Difference between revisions of "Similar and congruent triangles"

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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
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= Concept Map =
 
 
__FORCETOC__
 
__FORCETOC__
<mm>[[similar_and_congruent_triangles.mm]]</mm>
 
  
= Textbook =
 
To add textbook links, please follow these instructions to:
 
([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
 
  
=Additional Information=
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=== Additional Resources ===
==Useful websites==
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==Reference Books==
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====OER====
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*Web resources:
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**This videos is related congruence of triangle.
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{{#widget:YouTube|id=HwbTJNcZsuc}}    {{#widget:YouTube|id=PJ6TVVdIHpg}}    {{#widget:YouTube|id=5of3Z9hr_7A}}   
  
= Teaching Outlines =
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{{#widget:YouTube|id=mFpvOZP4aPI}}
  
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*Books and journals
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*Textbooks
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**NCERT Textbooks: [http://ncert.nic.in/textbook/pdf/gemh107.pdf#page=1&zoom=auto,-19,794 Class 7-Chapter 7:Congruence of Triangles] ,  [http://ncert.nic.in/textbook/textbook.htm?iemh1=7-15 Class 9: Triangles]
  
==Concept # 1. Congruent triangles ==
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*Syllabus documents
===Learning objectives===
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====Non-OER====
# Analyse and identify the structure of simple triangles
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*Web resources:
# Gather information about the similarities and differences between triangles
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**[https://www.mathsisfun.com/geometry/triangles-congruent-finding.html Math is fun] : This website gives descriptions and diagrams related to rules of congruency
# Comprehend the meaning of congruent triangles - Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
 
# Utilise the newly acquired knowledge in order to solve related problems.
 
# Ability to draw congruent triangles
 
# List four basic properties of congruent triangles.
 
# Properties of Congruent Triangles:<br>
 
* If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.
 
i.e. CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent".
 
* In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.
 
  
===Notes for teachers===
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*Books and journals
# The teacher can ask students to think of many objects which are mass-produced and that are found to be exactly the same size and shape like pens, CD-roms, cars etc.
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*Textbooks:
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**Karnataka Govt Text book – Class 8 :  [http://ktbs.kar.nic.in/new/website%20textbooks/class8/8th-english-maths-2.pdf Part 2]
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*Syllabus documents (CBSE, ICSE, IGCSE etc)
  
===Activity No # ===
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=== Learning objectives ===
{| style="height:10px; float:right; align:center;"
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*Analyse and identify the structure of simple triangles
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
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* Gather information about the similarities and differences between triangles
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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* Comprehend the meaning of congruent triangles - Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.  
|}
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* Utilise the newly acquired knowledge in order to solve related problems.
*Estimated Time : 30 minutes.
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* Ability to draw congruent triangles
*Materials/ Resources needed; Laptop, projector. geogebra file and a pointer.
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* Understand the properties of congruent Triangles 
*Prerequisites/Instructions, if any;
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# The students should have knowledge about triangles and its elements.
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=== Teaching Outlines ===
# They should know about angles , measurements and its types.
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====Concept 1: What are congruent triangles?====
# They should know the meaning of the terms equal and congruent.
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If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.
# They should have a fair understanding of congruent lines and angles.
 
*Multimedia resources; laptop
 
*Website interactives/ links/ / Geogebra Applets
 
*Process:
 
  
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i.e. CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent". In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.
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=====Activities #=====
  
*Developmental Questions:
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====== [[Identifying congruent shapes]] ======
# Measure the sides and the area of both triangles.
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Polygons are congruent if they are the same size and shape that is, if their corresponding angles and sides are equal. The activity helps in identifying congruent shapes.
# Drag the points of the triangles so that you obtain two triangles with the same dimensions.
 
