SIMILAR

1
SIMILAR
TRIANGLES
Presentation by Shweta Singh Huria, TGT(Maths),
KV,Sector-8, Rohini, New Delhi.
2
In geometry, two figures or objects are congruent
if they have the same shape and size, or if one
has the same shape and size as the mirror image
of the other This means that either object can be
repositioned and reflected (but not resized) so
as to coincide precisely with the other object.
So two distinct plane figures on a piece of paper
are congruent if we can cut them out and then
match them up completely. Turning the paper over
is permitted. Congruence of two triangles In
elementary geometry the word congruent is often
used as follows.The word equal is often used in
place of congruent for these objects. Two line
segments are congruent if they have the same
length. Two angles are congruent if they have the
same measure. Two circles are congruent if they
have the same diameter. In this sense, two plane
figures are congruent implies that their
corresponding characteristics are "congruent" or
"equal" including not just their corresponding
sides and angles, but also their corresponding
diagonals, perimeters and area
3
(No Transcript)
4
m your earlier classes. In Class IX, you have
studied congruence of triangles in detail. Recall
that two figures are said to be congruent, if
they have the same shape and the same size. In
this chapter, we shall study about those figures
which have the same shape but not necessarily the
same size. Two figures having the same shape (and
not necessarily the same size) are called similar
figures.
We shall study about those figures which have the
same shape but not necessarily the same size
5
Consider any two (or more) circles . Are they
congruent? Since all of them do not have the same
radius, they are not congruent to each other.
Note that some are congruent and some are not,but
all of them have the same shape.So they all are,
what we call, similar.Two similar figures have
the sameshape but not necessarily the same size.
Therefore, all circles are similar.What about two
(or more) squares or two (or more) equilateral
triangles /As observed in the case of circles,
here also all squares are similar and all
equilateral triangles are similar. From the
above, we can say that all congruent figures
are similar but the similar figures need not be
congruent.
6
we can say that all congruent figures are similar
but the similar figures need not be congruent.
7
For example, all circles are similar to each
other,all squares are similar to each other, and
all equilateral triangles are similar to each
other.
8
Similar triangles are triangles that have the
same shape but not necessarily the same size.
?ABC ? ?DEF
When we say that triangles are similar there are
several repercussions that come from it.
?A ? ?D
?B ? ?E
?C ? ?F
9
If we need to prove that a pair of triangles are
similar how many of those statements do we need?
Because we are working with triangles and the
measure of the angles and sides are dependent on
each other. We do not need all six. There are
three special combinations that we can use to
prove similarity of triangles.
1. SSS Similarity Theorem ? 3 pairs of
proportional sides
2. SAS Similarity Theorem ? 2 pairs of
proportional sides and congruent angles between
them
3. AA Similarity Theorem ? 2 pairs of
congruent angles
10
1. SSS Similarity Theorem ? 3 pairs of
proportional sides
?ABC ? ?DFE
11
2. SAS Similarity Theorem ? 2 pairs of
proportional sides and congruent angles between
them
m?H m?K
?GHI ? ?LKJ
12
The SAS Similarity Theorem does not work
unless the congruent angles fall between the
proportional sides. For example, if we have the
situation that is shown in the diagram below, we
cannot state that the triangles are similar. We
do not have the information that we need.
Angles I and J do not fall in between sides GH
and HI and sides LK and KJ respectively.
13
3. AA Similarity Theorem ? 2 pairs of
congruent angles
m?N m?R
?MNO ? ?QRP
m?O m?P
14
It is possible for two triangles to be similar
when they have 2 pairs of angles given.
m?T m?X
m?S m?Z
m?S 180?- (34? 87?)
?TSU ? ?XZY
m?S 180?- 121?
m?S 59?