• A ray is a one-dimensional, straight figure that begins at a single endpoint and extends in one direction indefinitely.
• Rays can be named by a single letter, their endpoint, or by two letters, their endpoint plus another point on the ray.
• A ray has a beginning point, but its length is infinite and cannot be measured.
• An angle is made up of two rays that share an endpoint, called the vertex.
• There are various types of angles, including right (90 degrees), acute (less than 90 degrees), straight (180 degrees), obtuse (greater than 90 degrees but less than 180 degrees), and reflex (greater than 180 degrees but less than 360 degrees).

Lines and Line Segments

• A line is a one-dimensional figure that extends indefinitely, contains an infinite number of points, and cannot be measured.
• Lines can be drawn on a plane, which is a two-dimensional figure that extends indefinitely in two directions (x and y).
• Lines contain an infinite number of points that have two values (x and y).
• There are three types of line pairs: parallel (never intersect), perpendicular (intersect at right angles), and transverse (cross two other lines in the same plane).
• The equation of a line is y = mx + b, where y and x are a point on the line, m is the slope, and b is where the line crosses the y-axis.

Intersecting Lines

• Lines and line segments are considered intersecting lines when they cross each other at one single point.
• The point where lines intersect is termed the point of intersection.
• Lines can intersect each other at any angle between 0 and 180 degrees.
• When intersecting lines cross at a right angle, they are called perpendicular lines.
• The slopes of perpendicular lines are negative reciprocals.

Perpendicularity

• Perpendicular means two lines or sides that form a right angle (90 degrees).
• Perpendicular line segments are line segments that intersect at a 90-degree angle.
• A perpendicular shape is a shape that has at least two sides that come together at a 90-degree angle.
• A right triangle has one right angle and two perpendicular lines.
• A rectangle and a square both have four right angles and four perpendicular lines.

Skew Lines

• Skew lines are defined as lines that are not parallel and do not intersect.
• Parallel lines are coplanar (lie in the same plane) and do not intersect.
• Skew lines are defined in three-dimensional space and have three conditions: not parallel, not intersecting, and not coplanar.
• The shortest distance between two skew lines is the line connecting them that is perpendicular to both.

Polygons

• A polygon is a two-dimensional closed region formed by at least three segments of a line that are consecutive.
• Two consecutive segments have a common endpoint, which are the vertices of the polygon.
• Polygons can be classified as regular (equilateral and equiangular) or irregular (not regular); concave or convex; and simple (segments intersect on vertices only) or complex (not simple).

Quadrilaterals

• A quadrilateral is a shape with four sides.
• The term "edges" refers to the sides of the shape.
• A vertex (plural: vertices) refers to the corners of a shape.
• Each vertex has an angle on the inside of the quadrilateral referred to as an interior angle.
• The angle outside of the quadrilateral is referred to as the exterior angle.
• In a quadrilateral, the sum of all four interior angles is always 360°.
• A convex quadrilateral is one where each of the four interior angles is less than 180°.
• A concave quadrilateral is one where at least one interior angle is greater than 180°.
• A complex quadrilateral is one where the sides intersect each other.

Geometric Concepts

• A plane is a flat surface that only occupies two dimensions and extends infinitely in those dimensions.
• There are no real-world examples of an actual geometric plane, but it can be modeled using a flat surface such as a chalkboard, table, or piece of paper.
• A plane has no thickness and only exists in two dimensions.
• There are different equations to describe a plane, including standard, normal, Hesse's normal, Cartesian, and parametric equations.

Plane Representation

• A plane can be drawn or modeled as a parallelogram with arrows pointing away from its sides to represent its infinite nature.
• A plane can be labeled or named using a single capital letter written in script or italics in one of its corners.
• A plane can also be named using any three or four points drawn on the edges of or within the parallelogram and labeled with letters.

Geometry Basics

• Geometry is a branch of mathematics that deals with shapes, measurements, and relationships of points, lines, and angles.
• A point is a location in space with no length or width, usually marked with a dot.
• A line is a straight series of points that extend in opposite directions infinitely, represented by an arrow at each end.
• A line segment is part of a line, has two endpoints, and does not extend infinitely.

Rays and Angles

• A ray is a one-dimensional, straight figure that begins at a single endpoint and extends in one direction indefinitely.
• Rays can be named by a single letter, their endpoint, or by two letters, their endpoint plus another point on the ray.
• A ray has a beginning point, but its length is infinite and cannot be measured.
• An angle is made up of two rays that share an endpoint, called the vertex.
• There are various types of angles, including right (90 degrees), acute (less than 90 degrees), straight (180 degrees), obtuse (greater than 90 degrees but less than 180 degrees), and reflex (greater than 180 degrees but less than 360 degrees).

Lines and Line Segments

• A line is a one-dimensional figure that extends indefinitely, contains an infinite number of points, and cannot be measured.
• Lines can be drawn on a plane, which is a two-dimensional figure that extends indefinitely in two directions (x and y).
• Lines contain an infinite number of points that have two values (x and y).
• There are three types of line pairs: parallel (never intersect), perpendicular (intersect at right angles), and transverse (cross two other lines in the same plane).
• The equation of a line is y = mx + b, where y and x are a point on the line, m is the slope, and b is where the line crosses the y-axis.

Intersecting Lines

• Lines and line segments are considered intersecting lines when they cross each other at one single point.
• The point where lines intersect is termed the point of intersection.
• Lines can intersect each other at any angle between 0 and 180 degrees.
• When intersecting lines cross at a right angle, they are called perpendicular lines.
• The slopes of perpendicular lines are negative reciprocals.

Perpendicularity

• Perpendicular means two lines or sides that form a right angle (90 degrees).
• Perpendicular line segments are line segments that intersect at a 90-degree angle.
• A perpendicular shape is a shape that has at least two sides that come together at a 90-degree angle.
• A right triangle has one right angle and two perpendicular lines.
• A rectangle and a square both have four right angles and four perpendicular lines.

Skew Lines

• Skew lines are defined as lines that are not parallel and do not intersect.
• Parallel lines are coplanar (lie in the same plane) and do not intersect.
• Skew lines are defined in three-dimensional space and have three conditions: not parallel, not intersecting, and not coplanar.
• The shortest distance between two skew lines is the line connecting them that is perpendicular to both.

Polygons

• A polygon is a two-dimensional closed region formed by at least three segments of a line that are consecutive.
• Two consecutive segments have a common endpoint, which are the vertices of the polygon.
• Polygons can be classified as regular (equilateral and equiangular) or irregular (not regular); concave or convex; and simple (segments intersect on vertices only) or complex (not simple).

Quadrilaterals

• A quadrilateral is a shape with four sides.
• The term "edges" refers to the sides of the shape.
• A vertex (plural: vertices) refers to the corners of a shape.
• Each vertex has an angle on the inside of the quadrilateral referred to as an interior angle.
• The angle outside of the quadrilateral is referred to as the exterior angle.
• In a quadrilateral, the sum of all four interior angles is always 360°.
• A convex quadrilateral is one where each of the four interior angles is less than 180°.
• A concave quadrilateral is one where at least one interior angle is greater than 180°.
• A complex quadrilateral is one where the sides intersect each other.

Learn about the fundamental concept of geometric planes, their characteristics and how they can be modeled in the real world.