# What are your observations about the areas of the triangles?
 
# What are your observations about the perimeters of the triangles ?
 
*Evaluation :
 
# Define congruent triangles.
 
*Question Corner:
 
# List the properties of congruent triangles.
 
  
===Activity No # ===
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====== [[Shapes that are congruent]] ======
{| style="height:10px; float:right; align:center;"
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Shapes can be combined together to form congruent shapes. Identifying such shapes for congruence is explored in this activity.
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
  
==Concept # 2. Tests for congruency of triangles.==
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====== [[Congruence of regular geometric shapes|Congruence of geometric shapes]] ======
===Learning objectives===
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This activity involves in identifying the congruent shapes among the shapes of different geometric shapes.<span></span><span></span>
Any triangle is defined by six measures (three sides, three angles). Triangles are congruent if:<br>
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====Concept # 2. Postulates for congruence of triangles.====
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Any triangle is defined by six measures (three sides, three angles). Triangles are congruent if:
 
# All three corresponding sides are equal in length. SSS (side side side) congruency postulate
 
# All three corresponding sides are equal in length. SSS (side side side) congruency postulate
 
# A pair of corresponding sides and the included angle are equal. -- SAS (side angle side) congruency postulate.
 
# A pair of corresponding sides and the included angle are equal. -- SAS (side angle side) congruency postulate.
Line 113: Line 78:
 
# A pair of corresponding angles and a non-included side are equal.-- AAS (angle angle side) congruency postulate.
 
# A pair of corresponding angles and a non-included side are equal.-- AAS (angle angle side) congruency postulate.
 
# HL (hypotenuse leg of a right triangle) :Two right triangles are congruent if the hypotenuse and one leg are equal.Also known as RHS postulate.
 
# HL (hypotenuse leg of a right triangle) :Two right triangles are congruent if the hypotenuse and one leg are equal.Also known as RHS postulate.
===Notes for teachers===
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=====Activity 1=====
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner:
 
# Why SSA and AAA doesn't work ?
 
  
===Activity No # ===
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====== [[Congruence in triangles – SSS Rule]] ======
{| style="height:10px; float:right; align:center;"
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Investigating the possibility of congruence if three sides of two triangles are congruent.
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
  
==Concept # 3. Theorms on congruent triangles ==
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===== Activity 2 =====
===Learning objectives===
 
===Notes for teachers===
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
  
===Activity No # ===
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====== [[Congruence in triangles – SAS Rule]] ======
{| style="height:10px; float:right; align:center;"
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Given two sides and an angle of two triangles are equal, are the two triangles congruent? This activity investigates the position of the given angle for the two triangles to be congruent. <span></span><span></span>
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
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====Concept # 4 What are similar triangles?====
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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Triangles are similar if corresponding sides are in the same ratio and  corresponding angles are equal. All regular polygons having the same number of sides are always similar. All squares and equilateral triangles are similar. All congruent figures are similar but all similar figures need not be congruent. 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
  
==Concept # 4.Similar triangles==
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=====Activities # =====
===Learning objectives===
 
# To develop an intuitive understanding of the concept “similarity of figures”.
 
# Triangles are similar if they have the same shape, but can be different sizes.
 
# Understand that 'corresponding' means matching and 'congruent' means equal in measure.
 
# To determine the correspondences between the pairs of similar triangles.
 
# The ratio of the corresponding sides is called the ratio of simultude or scale factor.
 
# Triangles are similar if their corresponding  angles are congruent and the ratio of their corresponding sides are in proportion.
 
# To develop an ability to state and apply the definition of similar triangles.
 
# recognize and apply “corresponding sides of similar triangles are proportional”.
 
  
===Notes for teachers===
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====== [[Introduction to similar triangles|Introduction to similar triangles]] ======
# The teacher can bring  different sized photographs got from same negative like stamp size, passport size and a post card size .
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The concept of similarity is introduced and investigated by comparing elements of two triangles.
# Compare them and say that all photos are look alikes and are proportionate. only the size differs.
 
# She can also mention about scale concept in graphical representation.
 
# Hence similar triangles are the same proportionate triangles but of different sizes.
 
# Two triangles are similar  if they have:
 
* all their angles equal
 
* corresponding sides in the same ratio
 
# In similar triangles, the sides facing the equal angles are always in the same ratio.  Application of this finds its use in finding  the unknown lengths in similar triangles . For this :<br>
 
Step 1: Find the ratio of corresponding sides in pairs of similar triangles.<br>
 
Step 2: Use that ratio to find the unknown lengths.<br>
 
===Activity No # 1. SIMILAR TRIANGLES===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time:45 minutes.
 
*Materials/ Resources needed:
 
Laptop, geogebra file, projector and a pointer.
 
