Difference between revisions of "Triangles"

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=Concept Map =
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<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#ffffff; vertical-align:top; text-align:center; padding:5px;">
[[File:5. Triangles.mm]]
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''[https://karnatakaeducation.org.in/KOER/index.php/%e0%b2%a4%e0%b3%8d%e0%b2%b0%e0%b2%bf%e0%b2%ad%e0%b3%81%e0%b2%9c%e0%b2%97%e0%b2%b3%e0%b3%81 ಕನ್ನಡ]''</div>
  
=Additional Resources=
+
===Concept Map===
==OER==
+
{{#drawio:mmTriangles|interactive}}
* Web resources:
 
# [https://www.brighthubeducation.com/middle-school-math-lessons/39674-triangle-properties-and-angles/ Bright Hub Education] - Basic concepts of triangles,Types of triangles,Angle sum property,Exterior and interior angle relation
 
# [http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 Folioz] - Inequalities in triangle, Investigate the relationship between the  sides and angles in a triangle
 
# [https://jsuniltutorial.weebly.com/ix-triangles.html JsunilTutoria]l - Test papers for triangles
 
* Books and journals
 
* Textbooks
 
* NCERT Textbooks – Class 9
 
# Karnataka Govt Text book – Class 8
 
* Syllabus documents
 
  
==Non-OER==
+
===Learning Objectives===
* Web resources:
+
* Identifying a triangle and understanding its formation
[http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 CPALMS -Introduction to triangles as a closed three sided figure,Inequalities in triangles]
+
* Recognizing parameters related to triangles
* Books and journals
+
* Understanding different formation of triangles based on sides and angles
* Textbooks
+
* Establishing relationship between parameters associated with triangles
* Syllabus documents (CBSE, ICSE, IGCSE etc)
 
  
=Learning Objectives=
+
===Teaching Outlines===
* Identify a triangle
 
* Recognize interior and exterior angles
 
* Classifying types of triangles
 
* Recognize the angle sum property
 
* Establish relation between interior and exterior angles
 
  
==Teaching Outlines==
+
==== Concept #: 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 element on which most other figures depend. Here we will be investigating triangles and related its properties
  
== Concept #1. Formation of a triangle, elements of a triangle and its measures ==
+
===== Activities # =====
# A triangle is a three sided closed figure.
 
# It is one of the basic shapes in geometry.
 
# It triangle is a polygon with three edges and three vertices.
 
# There are three angles in a triangle formed at the three vertices of the triangle.
 
# 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]] ======
[[File:Triangle formation.png|thumb|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. 
* '''Objectives'''
 
# Understand formation of triangles
 
# Recognize elements of triangle
 
# Introduce  concepts of exterior angle.
 
* '''Pre-requisites'''
 
# Prior knowledge of point, lines, angles, parallel linesResources needed
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil
 
# Geogebra files :  '''“[https://www.geogebra.org/m/bwsvgqqg#material/z4h42k8z Introduction to a triangle.ggb]”'''
 
* '''How to do'''
 
# Use the geogebra file to illustrate.
 
# How many lines are there? Are the lines meeting?
 
# Are the two lines parallel? How can you say they are parallel or not?
 
# How many angles are formed at the point of intersection?
 
# What is the measure of the total angle at the point of intersection of two lines?
 
# Of the four angles formed which of the angles are equal? What are they called?
 
# Do the three intersecting lines enclose a space? How does it look? It is called a triangle.
 
# What are the points of intersection of these three lines called?
 
# The line segments forming the triangle are called sides.
 
# How many angles are formed when three lines intersect with each other?
 
# How many angles are enclosed by the triangle?
 
* '''Evaluation at the end of the activity'''
 
# Can there be a closed figure with less than three sides?
 
# Can the vertices of the triangle be anywhere on a plane?
 
# What will happen if the three vertices are collinear?
 
  
=== Activity No # 2 :  Elements of a Triangle ===
+
====== [[Measures in triangle|Elements and measures in triangle]] ======
* '''Objectives'''
+
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.
# To understand the elements of a triangle
 
* '''Pre-requisites'''
 
# Prior knowledge of point, lines, angles, parallel lines
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil,6-8 strings (preferably in different colours)
 
# Geogebra files :  '''“[https://www.geogebra.org/m/bwsvgqqg#material/hyecxd9u Elements of a triangle.ggb]”'''
 
* '''How to do'''
 
