Difference between revisions of "Similar and congruent triangles"
Jump to navigation
Jump to search
m (Syncing Girija's changes done while migration was in progress) |
|||
| (43 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | <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;"> | |
| + | ''[http://karnatakaeducation.org.in/KOER/index.php/ತ್ರಿಭುಜಗಳ_ಸರ್ವಸಮತೆ ಕನ್ನಡ]''</div> | ||
<!-- This portal was created using subst:box portal skeleton --> | <!-- This portal was created using subst:box portal skeleton --> | ||
<!-- BANNER ACROSS TOP OF PAGE --> | <!-- BANNER ACROSS TOP OF PAGE --> | ||
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;" | {| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;" | ||
|- | |- | ||
| − | |style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http:// | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks] |
| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "| | + | | style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " | |
| − | [http:// | + | [http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank] |
|} | |} | ||
| − | While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''] | + | While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''] |
| + | |||
| − | |||
__FORCETOC__ | __FORCETOC__ | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | =Additional | + | === Additional Resources === |
| − | == | + | |
| − | == | + | ====OER==== |
| + | *Web resources: | ||
| + | **This videos is related congruence of triangle. | ||
| + | {{#widget:YouTube|id=HwbTJNcZsuc}} {{#widget:YouTube|id=PJ6TVVdIHpg}} {{#widget:YouTube|id=5of3Z9hr_7A}} | ||
| − | = | + | {{#widget:YouTube|id=mFpvOZP4aPI}} |
| + | *Books and journals | ||
| + | *Textbooks | ||
| + | **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] | ||
| − | == | + | *Syllabus documents |
| − | === | + | ====Non-OER==== |
| − | + | *Web resources: | |
| − | + | **[https://www.mathsisfun.com/geometry/triangles-congruent-finding.html Math is fun] : This website gives descriptions and diagrams related to rules of congruency | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | *Books and journals | |
| − | + | *Textbooks: | |
| + | **Karnataka Govt Text book – Class 8 : [http://ktbs.kar.nic.in/new/website%20textbooks/class8/8th-english-maths-2.pdf 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. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | === | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | ==Concept # 2. | + | ====== [[Congruence of regular geometric shapes|Congruence of geometric shapes]] ====== |
| − | + | 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: | + | ====Concept # 2. Postulates for congruence of triangles.==== |
| + | 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. | ||
| − | == | + | =====Activity 1===== |
| − | ===Activity | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | === | + | ====== [[Congruence in triangles – SSS Rule]] ====== |
| − | + | Investigating the possibility of congruence if three sides of two triangles are congruent. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | == | + | ===== Activity 2 ===== |
| − | |||
| − | |||
| − | ===Activity | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | === | + | ====== [[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. <span></span><span></span> | |
| − | + | ====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|Introduction to similar triangles]] ====== |
| − | + | The concept of similarity is introduced and investigated by comparing elements of two triangles. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | = | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | ==== Concept # 5. Tests for similarity ==== | |
| − | + | <span></span><span></span>Two triangles are said to be similar if any of the following equivalent conditions hold: | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | === | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | 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.” | ||
| − | === | + | =====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 : [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. | ||
| − | + | <span></span><span></span> | |
| − | + | [[Category:Triangles]] | |
| + | [[Category:Class 8]] | ||
| + | [[Category:Class 9]] | ||
Latest revision as of 07:04, 30 June 2022
| Philosophy of Mathematics |
While creating a resource page, please click here for a resource creation checklist
Additional Resources
OER
- Web resources:
- This videos is related congruence of triangle.
- Books and journals
- Textbooks
- NCERT Textbooks: Class 7-Chapter 7:Congruence of Triangles , Class 9: Triangles
- 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:
- 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 angles and the included side are equal. -- ASA (angle side angle) 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.
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:
- 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.
- 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.