*Prerequisites/Instructions, if any:
 
# The students should have prior knowledge of triangles , sides , angles , vertices .
 
# They should know meaning of the terms 'similar' and 'proportionate'.
 
# They should be able to identify  the corresponding sides.
 
# They should know how to find ratio.
 
# They should know to find area and perimeter of triangles.
 
*Multimedia resources: Laptop
 
*Website interactives/ links/ / Geogebra Applets
 
*Process:
 
  
 
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==== Concept # 5. Tests for similarity ====
*Developmental Questions:
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<span></span><span></span>Two triangles are said to be similar if any of the following equivalent conditions hold:
# Look at the shape of both triangles being formed? (look alikes )
 
# As I increase /decrease the size of triangles do you see that the measures are changing proportionately ?
 
# Can any one explain what exactly  proportionately means ?
 
# Can you identify the corresponding sides and angles ?
 
*Evaluation:
 
# Name the corresponding sides.
 
# Compare the perimeters of two similar triangles.
 
# What are equiangular triangles ?
 
*Question Corner:
 
# Compare the ratio of corresponding sides of similar triangles. What do you infer ?
 
# How can one draw similar triangles if only one triangles sides are given ?
 
# Discuss the applications of similar triangles in finding unknowns in real life situations.
 
# Give examples where one uses the concept of similarity.
 
 
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
 
 
==Concept # 5. Similarity postulates==
 
===Learning objectives===
 
Two triangles are said to be similar if any of the following equivalent conditions hold:
 
 
# The SSS similarity postulate states that if the sides of two triangles are in proportion, then the triangles are similar.
 
# The SSS similarity postulate states that if the sides of two triangles are in proportion, then the triangles are similar.
 
# The AA similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are said to be similar.
 
# The AA similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are said to be similar.
 
# SAS Similarity Postulate states, “If an angle of one triangle is congruent to the corresponding angle of another triangle and the sides that include this angle are proportional, then the two triangles are similar.”
 
# SAS Similarity Postulate states, “If an angle of one triangle is congruent to the corresponding angle of another triangle and the sides that include this angle are proportional, then the two triangles are similar.”
  
===Notes for teachers===
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=====Activities #=====
===Activity No # Similarity test (AA postulate)===
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======[[Similarity test - AA postulate]]======
{| style="height:10px; float:right; align:center;"
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In two triangles, if the angles are equal, then the sides opposite to the equal angles are in the same ratio and hence the two triangles are similar.
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time :45 minutes
 
*Materials/ Resources needed: Laptop, geogebra file, projector and a pointer
 
*Prerequisites/Instructions, if any
 
# The students should know the meaning of the terms congruent and similar.
 
# They should understand the terms corresponding sides and angles.
 
# They should have an idea of ratio and proportion.
 
*Multimedia resources :Laptop
 
*Website interactives/ links/ / Geogebra Applets
 
 
 
*Process:
 
# The teacher can initially have a warm up session regarding terms congruence, similarity and corresponding angles and ratio.
 
# She can then project the geogebra file and by moving the sliders she can change the side and angle measures and teach teh AA similarity postulate.
 
# Also she can let them understand that in similar triangles, the corresponding sides are proportional.
 
* Developmental Questions:
 
# What does congruent mean ?
 
# What does similarity mean ?
 
# How can we test whether the two given figures are similar or not ?
 
# In the above two triangles, what measures of both are same ?
 
# Identify the corresponding sides and angles.
 
# Is their ratio same ?
 
# What can you say about the two triangles ?
 
# Recall the similarity postulates.
 
# By what postulate are the two triangles similar ?
 
*Evaluation:
 
# Differentiate similarity and congruence.
 
*Question Corner:
 
# Can the tree and its shadow be considered as similar figures ?
 
# Can this similarity concept be used to find the height and depth of objects ? Frame any two of your own questions which can be solved using similarity postulates.
 
