Difference between revisions of "Similar and congruent triangles"

From Karnataka Open Educational Resources
Jump to navigation Jump to search
Line 55: Line 55:
  
 
=== Teaching Outlines ===
 
=== Teaching Outlines ===
−
*
 
−
 
 
====Concept 1: What are congruent triangles?====
 
====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.
 
If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle.
Line 69: Line 67:
 
====== [[Congruence of regular geometric shapes]] ======
 
====== [[Congruence of regular geometric shapes]] ======
 
<span></span><span></span>
 
<span></span><span></span>
−
====Concept # 2. Theorems related to congruency of triangles.====
+
====Concept # 2. Postulates for congruence of triangles.====
 
Any triangle is defined by six measures (three sides, three angles). Triangles are congruent if:
 
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
Line 82: Line 80:
 
====== [[Congruence in triangles – SAS Rule]] ======
 
====== [[Congruence in triangles – SAS Rule]] ======
 
<span></span><span></span>
 
<span></span><span></span>
−
====Concept # 4.Similar triangles====
+
====Concept # 4 What are similar triangles?====
−
=====Learning objectives=====
+
=====Activities =====
−
# To develop an intuitive understanding of the concept “similarity of figures”.
+
[[Introduction to similar triangles|'''Introduction to similar triangles''']]
−
# 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
 
−
# The teacher can bring  different sized photographs got from same negative like stamp size, passport size and a post card size .
 
−
# 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://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
 
−
<span> </span>
 
−
 
 
−
<span></span><div id="ggbContainer16f554cd0061d9b3758eec50a21dcbb3"></div><span></span>
 
−
*Process:
 
−
# The teacher can use this geogebra file to explain about similar triangles.
 
−
# Also she can help differentiate between congruent and similar triangles.
 
−
*Developmental Questions:
 
−
# 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.
 
−
*
 
  
−
====Concept # 5. Similarity postulates/theorms====
+
==== Concept # 5. Tests for similarity ====
−
Two triangles are said to be similar if any of the following equivalent conditions hold:
+
<span></span><span></span>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.”
  
−
======Activity No # Similarity test (AA postulate)======
+
=====Activities=====
 +
======[[Similarity test - AA postulate]]======
 
{| style="height:10px; float:right; align:center;"
 
{| 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;">
 
|<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://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
 
|}
 
|}
−
*Estimated Time :45 minutes
+
*<span></span><span></span>
−
*Materials/ Resources needed: Laptop, geogebra file, projector and a pointer
+
<nowiki>===== Solved problems/ key questions (earlier was hints for problems).[edit | edit source] =====  === Projects (can include math lab/ science lab/ language lab)[edit | edit source] ===  === Assessments - question banks, formative assessment activities and summative assessment activities ===</nowiki>
−
*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
 
−
<span> </span>
 
−
 
 
−
<span></span><div id="ggbContainer6cd0a400ff760d9a599f4b83605dba3b"></div><span></span>
 
−
*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.
 
−
* ===== Solved problems/ key questions (earlier was hints for problems).[edit | edit source] =====  === Projects (can include math lab/ science lab/ language lab)[edit | edit source] ===  === Assessments - question banks, formative assessment activities and summative assessment activities ===
 

Revision as of 14:59, 29 April 2019

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

While creating a resource page, please click here for a resource creation checklist.

Concept Map

[maximize]

Additional Resources

OER

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

  • Syllabus documents

Non-OER

  • 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
Shapes that are congruent
Congruence of regular 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.
Activities
Congruence in triangles – SSS Rule
Congruence in triangles – SAS Rule

Concept # 4 What are similar triangles?

Activities

Introduction to similar 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

===== Solved problems/ key questions (earlier was hints for problems).[edit | edit source] ===== === Projects (can include math lab/ science lab/ language lab)[edit | edit source] === === Assessments - question banks, formative assessment activities and summative assessment activities ===

Retrieved from "https://karnatakaeducation.org.in/KOER/en/index.php?title=Similar_and_congruent_triangles&oldid=30814"