Download Triangle Congruence

Lesson plan . Subject: Maths. Title: Triangle Congruence
Content : Introduction to Triangle Congruence..
Teaching aims: After completing the activity, students should be able to recognize the properties
and conditions for congruent triangles and the notation of corresponding sides and angles in
congruent triangles.
Learning Outcomes Know
what is congruence
and objectives
Be able to: determine if two triangles are congruent and provide the correct
They are writing on reason why the triangles are congruent.
Be aware: Proofs of problems
the board so that
learners are clear
about what they
should achieve by the
end of the lessons.
Assessment
Content and language:
Summative (standard test) and formative (informal) assessment.
Communication
Examples of communication activities:
 Tell the learners they have to work with a partner to explain “What
is congruence?”
 Solve with partner, simple problems proposed
Cognition
Know Congruence and the Criteria for triangles Congruence: SSS, ASA,
SAS and AAS
Knowing how to use Geogebra to resolve problems related to triangles.
Follow English instructions on solving problems .
Thinking Skills
Proofs of problems
Citizenship
Example of citizenship: lesson video in youtube of classe in other country.
Procedure
Learning Geometry through an Intuitive Approach
Language of
learning
Language Objectives:
After completing the activity, students should be able to:
recognize and understand the key English terms related to congruent
triangles, e.g., congruent figures, same size, same shape, congruent
triangles, fit into each other, rotate, overlap, coincide, corresponding sides,
corresponding angles, symbol, position.
Understand the English expressions for explaining the properties and
conditions for congruent triangles (e.g., If two triangles can fit into each
other perfectly, they are congruent triangles. When you cut the triangles out
and rotate them, they can exactly overlap/coincide with each other. The
overlapping sides are called ‘corresponding sides’ and the overlapping
angles are called ‘corresponding angles’.
use correct notations to indicate the conditions for two triangles to be
congruent,
e.g.,
-  SSS (side side side)
All three corresponding sides are equal in length.
See Triangle Congruence (side side side).
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 SAS (side angle side)
A pair of corresponding sides and the included angle are equal.
See Triangle Congruence (side angle side).
 ASA (angle side angle)
A pair of corresponding angles and the included side are equal.
See Triangle Congruence (angle side angle).
 AAS (angle angle side)
A pair of corresponding angles and a non-included side are equal.
See Triangle Congruence (angle angle side).
-follow English instructions on solving problems concerning this topic and
work on related problems written in English.
Resources
Use an online dictionary with an audio function to hear maths vocabulary pronounced,
e.g. http://www.amathsdictionaryforkids.com/dictionary.html
http://www.languageguide.org/im/numbers/eng/extra4.jsp?lang=it
Use a grammar reference book in order to practise producing complex sentences such as
conditional
Basic use of Geogebra (Software for mathematics)
Computer laboratory
Projector screen
Internet connection
Materials: rulers, compasses, protractors, pencils
and visualizer
Polygons with geogebra (italiano)
http://www.mathopenref.com/congruenttriangles.html
http://www.basic-mathematics.com/congruent-shapes.html
http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/congruencysimilarityrev1.shtml
http://www.geogebra.org/en/wiki/index.php/Triangles_English
http://www.youtube.com/watch?v=FaVABMOymkI
https://en.wikipedia.org/wiki/Congruence_%28geometry%29
http://www.mathsisfun.com/geometry/triangles-congruent.html
http://www.mathsisfun.com/geometry/triangles-congruent-finding.html
Mathematics – C. Meyrick, J. Roberts –Oxford (Content and Language
support))
http://www.leopoldopirelli.it/documenti/DISPENSE%20DI%20GEOMET
RIA-parte%20seconda%20-%20i%20triangoli.pdf
Constructing Congruent Triangles
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Subject : Maths
Title Triangle Congruence
“This morning I’m going to talk about congruence….
Main points
Know meaning of Congruence and the Criteria for triangles Congruence: SSS, ASA, SAS, AAS
Knowing how to use Geogebra to resolve problems related to triangles
Proofs of problems
Lesson 1
Triangle Congruence
Lesson 1
1. The teacher should introduce the idea of congruence by using circles, quadrilaterals,
rectangles and irregular hexagons as examples.
If two figures can fit into each other, that is, they have the same size and shape, they are
congruent figures.
1. Circles
Their radii are equal.
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Other examples of congruent shapes or figures
2. The teacher should then ask the students to give the conditions which make 2 triangles
congruent. (What are the conditions, so that the two triangles are congruent?
(Two triangles are congruent, if they have exactly the same three sides and exactly the same three
angles)
3. SSS: Using a visualizer, the teacher should demonstrate how to construct triangles with
sides a, b, c assigned with Geogebra.
 Build three sliders a, b, c with range min: 0 max: 10.
Set the values of the slider on the measures of the sides of the triangle to build, moving the
sliders.
 Draw the segment of length assigned a
 Draw the circle with center B and radius b
 Draw the circle with center A and radius c
 Determines the intersection points C and D of the two circumferences.
 Draw the triangle ABC.
4. The teacher should ask students to work in groups of four to construct triangles of given
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Sides (with Geogebra).
http://www.geogebra.org/en/upload/files/moloughlin/Congruent_Triangles___SSS.html
5. The teacher should then ask students to compare their triangles and see if they can
draw triangles with different shapes but with the same sides.
6. The teacher should then help the students to draw the conclusion: two triangles are
congruent if their corresponding sides are equal (SSS).

