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Contents:
 Similarity
 Similar figures
 Properties of similar figures
 Figures which are always similar
 Theorems of similar triangles
 Area and perimeter of similar triangles
 Congruency
 Congruent figures
 Theorems of congruent triangles
SIMILARITY
If two objects have the same shape, but one is the enlargement of the other then; they are called
"similar."
This is symbol that means “similar”
Real life examples of similar figures:
 Enlargement of a photograph, when a photograph is enlarged its size differ
from each other but their shape is similar.
 Shadow of a cardboard shape, if the shade is held in front if a light source, parallel to a
screen then a shadow of the shape will appear on the screen. The shape will be known as
a similar figure as it will have the same shape on the screen but a different size.
 Painting car, often car painters use a similar small model of their car to paint on the
model before painting the original car; to be more accurate.
Example: a real life solving problem
QUESTION:
Suppose you wanted to find the height of the tree. Unfortunately all that you have is a tape
measure, and you are too short to reach the top of the tree. How will you find the height of the
tree?
SOLUTION:
You can measure the length of the tree’s shadow. Then, measure the length of your shadow. If
you know how tall you are, then you can determine how tall the tree is. If your height is 6feet
and your shadows height is 2feet than you can find the height of tree.
If you are 6ft and the shadow is 2ft, it means that the shadow is 3 times smaller than you.
Therefore if the tree’s shadow is 10ft than the tree will be 3 times it.So, the tree will be 30ft.
Thus, the example above shows that the shadow of both the tree and you is a similar figure; as
only their size differs not their shape.
Similar figures
Are figures for which all corresponding angles are equal and all corresponding sides are
proportional. Corresponding sides are two sides of two similar figures which are relatively
positioned. The two triangles which are similar must be equi-angular.Also,when two figures are
similar; the ratios of the lengths of their corresponding sides are equal .
Example: which sides corresponds to which?
Properties of two figures to be Similar
1.) Perimeters of similar triangles:
Perimeters of
similar triangles are in the same ratio as their corresponding sides and this ratio is called the scale
factor.
There are two
similar triangles are
.
This ratio is called the scale factor.
Thus, the perimeters of two similar triangles are in the ratio of their scale factor.
2.) Areas of similar triangles:
The ratio of the areas of two similar triangles is equal to the ratio of the squares of the
corresponding sides, i.e. the square of the scale factor.
To prove that, draw perpendicular from A and P to meet seg.BC and seg.QR at D and S
respectively.
Since
Thus the areas of two similar triangles are in the same ratio as the square of their scale factors.
3.)Corresponding angles are the same
So in the figure above, ∠P =∠L, ∠Q =∠M, and ∠R =∠N.
From this, it follows that the corresponding exterior angles will also be the same.
4.)Corresponding sides are all in the same proportion
By definition each pair of corresponding sides are in the same proportion, or ratio.
Formally, in two similar triangles PQR and LMN :
So, for example, if in two similar polygons one side is twice the length of the corresponding side
in the other, Then all the other sides will be twice the length of their corresponding side also.
5.)Corresponding diagonals are in the same proportion
In each polygon the corresponding diagonals are in the same proportion. Their ratio is the same
as the ratio of the sides.
FIGURES WHICH ARE ALWAYS SIMILAR
1.)Circles are always similar because its angle is always 360◦ although its size vary from big to
small.
2.)All equilateral triangles are similar as all three angles of an equilateral triangle are equal. Thus
they add up to 180◦. Even if they vary in different size all the three angles will be equal.
3.) All squares are similar as all their four angles add up to 360◦,and each angle is 90◦. Thus, the
four sides of a square are proportionally corresponding.
All squares are similar
Theorems of similar figures
1.)a segment joining the midpoints of two sides of a triangle is parallel to the third side, and its
measure is one-half the measure of the third side.
Example:
R
Question:
find the measure of each angle for the figure given. The figure
represents letters S and T as midpoints; and R is 25◦, STR is 115◦.
a.) Q b.) RST C.) P d.)PST e.) QTS
T
Solution:
a.)Q=115◦ {as
corresponds
. Therefore; Q  STR }
b.)RST=40◦ {180◦-(115◦+25◦)=40◦
c.)P=40◦ {PRST}
d.)PST=140◦ {180◦-40◦=140◦}
e.)QTS=65◦ {180◦-115◦=65◦}
S
Q
P
PPP
P
2.) In similar triangles, corresponding altitudes are proportional to corresponding sides.