# Students work individually but in their groups.
 
# Take the strings and place them in such a way as to make a closed figure.
 
# What is the smallest number of strings with which you can form a closed figure?
 
# What is this figure called?
 
# Can you just draw the lines along the strings and see what you get?
 
# 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?
 
# When two line segments joined, what is it called?  (A vertex). How many vertices are there?
 
# Is there any angle formed when you made this figure? How many angles were formed?
 
# Show a simple Geogebra file with triangles – Use this file to demonstrate that every triangle has the elements - vertices, sides and angles
 
# How many triangles were formed?  Were there any strings left over?
 
# For each of the triangles trace the shape on the book and write down the elements of the triangle in the following format
 
{| class="wikitable"
 
|-
 
! Vertices !! Sides !! Angles
 
|-
 
|  ||  ||
 
|-
 
|  ||  ||
 
|-
 
|  ||  ||
 
|}
 
12.For each of the triangles observe (inspect visually) which is the longest side and which is the shortest side
 
{| class="wikitable"
 
|-
 
! Triangle name !! Largest angle !! Largest side
 
!Smallest angle
 
!Smallest side
 
|-
 
| || ||
 
|
 
|
 
|-
 
| || ||
 
|
 
|
 
|-
 
| || ||
 
|
 
|
 
|}
 
13. Allow the students to explore if there is any connection between the two?
 
  
14. After the students see the Geogebra file, they can attempt an alternative worksheet like below:
+
====== [[Interior and exterior angles in triangle]] ======
{| class="wikitable"
+
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.
|-
 
! 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'''
 
# Have the students been able to identify the elements in a triangle?
 
# Have they been able to extrapolate any connection between the angle and side in a triangle?
 
 
[[Category:Triangles]]
 
[[Category:Triangles]]
[[Category:Class 9]]
+
[[Category:Class 10]]
 +
 
 +
==== Concept #: 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.
 +
 
 +
'''Video resource:''' Explanation of Types of Triangles by NCERT.
 +
 
 +
{{Youtube|SXYZd536Vao
 +
}}
 +
 
 +
===== Activities # =====
 +
 
 +
====== [[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 #: 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.
 +
 
 +
===== Activities #  =====
 +
 
 +
====== [[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.
 +
 
 +
'''Video resource:''' Classroom activity for angle sum property of Triangle by NCERT
 +
 
 +
{{Youtube|BRDAXvQlzt0
 +
}}
 +
 
 +
====== [[Angle sum property of a Triangle]] ======
 +
 
 +
====== [[Relation between interior and exterior angles in triangle|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.
 +
 
 +
====== [[Exterior angle property of a Triangle]] ======
 +
 
 +
==== Concept #: 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
 +
 
 +
===== Activities # =====
 +
 
 +
====== [[Construction of a triangle with 3 sides|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 included angle|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 triangle with a side and hypotenuse|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 side, angle and sum of other two sides|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 side, angle and difference of other two sides|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 #: Concurrency in triangles ====
 +
A set of lines are said to be concurrent if they all intersect at the same point. In the figure below, the three lines are concurrent because they all intersect at a single point P. The point P is called the "point of concurrency". This concept appears in the various centers of a triangle.
 +
 
 +
[[File:concurrent lines.jpeg|100px|link=http://karnatakaeducation.org.in/KOER/en/index.php/File:Concurrent_lines.jpeg]]
 +
 
 +
All non-parallel lines are concurrent.
 +
 
 +
Rays and line segments may, or may not be concurrent, even when not parallel.
 +
 
 +
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.
 +
 
 +
The medians, altitudes, perpendicular bisectors and angle bisectors of a triangle are all concurrent lines. Their point of intersections are called centroid, orthocentre, circumcentre and incentre  respectively. Concurrent lines are especially important in triangle geometry, as the three-sided nature of a triangle means there are several special examples of concurrent lines, including the centroid, the circumcenter and the orthocenter.
 +
 
 +
These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle  are collinear  that is they lie on the same straight line  called the Euler line.
 +
======Activities #======
 +
======[[Exploring concurrent lines from given surroundings]]======
 +
Interactive activity to introduce concurrent lines using examples from our surroundings.
 +
=====Concept #: Concurrency of medians in triangles.=====
 +
Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.
 +
 