Congruent Triangles SSS
Lesson 2 SAS
1. The teacher should continue to ask students to draw triangles with two given sides and
a given included angle. Similar to the previous lesson, students should compare their
triangles with those of their group-mates and the teacher should guide students to come to
the conclusion that SAS is a sufficient condition for two triangles to be congruent.
1.
Congruent Triangles SAS Interactive sheet to show how SAS is sufficient for two
triangles to be congruent. By Michael O'Loughlin
2.
Congruent Triangles SAS
(SAS: “Drawing the segments AB and CD, to fix the extent of two sides, and an angle
α (EFG) to fix the measurement of the angle between them, draws a point F’ as the first vertex of
the new triangle.
Then trace a vector from F to the new point to make a translation of the angle to the vertex, you
know that the translation is a rigid transformation that guarantees the congruence Make now a
translation of the sides of the angle.
On the sides of the new angle you have to take two segments congruent to AB and CD respectively,
to make it drawing two circles with center F' and rays AB and CD, using the compass function.
The intersection points between the sides of the angle and the circumferences so constructed are the
other vertices of the triangle.
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The triangle thus obtained is unique.)
Lesson 3 ASA
1. After students have understood the idea of “SAS”, the teacher should ask them to draw
triangles with two given angles and one given side between the two angles.

Congruent Triangles ASA Interactive sheet to show how ASA is sufficient for two
triangles to be congruent. By Michael O'Loughlin

Congruent Triangles ASA
2. Students should then compare their triangles with their group-mates to see if the
triangles are congruent..
3. The teacher should then draw the conclusion: two triangles are congruent
if 2 corresponding angles and the included side are equal (ASA).
Lesson 4
1. Using the examples in the worksheet, the teacher can explain to the students what
corresponding sides and angles are. The teacher should stress that it is important to write
the name of the congruent triangles correctly (according to the corresponding sides and
angles).
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2. In order to check students’ understanding, the teacher should ask the students to
practise doing the exercise Part A (if the students are not very familiar with the
concept, the teacher can provide more examples and practise on the blackboard).
3. When the teacher thinks that students have understood the idea, he/she can ask the
students to finish the classwork in Part B, which is an exercise to check their
understanding of this lesson and also the previous one
Notation of Congruence and Congruent Triangles
Definition of Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size.
Congruent Triangles
Triangles are congruent when they have
exactly the same three sides and exactly the same three angles.
What is "Congruent" ... ?
It means that one shape can become another using Turns, Flips and/or Slides:
Rotation
Turn!
Reflection
Flip!
Translation
Slide!
Triangle Congruence
Determining congruence.
Sufficient evidence for congruence between two triangles can be shown through the following
comparisons:
 SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the
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included angles are equal in measurement, then the triangles are congruent.
 SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the
triangles are congruent.

ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement,
and the included sides are equal in length, then the triangles are congruent
For example:
is congruent to:
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AAS theorem:
If a pair
of corresponding angles and a non-included side are equal
Example:

Proof - Triangle Congruence
https://www.khanacademy.org/math/geometry/congruenttriangles/cong_triangle/v/congruent-triangle-proof-example
1. Problem: Is triangle PQR congruent to
triangle STV by SAS? Explain.
Solution: Segment PQ is congruent
to segment ST because
PQ = ST = 4.
Angle Q is congruent to
angle T because
angle Q = angle T = 100 degrees.
Segment QR is congruent
to segment TV because QR = TV = 5.
Triangle PQR is congruent
to triangle STV by Side-Angle-Side.
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1. Problem: Show that triangle QYN is congruent
to triangle QYP.
Solution: Segment QN is congruent to
segment QP and segment YN is
congruent to segment YP because that
information is given in the figure.
Segment YQ is congruent to segment
YQ by the Reflexive Property of Congruence, which says any figure is
congruent to itself.
Triangle QYN is congruent to triangle
QYP by Side-Side-Side.
PART A____________________________________________________________
Question
Congruent shapes
1. If two shapes are congruent, they are identical in both shape and size.
Remember: Shapes can be congruent even if one of them has been rotated or reflected.
Question
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Which of the shapes in the illustration above are congruent?
Answer
Did you get the following pairs?

A and G

D and I

E and J

C and H
Remember: Shapes can be congruent even if one of them has been rotated (as in A and G)
or reflected (as in C and H).
The symbol
means 'is congruent to'.
2. For each of the following pairs of triangles, state whether they are congruent. If they are,
give a reason for your answer (SSS, SAS, AAS).
Pair 1
Pair 2
Answer
1. Yes. SSS
2. No. The side of length 7cm is not in the same position on both triangles.
Therefore, it is not AAS.
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PART B__________________VIDEO TUORIAL
http://www.youtube.com/watch?v=FaVABMOymkI
This is the 8th video in the series Basic Trigonometry, suitable for GCSE and High school math.
This video looks at triangle congruency. How one triangle is an exact copy of another in terms of its
sides and angles. The five tests to show that one triangle is congruent to another are each explained.
The special case of right-angled triangle congruency is dealt with in some detail, using Geogebra
constructions. Three worked solutions to congruency problems are included
Writing Two Column Proof - Triangle Congruence
http://youtu.be/vsluxs0B9Gg
http://youtu.be/rSNYxIYI8pc
Thx for your comments, glad if it helps folks.. Different teachers have different requirements. I
choose to first identify them as vertical, then use a vertical angles theorem that says "if two angles
are vertical, then they are congruent."
KEY WORDS
Triangle Geometry and ICT (video, geogebra)
interActive Worksheets
3.
Understanding SAS 3 InterActivities for understanding the SAS side-angle-side property.
By Linda Fahlberg-Stojanovska User:LFS
4.
Congruent Triangles ASA Interactive sheet to show how ASA is sufficient for two
triangles to be congruent. By Michael O'Loughlin
5.
Congruent Triangles SAS Interactive sheet to show how SAS is sufficient for two
triangles to be congruent. By Michael O'Loughlin
6.
Congruent Triangles SSS Interactive sheet to show how SSS is sufficient for two
triangles to be congruent. By Michael O'Loughlin
7.
Types of Triangles
GGB download Creating the various types of triangles according
to sides and according to angles. By Ken Frank
8.
Naming triangles, sides and angles Basic
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9.
The vocabulary of Triangles
vocabulary. By Ken Frank
GGB download An illustration of basic triangle
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