Triangles

Concept Map

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Additional Resources

OER

• Web resources:

1. Bright Hub Education - Basic concepts of triangles,Types of triangles,Angle sum property,Exterior and interior angle relation
2. Folioz - Inequalities in triangle, Investigate the relationship between the  sides and angles in a triangle
3. JsunilTutorial - Test papers for triangles

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

1. Karnataka Govt Text book – Class 8

• Syllabus documents

Non-OER

• Web resources:

CPALMS -Introduction to triangles as a closed three sided figure,Inequalities in triangles

• Books and journals
• Textbooks
• Syllabus documents (CBSE, ICSE, IGCSE etc)

Learning Objectives

• Identify a triangle
• Recognize interior and exterior angles
• Classifying types of triangles
• Recognize the angle sum property
• Establish relation between interior and exterior angles

Concept #1. Formation of a triangle, elements of a triangle and its measures

1. A triangle is a three sided closed figure.
2. It is one of the basic shapes in geometry.
3. It triangle is a polygon with three edges and three vertices.
4. There are three angles in a triangle formed at the three vertices of the triangle.
5. Interior and exterior angles in a triangle at a vertex, together form a linear pair.

Activity No # 1 : Formation of a triangle

Formation of a triangle

• Objectives

1. Understand formation of triangles
2. Recognize elements of triangle
3. Introduce concepts of exterior angle.

• Pre-requisites

Prior knowledge of point, lines, angles, parallel linesResources needed

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil
3. Geogebra files : “Introduction to a triangle.ggb”

• How to do

1. Use the geogebra file to illustrate.
2. How many lines are there? Are the lines meeting?
3. Are the two lines parallel? How can you say they are parallel or not?
4. How many angles are formed at the point of intersection?
5. What is the measure of the total angle at the point of intersection of two lines?
6. Of the four angles formed which of the angles are equal? What are they called?
7. Do the three intersecting lines enclose a space? How does it look? It is called a triangle.
8. What are the points of intersection of these three lines called?
9. The line segments forming the triangle are called sides.
10. How many angles are formed when three lines intersect with each other?
11. How many angles are enclosed by the triangle?

• Evaluation at the end of the activity

1. Can there be a closed figure with less than three sides?
2. Can the vertices of the triangle be anywhere on a plane?
3. What will happen if the three vertices are collinear?

Activity No # 2 : Elements of a Triangle

• Objectives

1. To understand the elements of a triangle

• Pre-requisites

Prior knowledge of point, lines, angles, parallel lines

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil,6-8 strings (preferably in different colours)
3. Geogebra files : “Elements of a triangle.ggb”

• How to do

1. Students work individually but in their groups.
2. Take the strings and place them in such a way as to make a closed figure.
3. What is the smallest number of strings with which you can form a closed figure?
4. What is this figure called?
5. Can you just draw the lines along the strings and see what you get?
6. When you drew, what did you draw?  (Was it a line or was it an angle or was it a line segment?). It is a lime segment – how many line segments are there?
7. When two line segments joined, what is it called?  (A vertex). How many vertices are there?
8. Is there any angle formed when you made this figure? How many angles were formed?
9. Show a simple Geogebra file with triangles – Use this file to demonstrate that every triangle has the elements - vertices, sides and angles
10. How many triangles were formed?  Were there any strings left over?
11. For each of the triangles trace the shape on the book and write down the elements of the triangle in the following format

Vertices	Sides	Angles

1. For each of the triangles observe (inspect visually) which is the longest side and which is the shortest side

Triangle name	Largest angle	Largest side	Smallest angle	Smallest side

1. Allow the students to explore if there is any connection between the two?

1. After the students see the Geogebra file, they can attempt an alternative worksheet like below:

Side 1	Angle 1 (opposite angle)	! Side 2	Angle 2 (opposite angle)	! Side 3	Angle 3 (opposite angle)	! Largest side and angle ex-side1,angle1	! Smallest side and angle ex-side3,angle3

Evaluation at the end of the activity

1. Have the students been able to identify the elements in a triangle?

2. Have they been able to extrapolate any connection between the angle and side in a triangle?

Activity No # 3  : Measures associated in a triangle

• Objective:

1. To learn the different measurements in a triangle and the associated properties

• Pre-requisites

Prior knowledge of point, lines, angles, parallel lines

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil.
3. Geogebra files : “Measures in a triangle.ggb”