 +
From Latin: centrum - &quot;center&quot;, and Greek: -oid -&quot;like&quot;  The centroid of a triangle is the point through which all the mass of a triangular plate seems to act.  Also known as its 'center of gravity' , 'center of mass' , or barycenter. Refer to the figure . Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid. In the diagram, the medians of the triangle are shown as dotted blue lines.
 +
 
 +
[[Image:KOER%20Triangles_html_m404a4c0b.gif|link=]]
 +
======Activities #======
 +
======[[Marking centroid of a triangle|Marking centroid of the triangle]]======
 +
This is a hands on activity to explore concurrent lines formed in a triangle when vertices are joined to the midpoints of the opposite side.
 +
======[[Medians and centroid of a triangle]]======
 +
The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid  and you’ll explore several geometric relationships related to centroid and medians.
 +
=====Concept #: Concurrency of altitudes in triangles=====
 +
The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - &quot;straight, true, correct, regular&quot; The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.
 +
======Activities #======
 +
======[[Altitudes and orthocenter of a triangle]]======
 +
An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side.  A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.
 +
=====Concept #:  Concurrency of Perpendicular bisectors in a triangle=====
 +
The perpendicular bisector of a triangle is the perpendicular drawn to a line segment which divides it into two equal parts. The point where the three perpendicular bisectors of a triangle meet is called the circumcentre of a triangle. The circumcentre of a triangle is equidistant from all the three sides vertices of the triangle. This common distance is the crcumradius. The circumcentre is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. The circumcentre of a triangle lies inside or on a side or outside the triangle according as the triangle is acute or right angled or obtuse. The circumcentre of a right angled triangle is the mid-point of its hypotenuse. Latin: circum - "around" centrum - "center"
 +
 
 +
One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
 +
======Activities #======
 +
[[Perpendicular bisectors and circumcenter of a triangle|'''Perpendicular bisectors and circumcenter of a triangle''']]
 +
 
 +
Circumcentre for different types of triangles is investigated with this activity and this further explores several geometric relationships related to the circumcentre and perpendicular bisectors.
 +
=====Concept #: Concurrency of angle bisectors in triangles.=====
 +
The ray which bisects an angle is called the angle bisector of a triangle. The point of concurrence of angle bisectors of a triangle is called as incentre of the triangle. The incentre always lies inside the triangle. The distance from incentre to all the sides are equal and is referred to as inradius. The circle drawn with inradius is called incircle and touches all sides of the triangle.
 +
 
 +
If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.
 +
======Activities #======
 +
======[[Angular bisectors and incenter of a triangle]]======
 +
The intersecting point of three lines which are the bisectors of three angles of a triangle that is the incenter and it's properties are examined.
  
=== Activity No # : Measures associated in a triangle ===
+
==== Concept #: [[Similar and congruent triangles]] ====
* '''Objective''':
+
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
# To learn the different measurements in a triangle and the associated properties
 
* '''Pre-requisites'''
 
Prior knowledge of point, lines, angles, parallel lines
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil.
 
# Geogebra files :  '''“[https://www.geogebra.org/m/bwsvgqqg#material/jjmbu9ed Measures in a triangle.ggb]”'''
 
* '''How to do'''
 
# 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
 
# What is the sum of the angles in a triangle?
 
# 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?
 
# Side of triangle
 
# 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)
 
# 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.
 
# 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''':
+
==== Concept #: [https://karnatakaeducation.org.in/KOER/en/index.php/Basic_Proportionality_Theorem Basic Proportionality Theorem] ====
# Have the students been able to measure? Do they have an idea of what are the measurements possible?
+
The concept of Thales theorem has been introduced in similar triangles. If the given two triangles are similar to each other then,
# Have they been able to generalize the sum of angle?
 
# Have they been able to generalize any result about sides of a triangle?
 
# Are the students able to recognize a triangle in a general manner?
 
# Are they able to recognize types of triangles?
 
  
=== Activity No # 4  : Interior and Exterior angles in a triangle ===
+
*    Corresponding angles of both the triangles are equal
* '''Objectives'''
+
*    Corresponding sides of both the triangles are in proportion to each other
# Identify all angles when a triangle is formed
 
# Understand the relation between various angles that are formed in a triangle.
 
* '''Pre-requisites'''
 
# Prior knowledge of point, lines, angles
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil.
 
# Geogebra files :  '''“[https://www.geogebra.org/m/bwsvgqqg#material/q6fnttmn Angles of a triangle.ggb]”'''
 
* '''How to do'''
 
