Triangles

OER

 * Web resources:
 * Wikipedia, the free encyclopedia : The website gives a comprehensive information on triangles from basics to in depth understanding of the topic.
 * cbsemathstudy.blogspot.com : The website contains worksheets that can be downloaded. Worksheets for other chapters can also be searched for which are listed based on class.


 * Books and journals
 * Textbooks
 * NCERT Textbooks – Class 9


 * Syllabus documents

Non-OER

 * Web resources:
 * Bright hub education : The website describes a lesson plan for introducing triangles and lists classroom problems at the end of the lesson that can be solved for better understanding.
 * CPALMS : The website contains lesson plan, activities and worksheets associated with triangles.
 * UrbanPro : This website gives downloadable worksheets with problems on triangle construction.
 * AglaSem Schools : This website lists important questions for math constructions.


 * Books and journals
 * Textbooks:
 * Karnataka Govt Text book – Class 8 : Part 1, Part 2
 * Syllabus documents (CBSE, ICSE, IGCSE etc)

Learning Objectives

 * Identifying a triangle and understanding its formation
 * Recognizing parameters related to triangles
 * Understanding different formation of triangles based on sides and angles
 * Establishing relationship between parameters associated with triangles

Concept 1: Formation of a triangle, elements of a triangle and its measures
The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on.These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic parts on which most other figures depend. Here we will be investigating triangles and related its properties

Formation of a triangle
Introducing formation of a shape with least number of lines and the space enclosed by these lines form a geometric shape.The key geometric concepts that are related with this are explained.

Elements and measures in triangle
The components that make a triangle are investigated. Measuring these components gives a better understanding of properties of triangles. Relation between these components are conceptualized.

Interior and exterior angles in triangle
Interior angles are angles that are formed with in the closed figure by the adjacent sides.An exterior angle is an angle formed by a side and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles.

Concept 2: Types of triangles based on sides and angles
Variations in elements that make a triangle results in distinct triangles. Recognizing these variations helps in interpreting changes that are possible with in a triangle.

Types of triangles based on sides
A triangle can be drawn with different measures of sides and these sides determine the kind of triangle formed.

Types of triangles based on angles
A triangle can be drawn with different measures of angles which also determine the kind of triangle formed.

Concept 3: Theorems and properties
Properties of triangles are logically proved by deductive method. Relations ships between angles of a triangle when a triangle is formed are recognized and understood.

Angle sum property
Interior angles of a triangle are in relation and also determine the type of angles that can forms a triangle. This also helps in determining an unknown angle measurement.

Exterior angle theorem
Interior angle and the corresponding angle form a linear pair. This exterior angle in relation to the remote interior angles and their dependencies are deducted with the theorem.

Concept 4: Construction of triangles
Constructing geometric shapes to precision using a scale and a compass helps in understanding of properties of the shape. Constructing geometric shapes with minimum number of parameters enhances thinking skills.

The following constructions are based on three essential parameters that are required for construction following the SSS, SAS and ASA theorems

Construction of a triangle with three sides
Investigating formation of a unique triangle with the given parameters as the three sides. Constriction follows SSS congruence rule.

Construction of a triangle with two sides and an angle
Construing of a triangle when two of the sides and an angle of a triangle are known and recognizing the role of the given angle, this construction follows SAS congruence rule for the given parameters.

Construction of a triangle with two angles and included side
Construction of a triangle when two angles of a triangle and a side are known and understanding the side given can only be constructed between the two angles to form a unique triangle. Construction follows ASA congruence rule.

Construction of a right angled triangle
Right angle is one of the angles of the triangle the and assimilating other parameters that are required to complete construction. Construction of a triangle based on RH congruence rule.

Construction of a triangle with a side, an angle and sum of two sides
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Construction of a triangle with given parameters sum of two sides and an angle follows SAS congruence rule.

Construction of a triangle with a side, an angle and difference to two sides
Difference of two sides and an angle are parameters with which a triangle construction is possible, this construction of triangle follows SAS congruence rule.

Construction of a triangle with perimeter and base angles
Constructing a unique triangle with perimeter and with other two parameters the base angles of the triangle follows construction on a triangle using SAS congruence rule.

Concept 5: Concurrency in triangles
Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is the point of concurrency

In a triangle, the following sets of lines are concurrent:

The three medians.

The three altitudes.

The perpendicular bisectors of each of the three sides of a triangle.

The three angle bisectors of each angle in the triangle.

Concept 6: Similar and congruent triangles
Two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, they are congruent. Two triangles are said to be congruent to one another only if their corresponding sides and angles are equal to one another