Lines and Angles
|Philosophy of Mathematics
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- Web resources:
- CK12 - This website gives basic facts about parallel lines with examples and review questions.
- Education.com - An exemplar for a lesson plan for lines and angles is on the web site.
- Project Maths - A detailed lesson plan for the topic is available in pdf format.
- Teachers.net - An exemplar for discussion based approach for to introduce lines and angles is shown.
- CLAMS - This is an interactive lesson exploring Parallel and Perpendicular lines.
- CK – 12 - Basic facts of parallel lines are discussed and introduced.
- JsunilTutorial - Worksheet for the chapter can be downloaded.
- Books and journals
- Syllabus documents
- YouTube Videos
Watch Video on Vertical Opposite angles and Pair of Linear Equations
Watch video on parallel lines and transversal
- Introducing lines and line segments
- Understanding formation of angles and their types
- Differentiating parallel lines and skew lines
- Recognizing angles pairs of angles formed in parallel lines
Concept 1: Angles
An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. angles can be classified according to the size of the angle.
The standard angle concept based on the relative inclination of two lines meeting at a point irrespective of length of arms is discussed in an exploratory method.
We will learn the following types of angles: right angles, acute angles, obtuse angles, straight angles, reflex angles and complete angle.
Concept 2: Pairs of angles
In geometry, certain pairs of angles can have special relationships. Some examples are complementary angles, supplementary angles, vertical angles, alternate interior angles, alternate exterior angles, corresponding angles and adjacent angles.
Adjacent angles are two angles that have a common vertex and a common side. The vertex of an angle is the endpoint of the rays that form the sides of the angle.
Complementary angles are two angles with a sum of 90°. A common case is when they form a right angle.
Supplementary angles are two angles with a sum of 180°. A common case is when they lie on the same side of a straight line.
When two straight lines intersect each other four angles are formed.The pair of angles which lie on the opposite sides of the point of intersection are vertically opposite angles.
Linear pair axiom : If a ray stands on a line, then the sum of two adjacent angles so formed is 180o
Two adjacent angles are said to be form a linear pair of angles, if their non-common arms are two opposite rays. Linear pair axiom of theorems are if a ray stands on a line , then the sum of two adjacent angles so formed is 180 degree
Linear pair axiom : If the sum of two adjacent angles is 180o, then the non-common arms of the angles form a line
Concept 3: Parallel lines
Parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel.
Parallel lines are lines that are always the same distance apart. Because they are always the same distance from one another, parallel lines will never intersect.
Two parallel lines are cut by a transversal the angles formed are alternate angles, corresponding angles co-interior angles and vertically opposite angles.
The relation between angles that are formed in multiple parallel lines is investigated with geogebra sketch.