# 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?
 
# 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?
 
# How many angles are inside the triangle and how many are outside the triangle
 
# Can you find an exterior angle that is equal to the interior angle of a triangle at each vertex?Why are they equal?
 
# Identify the exterior angles that are equal? Justify why they are equal.
 
# 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.
 
# 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.
 
# Vary the position of the lines to check if interior and exterior angles form a linear pair.
 
Note the measure of angles 
 
{| class="wikitable"
 
!Triangle
 
!Angle A
 
!Angle B
 
!Angle C
 
!Exterior angle
 
!Angle A + Angle B
 
|-
 
|Triangle1
 
|
 
|
 
|
 
|
 
|
 
|-
 
|Triangle2
 
|
 
|
 
|
 
|
 
|
 
|-
 
|Triangle3
 
|
 
|
 
|
 
|
 
|
 
|} 
 
* '''Evaluation at the end of the activity'''
 
# Are  students able to recognize interior and exterior angles in a  triangle
 
# 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 ==
+
==== Solved problems/ key questions (earlier was hints for problems) ====
'''Types of triangles based on angles in the triangle'''
 
# Acute triangles are triangles in which the measures of all three angles are less than 90<sup>o</sup>.
 
# Obtuse triangles are triangles in which the measure of one angle is greater than 90<sup>o</sup>.
 
# Right triangles are triangles in which the measure of one angle equals 90 degrees.
 
'''Types of triangles based on sides in the triangle'''
 
# Equilateral triangles are triangles in which all three sides are the same length.
 
# Isosceles triangles are triangles in which two of the sides are the same length.
 
# Scalene triangles are triangles in which none of the sides are the same length.
 
  
=== Activity No # 5 : Types of triangles based on sides ===
+
===Additional Resources===
* '''Objectives'''
 
# Recognize  the triangles based on the measures of the sides
 
* '''Pre-requisites'''
 
# Prior knowledge of point, lines, angles, elements of triangle
 
* '''Resources needed'''
 
# Digital : Computer, geogebra  application, projector.
 
# Non  digital : Worksheet  and pencil.
 
# Geogebra files :  “'''[https://ggbm.at/zhhmxqkh 5. Types of  triangle by sides.ggb]'''”
 
* '''How to do'''
 
# Students should recognize the elements of a triangle – sides and  angles.
 
# What are the measures of the sides of a triangle, are the measurements of the sides equal or different.
 
# 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.
 
# Children can note the measures of the sides in the worksheet for different triangles to conclude the type of triangles based on the sides.
 
# 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.
 
{| class="wikitable"
 
!Observation
 
!Side 1
 
!Side 2
 
!Side 3
 
!What can you say about sides?
 
!Type of triangle formed
 
|-
 
|Triangle 1
 
|
 
|
 
|
 
|
 
|
 
|-
 
|Triangle 2
 
|
 
|
 
|
 
|
 
|
 
|-
 
|Triangle 3
 
|
 
|
 
|
 
|
 
|
 
|}
 
  
* '''Evaluation at the end of the activity'''
+
==== Resource Title and description ====
# Are children able to recognize the types of triangles when the sides are specified.
+
1.[http://www.mathopenref.com/tocs/triangletoc.html Triangles]- This resource contain all information related to triangle like definition, types of triangle, perimeter, congruency etc.,
# Have children been able to conclude when a triangle is formed given the three sides.
 
  
=== Activity No # 6  : Types of triangles based on angles ===
+
2. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRIA Triangles]- This helps to know the aplication of geometry in our daily life, it contain videos and interactives.
* '''Objectives'''
 
# Recognize the triangles based on the measures of the sides
 
* '''Pre-requisites'''
 
# Prior knowledge of point, lines, angles, elements of triangle
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil.
 
# Geogebra files :  “6. Types of triangle by angle.ggb”
 
* '''How to do'''
 
# Identify  the angle types in the triangle.
 
# Can  the angles in the triangle be of different type – obtuse or right  angle.
 
# 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.
 
# Is  it possible to have a triangle with two right angle or two obtuse  angle. Why or why not?
 
# What  kind of a triangle is formed when all the angles are equal? For  what measure of the angle such a triangle is formed?
 
