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CONGRUENT TRIANGALE
This chapter focuses on students ability to equalize different measurement of two
figures with same shape and it leads to the congruency of figures and thereby the
formation of the general principles and geometrical constructions. The main aim
of this chapter is to develop the concepts and aptitude which enable know about
congruency the students should acquire to know about congruency.
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Look at these pictures
Figure (1)
Figure(2)
Notice the difference
In figure (1) stack of coins are placed on over the other.
In figure (2) books are placed one over the other but they are not fit exactly. Each
one has different sizes of note books so they will not fit exactly.
In geometry, figures such as triangles, rectangles, circles and so on, which can
be placed one over the other so as to fit exactly are said to be congruent figures .
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TRIANGLE
MATCH
Take a look at these triangles
A
B
C
P
Q
To specify the equal sides and angles, let’s name the triangles as
∆PQR
Equal sides
R
∆ ABC and
Equal angles
AB = PQ
<ACB = <QRP
BC = PR
< BAC = < PQR
AC = QR
< ABC = <QPR
From the table, we see that the pair of sides and the pair of angles opposite to
these sides are equal .
If two triangles are congruent, then the sides and angles of one are equal to the
sides and angles of the other , angles opposite to equal sides are equal and sides
opposite to equal angles are equal .
Assignment
Cut out a pair of eerkkil bits , each 3 c.m long and another pair , each 2 c.m
long . How many different kinds of rectangles can you make with these ?
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WHEN SIDES ARE EQUAL
4CM
5CM
5CM
6CM
4CM
6CM
Make a copy of one of these on a piece of tracing paper and place it over the
others . These triangles are all congruent .
Here we see
If the three sides of a triangle are equal to the three sides of another triangle ,
then three triangles are congruent.
Assignment
(1) Check the two triangles given below are congruent and why ?
6CM
8CM
7CM
6CM
7CM
8CM
EVEN IF ANGLES ARE EQUAL
We can draw triangles with the same three angles in different size , can’t we ?
600
300
□
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600
300
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If the sides of two triangles are equal , their angles are also equal , but simply
because angles are equal , the sides need not be equal .
Look at this picture
Assignment
(1) Check the two triangles given below are congruent .why?
1000
500
1000
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300
500
300
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TWO SIDES AND AN ANGLE
We can draw a triangle ,if two side and the angle included between them are
specified
We can draw in different ways
5CM
600
3CM
3CM
3CM
600
600
5CM
5CM
Make a copy of one of these on a piece of tracing paper and place it over the
others . These triangles are congruent .
Here we see
If two sides of a triangle and their included angle are equal to two sides of
another triangle and their included angle , then these triangles are congruent .
Look at this pictures
300
300
4cm
4cm
5cm
5cm
These are not congruent.
Just because two sides and some angles of a triangle are equal to two sides and
some angles of another triangle , the two triangles need not be congruent.
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Assignment
(1) check the two triangle given below are congruent?
700
5CM
5CM
3CM
700
(2) Check the two triangles given below are congruent and why ?
2CM
920
920
4CM
2CM
4CM
ONE SIDE AND TWO ANGLES
We can draw a triangle , if one of the sides and the two angles on it are specified
Some such triangles are shown below
4cm
450
300
300
450
4cm
From these we see
If one side and the two angles on it of a triangles are equal tone side and the
angles on it of another triangle then these triangles are congruent .
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Assignment
(1) Check the two the triangles given below are congruent. Why ?
4cm
300
4CM
300
450
450
(2) Check the two triangles given below are congruent .why?
300
600
5CM
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300
600
6CM
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RIGHT ANGLED TRIANGLES
P
Q
A
R
B
C
In the right angled triangles ABC ,PQR shown below PR = AB and PQ = BC
Let’s take a look at the third sides . Is there any relation between the sides QR
and AC ?
AB is the hypotenuse of the right angled triangle ABC . So by Pythagoras Theorem
AC2 = AB2 –BC2
Similarly , since PR is the hypotenuse of the right angled triangle PQR we get
QR2 = PR2 –PQ2
=
AB2 – BC2
= AC2
From this we find QR = AC
Then the three sides of triangle PQR are equal to the sides of triangle ABC .So
∆ ABC = ∆ PQR
General result we have here
If the hypotenuse and one other side of a right angled triangle are equal to the
hypotenuse and one other side of another right angled triangle then these two
triangles are congruent.
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Assignment
(1) Check the two triangles are congruent . Why?
3C .M
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5 C.M
5CM
3C.M
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