Download TRIANGLES

CONGRUENT
TRIANGLES
In this tutorial, we will learn about
congruent triangles, principles of
congruent triangles and their
applications.
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At the and of this lesson you will be able to
:
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define the concept of congruent shape.
explain the congruence of two triangles.
determine the sufficient conditions for congruence
of two triangles.
use the cogruence of triangles for solving problems.
TABLE OF CONTENTS
The topicis divided into the following subsections:
 Introduction
 Congruence of Triangles
 Rules of Congurent Triangles
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The Side-Side-Side (SSS) Congruence Rule
The Side-Angle-Side(SAS)Congruence Rule
The Angle-Side-Angle (ASA) Congruence Rule
The Angle-Angle-Side(AAS) Congruence Rule
The Right Angle-Hypotenuse-Side (RHS) Congruence Rule
Summary
References
Introduction
Table of contents
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If two figures have the same size and shape, then they are said
to be congruent figures.
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Square ABCD is congruent to square EFGH as their
corresponding sides and angles are equal.
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Note:
Congruent figures are exact duplicates of each other. One
could be fitted over the other so that their corresponding parts
coincide.The concept of congruence applies to figures of any
type.
Congruence of Triangles
Table of contents
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Two triangles are said to be congruent if all the sides
and the angles of one triangle are respectively equal to
corresponding sides and angles of other triangle.
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The equal sides and angles may not be in the same
position (if there is a turn or a flip), but they will be
there.
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△ABC and △DEF have the same size and shape. In fact,
if you could “pick up” triangle ABC you fit it exactly over
triangle DEF so that the vertices and sides coincide. We
say △ABC is congruent to △DEF.
In symbolic form: △ABC ≅ △DEF
Table of contents
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The order of the points in the names of the triangles is
important. △ABC and △DEF says that the triangles will
coincide when A is placed on D,B on E, and C on F. ( We
would not say that △ABC ≅ △EFD.)
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The congruent triangles ABC and DEF have equal
corresponding sides (AB=DE, BC=EF,CA=FD) and equal
corresponding angles ( <A=<D, <B=<E, <C=<F)
Rules of Congruent Triangles
Table of contents
1) The Side-Side-Side (SSS) Congruence Rule
2) The Side-Angle-Side(SAS)Congruence Rule
3) The Angle-Side-Angle (ASA) Congruence Rule
4) The Angle-Angle-Side(AAS) Congruence Rule
5) The Right Angle-Hypotenuse-Side (RHS) Congruence
Rule
Table of contents
1)The Side-Side-Side (SSS) Congruence Rule : If three sides of
one triangle are equal to three sides of the other triangle, then
the two triangles are congruent.
△ABC ≅ △DEF
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Question : Find the value of each of pronumerals in the given
pair of triangles.
△ABC ≅ △DEF (SSS)
Click here to see the solution
x = 89, y = 58, z = 33 (corresponding
angles of congurent triangles)
Table of contents
2) The Side-Angle-Side (SAS) Congruence Rule : If two
sides and the included angle of one triangle are congurent to
the corresponding parts of another triangle, the triangles are
congruent.
△ABC ≅ △DEF
Click here to review
the
included angle
Question:
(Click the space
to see the
solution.)
Table of contents
We should use Vertical Angles
Theorem. The teorem states
that vertical angles are
congruent, so we know that
<ACB=<DCE. Now we have two
pairs of corresponding,
congruent sides and congruent
included angles. (SAS
congruence)
Table of contents
3)The Angle-Side-Angle (ASA) Congruence Rule: If two
angles and the included side of one triangle are congruent to
the corresponding parts of another triangle, the triangles are
congruent.
△ABC ≅ △DEF
Table of contents
Question: Use the data in the diagram to prove that △ABD ≅
△CDB
In △ABD and △CDB,
a=b (Alternate Angles)
BD=DB
(Common
Side)
Click
here
to see the
solution
x=y (Alternate Angles)
So, △ABD ≅ △CDB (ASA).
Table of contents
4)The Angle-Angle-Side(AAS) Congruence Rule: If two angles
and a non-included side of one triangle are equal to two angles and
a non-included side of another triangle, then the triangles are
congruent.
△ ACB ≅ △A’C’B’
Compare AAS with AAS :
Compare AAS with ASA :
Compare AAS with AAS :
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For the ASA rule the given
side must be included and
for AAS rule the side given
must not be included. The
trick is we must use the
same rule for both the
triangles that we are
comparing
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Question: Which of the following conditions would be sufficent for
the above triangles to be congruent?
A
a=e , x=u, c=f
B
x=u, y=t, z=s
C
a=e, y=s, z=t
D
a=f, y=t, z=s
Solution for c)
Step 1: a = e gives the S
y = s gives the A
z = t gives the A
Step Click
2: a and
e aretonon-included
sides.
here
see the solution!
Follows the AAS rule.
Answer: a = e, y = s, z = t is sufficient show
that the above are congruent triangles.
Table of contents
5)The Right Angle-Hypotenuse-Side (RHS) Congruence
Rule: If the hypotenuse and one leg of a right triangle are
equal to the hypotenuse and one leg of another right triangle,
then the two right triangles are congruent.
△BCA ≅ △EFD
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Question : According to given triangles, decide following
statements are true/false.
•△ABC ≅ △DEF
TRUE
FALSE
•x=50 y=40
TRUE
FALSE
△ABC ≅ △DEF (RHS congruent
rule)
Click for the explanation
X=40, y=50 (Corresponding
angles of congruent triangles)
There is a useful simulation about
Congruence Rules:
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http://www.mathopenref.com/congruentsss.ht
ml
summary
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Table of contents
In this lesson you learned that:
Congruent figures have the same size and shape.
Congruent triangles have the same size and the same
shape. The corresponding sides and
the corresponding angles of congruent triangles are
equal.
There are five types of congruence rules.
(SSS,SAS,ASA,AAS, RHS).
How to use them by solving questions.
references
The following resources have been used in this
tutorial:
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The Math Curriculum of The Ministry of Education
http://www.mathsteacher.com.au/year9/ch13_geometry/
07_congruent/triangles.htm
http://www.onlinemathlearning.com/congruenttriangles.html#aas
http://www.excellup.com/classnine/mathnine/triangletheo
rem.aspx
http://www.mathopenref.com/congruentsss.html