• How to do

1. Show the Geogebra file and ask students to record the values of angles and sides that are seen and ask if there is any connection between the side and the angle
2. What is the sum of the angles in a triangle?
3. Students make triangles picking any three strings from the set of strings they have been given. Is there any time when a triangle is not possible?
4. Side of triangle
5. Use the file “1b. Measures in a triangle.ggb”. This file can be used to help students’ conception of triangle in a generalized manner. This file can be used to illustrate revise the points about vertically opposite angles, adjacent angles etc,. Help students identify the triangle. (Use the transaction notes for this file as needed)
6. Ask the students to make another 3 triangles from the strings they have been given. Students should make a triangle in which all angles are acute and one in which one angle is obtuse. Use the following Geogebra file for types of triangles by angle.Have the students explore the types of angles in a triangle.
7. Ask the students to make another set of two or three triangles with the strings they have been given. Is there anything you can say about the sides of the triangle? Show the Geogebra file called Types of triangle by sides.

Evaluation:

1. Have the students been able to measure? Do they have an idea of what are the measurements possible?
2. Have they been able to generalize the sum of angle?
3. Have they been able to generalize any result about sides of a triangle?
4. Are the students able to recognize a triangle in a general manner?
5. Are they able to recognize types of triangles?

Activity No # 4  : Interior and Exterior angles in a triangle

• Objectives

1. Identify all angles when a triangle is formed
2. Understand the relation between various angles that are formed in a triangle.

• Pre-requisites

Prior knowledge of point, lines, angles

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil.
3. Geogebra files : “Angles of a triangle.ggb”

• How to do

1. Ask students how many lines are there? They should be able to identify the points of intersection of the lines. How many points of intersection are formed?
2. How many angles are formed at an intersecting point? How many angles in total at the three points of intersection?What is the total angle measure at each intersecting point?
3. How many angles are inside the triangle and how many are outside the triangle?
4. Can you find an exterior angle that is equal to the interior angle of a triangle at each vertex?Why are they equal?
5. Identify the exterior angles that are equal? Justify why they are equal.
6. Establish that there are 2 angles which are exterior of the triangle that are equal and are formed when the sides of the triangle is extended at the vertex.
7. Students to analyze the interior and exterior angle at each point to find a relation between the interior angle and one of the exterior angles at the vertex. Students should be able to recognize the linear pair formed by interior angle and exterior angle.
8. Vary the position of the lines to check if interior and exterior angles form a linear pair.

Note the measure of angles 

• Evaluation at the end of the activity

1. Are students able to recognize interior and exterior angles in a triangle
2. Have the students able to find a relation between the interior angle and exterior angle that are formed at each vertex?

Concept 2: Types of Triangles based on sides and angles

Types of triangles based on angles in the triangle

1. Acute triangles are triangles in which the measures of all three angles are less than 90^o.
2. Obtuse triangles are triangles in which the measure of one angle is greater than 90^o.
3. Right triangles are triangles in which the measure of one angle equals 90 degrees.

Types of triangles based on sides in the triangle

1. Equilateral triangles are triangles in which all three sides are the same length.
2. Isosceles triangles are triangles in which two of the sides are the same length.
3. Scalene triangles are triangles in which none of the sides are the same length.

Activity No # 5 : Types of triangles based on sides 

• Objectives

1. Recognize the triangles based on the measures of the sides

• Pre-requisites

1. Prior knowledge of point, lines, angles, elements of triangle

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil.
3. Geogebra files : “5. Types of triangle by sides.ggb”

• How to do

1. Students should recognize the elements of a triangle – sides and angles.
2. What are the measures of the sides of a triangle, are the measurements of the sides equal or different.
3. Establish that different triangles are formed with the different measures of the sides: when all sides are different, when any two sides are equal and when all sides are equal.
4. Children can note the measures of the sides in the worksheet for different triangles to conclude the type of triangles based on the sides.
5. Is the triangle formed for any measure of the sides. When does the triangle not form. What is the relation between the 3 sides for the triangle to be formed.

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• Evaluation at the end of the activity

1. Are children able to recognize the types of triangles when the sides are specified.
2. Have children been able to conclude when a triangle is formed given the three sides.