# Can  a triangle be formed with a reflex angle.
 
# Measure  the angles for different triangle, recognize the types of angles  and conclude the type of triangle.
 
{| class="wikitable"
 
!Observation
 
!Angle 1 measure and
 
  
its Type
+
3. [http://www.mathopenref.com/tocs/congruencetoc.html Congruence]
!Angle 2 measure and
 
  
its Type
+
4. [http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.SIM Similarity and Congruence]
!Angle 3 measure and  
 
  
its Type
+
5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S
!What can you say about angles?
 
!Type of triangle formed
 
|-
 
|Triangle 1
 
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|Triangle 2
 
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|Triangle 3
 
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|}
 
* '''Evaluation at the end of the activity'''
 
# Are children able to recognize the types of triangles based on the angles in a triangle.
 
  
== Concept 3: Angle Sum Property ==
+
{{#widget:YouTube|id=GfP8M5GwcdQ}}
  
=== Activity No # 7 :Angle sum property  ===
+
====OER====
* '''Objectives'''
+
* Web resources:
# To establish the angle sum property of a triangle
+
** [https://en.wikipedia.org/wiki/Triangle Wikipedia, the free encyclopedia] : The website gives a comprehensive information on triangles from basics to in depth understanding of the topic.
# To help visualization of the geometric proof
+
** [https://jsuniltutorial.weebly.com/ix-triangles.html 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.
* '''Pre-requisites'''
 
# Prior understanding of point, lines and angles, adjacent angles, vertically opposite angles, linear pair, parallel lines, alternate angles,      corresponding angles.
 
* '''Resources needed'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil.
 
# 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'''
 
# Use the file - “7a. Angles in a right triangle.ggb”
 
# Ask students what is the kind of triangle they observe.
 
# Draw a parallel line to the side containing the right angle through the opposite vertex by dragging point D along the x-axis
 
# Students should be able to recognize the corresponding angles formed when the parallel line is drawn.
 
# Students should be able to recognize the alternate angles formed. Is the alternate angle same as one of the angles of the triangle.
 
# So what can you say about the all the angles of the triangle?
 
# With the file - “7b. Angle sum property proof.ggb”
 
# Draw a line perpendicular to a side passing through the opposite vertex. How many triangles are formed. What kinds of triangles are formed?
 
# In each of the two triangles if on angle is 90<sup>o</sup>, what will be the sum of the other two angles. What is the sum of these angles?
 
# Children can record the values of the angles of a triangle in the worksheet
 
{| class="wikitable"
 
!Observation
 
!Angle 1
 
!Angle 2
 
!Angle 3
 
!Angle 1 + Angle 2 + Angle 3
 
!What can you say about sum of angles?
 
|-
 
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|-
 
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|}
 
# With the file – “7c. Angle sum property of a triangle.ggb”
 
# Ask students what happens when the three angles of the triangle are placed adjacent to each other.
 
# What can you say about the line drawn?
 
# Is it parallel to one of the sides?
 
# What can you say about the pairs of angles – look at the matching colors.
 
# Once the parallel line reaches the vertex, how many angles are formed?
 
# Students should be able to identify the two angles moving are corresponding angles as the line moving is parallel to one of the sides.
 
# Students can see that when the three angles of the triangle are placed adjacent  to each other they form a straight line.
 
* Evaluation at the end of the activity
 
# Have  students able to conclude if the sum of angles in any triangle is  180<sup>o</sup>?
 
  
== Concept 4: Relation between interior and exterior angles in a triangle ==
+
* Books and journals
 +
* Textbooks
 +
** NCERT Textbooks – [http://ncert.nic.in/textbook/textbook.htm?iemh1=7-15 Class 9]
  
=== Activity No # 8 : Relation between interior and exterior angles in a triangle ===
+
* Syllabus documents
* '''Objectives'''
 