 
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
 
 
==Concept # 6. Theorms on Similar triangles ==
 
===Learning objectives===
 
===Notes for teachers===
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
 
 
===Activity No # ===
 
{| style="height:10px; float:right; align:center;"
 
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
 
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
*Estimated Time
 
*Materials/ Resources needed
 
*Prerequisites/Instructions, if any
 
*Multimedia resources
 
*Website interactives/ links/ / Geogebra Applets
 
*Process/ Developmental Questions
 
*Evaluation
 
*Question Corner
 
 
 
 
 
= Hints for difficult problems =
 
 
 
= Project Ideas =
 
  
= Math Fun =
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=== Projects (can include math lab/ science lab/ language lab) ===
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'''''Laboratory Manuals''' - Mathematics : [https://ncert.nic.in/pdf/publication/sciencelaboratorymanuals/classIXtoX/mathematics/lelm405.pdf Click here] to refer to activity 13,15,16,17 and 18'' which explains the similarity of two Triangles.
  
'''Usage'''
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<span></span><span></span>
  
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
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[[Category:Triangles]]
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[[Category:Class 8]]
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[[Category:Class 9]]

Latest revision as of 07:04, 30 June 2022

ಕನ್ನಡ

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

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Additional Resources

OER

  • Web resources:
    • This videos is related congruence of triangle.

  • Syllabus documents

Non-OER

  • Web resources:
    • Math is fun : This website gives descriptions and diagrams related to rules of congruency
  • Books and journals
  • Textbooks:
    • Karnataka Govt Text book – Class 8 : Part 2
  • Syllabus documents (CBSE, ICSE, IGCSE etc)

Learning objectives

  • Analyse and identify the structure of simple triangles
  • Gather information about the similarities and differences between triangles
  • Comprehend the meaning of congruent triangles - Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
  • Utilise the newly acquired knowledge in order to solve related problems.
  • Ability to draw congruent triangles
  • Understand the properties of congruent Triangles

Teaching Outlines

Concept 1: What are congruent triangles?

If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.

i.e. CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent". In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.

Activities #
Identifying congruent shapes

Polygons are congruent if they are the same size and shape that is, if their corresponding angles and sides are equal. The activity helps in identifying congruent shapes.

Shapes that are congruent

Shapes can be combined together to form congruent shapes. Identifying such shapes for congruence is explored in this activity.

Congruence of geometric shapes

This activity involves in identifying the congruent shapes among the shapes of different geometric shapes.

Concept # 2. Postulates for congruence of triangles.

Any triangle is defined by six measures (three sides, three angles). Triangles are congruent if:

  1. All three corresponding sides are equal in length. SSS (side side side) congruency postulate
  2. A pair of corresponding sides and the included angle are equal. -- SAS (side angle side) congruency postulate.
  3. A pair of corresponding angles and the included side are equal. -- ASA (angle side angle) congruency postulate.
  4. A pair of corresponding angles and a non-included side are equal.-- AAS (angle angle side) congruency postulate.
  5. HL (hypotenuse leg of a right triangle) :Two right triangles are congruent if the hypotenuse and one leg are equal.Also known as RHS postulate.
Activity 1
Congruence in triangles – SSS Rule

Investigating the possibility of congruence if three sides of two triangles are congruent.

Activity 2
Congruence in triangles – SAS Rule

Given two sides and an angle of two triangles are equal, are the two triangles congruent? This activity investigates the position of the given angle for the two triangles to be congruent.

Concept # 4 What are similar triangles?

Triangles are similar if corresponding sides are in the same ratio and corresponding angles are equal. All regular polygons having the same number of sides are always similar. All squares and equilateral triangles are similar. All congruent figures are similar but all similar figures need not be congruent. 

Activities #
Introduction to similar triangles

The concept of similarity is introduced and investigated by comparing elements of two triangles.

Concept # 5. Tests for similarity

Two triangles are said to be similar if any of the following equivalent conditions hold:

  1. The SSS similarity postulate states that if the sides of two triangles are in proportion, then the triangles are similar.
  2. The AA similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are said to be similar.
  3. SAS Similarity Postulate states, “If an angle of one triangle is congruent to the corresponding angle of another triangle and the sides that include this angle are proportional, then the two triangles are similar.”
Activities #
Similarity test - AA postulate

In two triangles, if the angles are equal, then the sides opposite to the equal angles are in the same ratio and hence the two triangles are similar.

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

Laboratory Manuals - Mathematics : Click here to refer to activity 13,15,16,17 and 18 which explains the similarity of two Triangles.