Activity No # 6  : Types of triangles based on angles

• Objectives

1. Recognize the triangles based on the measures of the sides

• Pre-requisites

1. Prior knowledge of point, lines, angles, elements of triangle

• Resources needed

1. Digital : Computer, geogebra application, projector.
2. Non digital : Worksheet and pencil.
3. Geogebra files : “6. Types of triangle by angle.ggb”

• How to do

1. Identify the angle types in the triangle.
2. Can the angles in the triangle be of different type – obtuse or right angle.
3. Establish the types of triangles based on the types of angles that form the triangle – when all angles are acute angles, when one of the angle is a right angle and when one of the angle is obtuse angle.
4. Is it possible to have a triangle with two right angle or two obtuse angle. Why or why not?
5. What kind of a triangle is formed when all the angles are equal? For what measure of the angle such a triangle is formed?
6. Can a triangle be formed with a reflex angle.
7. Measure the angles for different triangle, recognize the types of angles and conclude the type of triangle.

• Evaluation at the end of the activity

1. Are children able to recognize the types of triangles based on the angles in a triangle.

Concept 3: Angle Sum Property

Activity No # 7 :Angle sum property

• Objectives

1. To establish the angle sum property of a triangle
2. To help visualization of the geometric proof

• Pre-requisites

Prior understanding of point, lines and angles, adjacent angles, vertically opposite angles, linear pair, parallel lines, alternate angles, corresponding angles.

• Resources needed

1. Digital : Computer, geogebra application, projector.

2. Non digital : Worksheet and pencil.

3. Geogebra files : “7a. Angles in a right triangle.ggb” , “7b. Angle sum property proof.ggb” , “7c. Angle sum property of a triangle.ggb”

• How to do

1. Use the file - “7a. Angles in a right triangle.ggb”
2. Ask students what is the kind of triangle they observe.
3. Draw a parallel line to the side containing the right angle through the opposite vertex by dragging point D along the x-axis
4. Students should be able to recognize the corresponding angles formed when the parallel line is drawn.
5. Students should be able to recognize the alternate angles formed. Is the alternate angle same as one of the angles of the triangle.
6. So what can you say about the all the angles of the triangle?
7. With the file - “7b. Angle sum property proof.ggb”
8. Draw a line perpendicular to a side passing through the opposite vertex. How many triangles are formed. What kinds of triangles are formed?
9. In each of the two triangles if on angle is 90^o, what will be the sum of the other two angles. What is the sum of these angles?
10. Children can record the values of the angles of a triangle in the worksheet

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1. With the file – “7c. Angle sum property of a triangle.ggb”
2. Ask students what happens when the three angles of the triangle are placed adjacent to each other.
3. What can you say about the line drawn?
4. Is it parallel to one of the sides?
5. What can you say about the pairs of angles – look at the matching colors.
6. Once the parallel line reaches the vertex, how many angles are formed?
7. Students should be able to identify the two angles moving are corresponding angles as the line moving is parallel to one of the sides.
8. Students can see that when the three angles of the triangle are placed adjacent to each other they form a straight line.

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• Evaluation at the end of the activity

1. Have students able to conclude if the sum of angles in any triangle is 180^o?

Concept 4: Relation between interior and exterior angles in a triangle

Activity No # 8 : Relation between interior and exterior angles in a triangle

• Objectives

1. To show interior angles of a triangle have a relation with its exterior angles.

• Pre-requisites

Prior understanding of point, lines and angles, adjacent angles, vertically opposite angles, linear pair

• Resources needed

1.Digital : Computer, geogebra application, projector.

2. Non digital : Worksheet and pencil.

3. Geogebra files : “8a. EA= Sum of opposite IAs in a triangle proof.ggb” ,“8b. EA= Sum of opposite IAs in a triangle.ggb” ,“8c. EA= Sum of opposite IAs in a triangle demo.ggb”

• How to do 1. In the triangle students should identify the angles of the triangle. 2. Extend one side, students should recognize the exterior angle formed. 3. What is the sum of the angles of a triangle? 4. Students should be able to recognize the alternate angle formed for one of the interior angle(Angle BAC) 5. Drag the parallel line to the opposite vertex, to place the alternate angle next to the angle at the opposite vertex. 6. Compare the angles formed and the exterior angle, do they have a relation. 7. How are the two angles together related to the exterior angle? 8. Do you notice any relation between the exterior angle and the interior angles 9. If you know the measure of interior angle can you find the corresponding exterior angle? 10. The other two files can be used to demonstrate the the relation between the exterior angle and opposite interior angles.

Note the measure of angles

Work sheet

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• Evaluation at the end of the activity 1. Have the students able to identify the relation between  exterior and interior opposite angles of a triangle?