# 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'''
 
# Digital : Computer, geogebra application, projector.
 
# Non digital : Worksheet and pencil.
 
# 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''' 
 
# In the triangle students should identify the angles of the triangle. 
 
# Extend one side, students should recognize the exterior angle formed. 
 
# What is the sum of the angles of a triangle? 
 
# Students should be able to recognize the alternate angle formed for one of the interior angle(Angle BAC) 
 
# Drag the parallel line to the opposite vertex, to place the alternate angle next to the angle at the opposite vertex. 
 
# Compare the angles formed and the exterior angle, do they have a relation. 
 
# How are the two angles together related to the exterior angle? 
 
# Do you notice any relation between the exterior angle and the interior angles 
 
# If you know the measure of interior angle can you find the corresponding exterior angle? 
 
# 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
+
====Non-OER====
{| class="wikitable"
+
* Web resources:
!Triangle
+
** [https://www.brighthubeducation.com/middle-school-math-lessons/39674-triangle-properties-and-angles/ 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.
!Angle A
+
** [http://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261 CPALMS] : The website contains lesson plan, activities and worksheets associated with triangles.
!Angle B
+
** [https://www.urbanpro.com/cbse-class-9-maths-construction UrbanPro] : This website gives downloadable worksheets with problems on triangle construction.
!Angle C
+
** [https://schools.aglasem.com/59755 AglaSem Schools] : This website lists important questions for math constructions.
!Exterior angle
+
** [http://www.nios.ac.in/media/documents/SecMathcour/Eng/Chapter-12.pdf National Institute of Open Schooling]:  This website has good reference notes on concurrency of lines in a triangle for both students and teachers.
!Angle A + Angle B
+
** [http://www.mathopenref.com/ Math Open Reference] : This website gives activities that can be tried and manipulated online for topics on geometry.
|-
+
 
|Triangle1
+
* Books and journals
|
+
* Textbooks:
|
+
** Karnataka Govt Text book – Class 8 : [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-english-maths-1.pdf Part 1] , [http://ktbs.kar.nic.in/New/website%20textbooks/class8/8th-kannada-maths-2.pdf Part 2]
|
+
* Syllabus documents (CBSE, ICSE, IGCSE etc)
|
+
 
|
+
=== Projects (can include math lab/ science lab/ language lab) ===
|-
+
'''''Laboratory Manuals''' - Mathematics : [https://ncert.nic.in/pdf/publication/sciencelaboratorymanuals/classIXtoX/mathematics/lelm402.pdf Click here] to refer activity 15,16,18 and 20'' which explains the properties of Triangle.
|Triangle2
+
 
|
+
=== Assessments - question banks, formative assessment activities and summative assessment activities ===
|
+
# [[:File:Introducction to Triangles.pdf|Introduction to triangles]]
|
+
# [[:File:Types of Triangle by sides.pdf|Types of Triangle by sides]]
|
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# [[:File:Properties of Triangles.pdf|Properties of Triangle]]
|
+
[[Category:Class 9]]
|-
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[[Category:Class 8]]
|Triangle3
 
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|}
 
* Evaluation at the end of the activity 
 
# Have the students able to identify the relation between  exterior and interior opposite angles of a triangle?
 

Latest revision as of 12:03, 3 July 2023

ಕನ್ನಡ

Concept Map

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

Teaching Outlines

Concept #: 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 element on which most other figures depend. Here we will be investigating triangles and related its properties

Activities #
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 #: 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.

Video resource: Explanation of Types of Triangles by NCERT.


Activities #
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 #: 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.

Activities #
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.

Video resource: Classroom activity for angle sum property of Triangle by NCERT


Angle sum property of a Triangle
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.

Exterior angle property of a Triangle

Concept #: 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

Activities #
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 #: Concurrency in triangles

A set of lines are said to be concurrent if they all intersect at the same point. In the figure below, the three lines are concurrent because they all intersect at a single point P. The point P is called the "point of concurrency". This concept appears in the various centers of a triangle.

Concurrent lines.jpeg

All non-parallel lines are concurrent.

Rays and line segments may, or may not be concurrent, even when not parallel.

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.

The medians, altitudes, perpendicular bisectors and angle bisectors of a triangle are all concurrent lines. Their point of intersections are called centroid, orthocentre, circumcentre and incentre respectively. Concurrent lines are especially important in triangle geometry, as the three-sided nature of a triangle means there are several special examples of concurrent lines, including the centroid, the circumcenter and the orthocenter.

These points of concurrencies, orthocenter, centroid, and circumcenter of any triangle are collinear that is they lie on the same straight line called the Euler line.

Activities #
Exploring concurrent lines from given surroundings

Interactive activity to introduce concurrent lines using examples from our surroundings.

Concept #: Concurrency of medians in triangles.

Median of a triangle is a line segment from a vertex to the midpoint of the opposite side. A triangle has three medians. Each median divides the triangle into two smaller triangles of equal area. The medians of a triangle are concurrent and the point of concurrence is called the centroid. The centroid is always inside the triangle. The centroid is exactly two-thirds the way along each median. i.e the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

From Latin: centrum - "center", and Greek: -oid -"like" The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter. Refer to the figure . Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid. In the diagram, the medians of the triangle are shown as dotted blue lines.

KOER Triangles html m404a4c0b.gif

Activities #
Marking centroid of the triangle

This is a hands on activity to explore concurrent lines formed in a triangle when vertices are joined to the midpoints of the opposite side.

Medians and centroid of a triangle

The centroid of a triangle is where the three medians intersect. This activity will show you how to find the centroid and you’ll explore several geometric relationships related to centroid and medians.

Concept #: Concurrency of altitudes in triangles

The distance between a vertex of a triangle and the opposite side is called the altitude of the triangle. Altitude also refers to the length of the segment. Altitudes can be used to compute the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. A triangle has 3 altitudes. The intersecting point of 3 altitudes of a triangle is known as orthocentre of the triangle. This point may be inside, outside, or on the triangle. If the triangle is obtuse, it will be outside. If the triangle is acute, the orthocentre is inside the triangle. The orthocenter on a right triangle would be directly on the 90° vertex. From Greek: orthos - "straight, true, correct, regular" The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.

Activities #
Altitudes and orthocenter of a triangle

An altitude of a triangle is a line segment that is drawn from the vertex to the opposite side and is perpendicular to the side.  A triangle can have three altitudes. Point of intersection of these lines for different types of triangles is explored.

Concept #: Concurrency of Perpendicular bisectors in a triangle

The perpendicular bisector of a triangle is the perpendicular drawn to a line segment which divides it into two equal parts. The point where the three perpendicular bisectors of a triangle meet is called the circumcentre of a triangle. The circumcentre of a triangle is equidistant from all the three sides vertices of the triangle. This common distance is the crcumradius. The circumcentre is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. The circumcentre of a triangle lies inside or on a side or outside the triangle according as the triangle is acute or right angled or obtuse. The circumcentre of a right angled triangle is the mid-point of its hypotenuse. Latin: circum - "around" centrum - "center"

One consequence of the Perpendicular Bisector Theorem is that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

Activities #

Perpendicular bisectors and circumcenter of a triangle

Circumcentre for different types of triangles is investigated with this activity and this further explores several geometric relationships related to the circumcentre and perpendicular bisectors.

Concept #: Concurrency of angle bisectors in triangles.

The ray which bisects an angle is called the angle bisector of a triangle. The point of concurrence of angle bisectors of a triangle is called as incentre of the triangle. The incentre always lies inside the triangle. The distance from incentre to all the sides are equal and is referred to as inradius. The circle drawn with inradius is called incircle and touches all sides of the triangle.

If a point is on the bisector of an angle, then it is equidistant from the two arms of the angle.

Activities #
Angular bisectors and incenter of a triangle

The intersecting point of three lines which are the bisectors of three angles of a triangle that is the incenter and it's properties are examined.

Concept #: 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

Concept #: Basic Proportionality Theorem

The concept of Thales theorem has been introduced in similar triangles. If the given two triangles are similar to each other then,

  •    Corresponding angles of both the triangles are equal
  •    Corresponding sides of both the triangles are in proportion to each other

Solved problems/ key questions (earlier was hints for problems)

Additional Resources

Resource Title and description

1.Triangles- This resource contain all information related to triangle like definition, types of triangle, perimeter, congruency etc.,

2. Triangles- This helps to know the aplication of geometry in our daily life, it contain videos and interactives.

3. Congruence

4. Similarity and Congruence

5. The below video gives information about Angle Sum Property by Gireesh K S and Suchetha S S

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
  • 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.
    • National Institute of Open Schooling: This website has good reference notes on concurrency of lines in a triangle for both students and teachers.
    • Math Open Reference : This website gives activities that can be tried and manipulated online for topics on geometry.
  • Books and journals
  • Textbooks:
  • Syllabus documents (CBSE, ICSE, IGCSE etc)

Projects (can include math lab/ science lab/ language lab)

Laboratory Manuals - Mathematics : Click here to refer activity 15,16,18 and 20 which explains the properties of Triangle.

Assessments - question banks, formative assessment activities and summative assessment activities

  1. Introduction to triangles
  2. Types of Triangle by sides
  3. Properties of Triangle