Example:
D
Question:
find the value of x and y.
Solution:
ACD  EGH
X
A
=
set up a proportion
9x = 84 cross products are equal
x=9
8
B
=
12y = 72
y=6
H
12
22
2
9
7
Y
C
E
F
G
3.) If a segment is parallel to one side of a triangle and intersects the other sides in two points,
then the tr iangle formed is similar to the original triangle.
4.) If a segment is parallel to one side of a triangle and intersects the other sides in two points,
then the segment divides those two sides proportionally.
Example:
P
Question:
find PT and PR
4
T
S
Solution:
=
X
12
7
7x = 48; therefore x = 6
the answer will be, PT = 6 and PR = 12 + 6
Q
R
=18
5.) If three parallel lines intersects two transversals, then they divide the transversals
proportionally.
6.) In similar triangles, corresponding medians are proportional to corresponding sides.
Example:
V
Question:
are the triangles UVW and KLM similar?
Solution:
find the ratios of corresponding sides.
=
=
=
=
The sides that include V and L are
proportional.
V  L Given from the figure
the answer will be,UVW  KLM
15
9
U
W
L
12
2
K
20
M
Area of similar triangles
The ratio of the areas of similar triangles is equal to the square of the ratio of corresponding
sides.
In the figure below, if triangles ABC and XYZ are similar then,
A
X
=
=
B
C
Y
find the area of triangle XYZ given that the area of triangle ABC is 12
.
D
=
W
Z
=
=
K being the scale factor.
Example:
Question:
X
In triangle XYZ, Z = 60◦ and X
A
=50◦;and In triangle ABC,B = 70◦and
C=60◦. Hence
B
C
Y
Z
=5cm and
=10cm.
Solution:
In triangle XYZ, Y = 70◦;and In triangle ABC,A = 50◦. Hence the two triangles are similar
because they are equiangular ,BC and YZ correspond, therefore
K=
=
=
=4
=2
=4
Area of XYZ = 4 12
= 48
Exercise:
1.) In the following figure, the triangle ABC and EFG are similar. If the area of ABC is
E
the area of EFG.
A
, calculate
6cm
2cm
M
B
C
F
G
2.) In the triangle LPR, MN is parallel to PR. The area of triangle LMN=
LPR=
.
A.)Calculate the area of MNRP
L
M
B.)Calculate the ratio
fraction in its lowest terms.
N
8cm
P
and the area of triangle
C.)If MN= 8cm, find the length of PR.
R
as a
Perimeters of similar triangles
In similar triangles, that is corresponding angles are the same and corresponding sides are
proportional then, ratio of corresponding sides = ratio of their perimeter.
Triangle DEF is bigger than triangle ABC.
Congruency
It is when two objects are of the same shape and size. Then, this objects are known as
‘congruent’.

Real life examples of congruent figures:
 Human’s eyes, every human’s both eyes are of the same shape and size.
Congruent figures
Two figures are congruent if they have the same shape and size. When two triangles are
congruent, you can fit one on top of the other so that the two figures match exactly. The sides
and angles that match are corresponding parts. The vertices that math are corresponding
vertices.
B
A
C
CORRESPONDING VETRICES
CORRESPONDING PARTS
A corresponds to D
B corresponds toE
C corresponds toF
A corresponds to D
B corresponds to E
C corresponds to F
corresponds to
corresponds to
corresponds to
When writing a congruement statement, list the corresponding vertices in the same order.
Below are, all the ways for the triangles above. Note that all these congruency statements imply
the same correspondence.
ABCDEF
BACEDF
CABFDE
ACB DFE
BCA EFD
CBAFED
Congruent triangles are triangles whose vertices can be made to correspond in such a way that
the corresponding parts of the triangles are congruent.
Every triangle has six parts; three sides and three angles. If two triangles are congruent, then six
pairs of corresponding parts are congruent. Also, if the vertices can be matched so that all six
pairs of corresponding parts are congruent, then the triangles are congruent,
Theorems of congruent figures
1.) If two angles and a nonincluded side of one triangle are congruent to two angles and the
corresponding nonincluded side of another triangle,then the two triangles are congruent .
2.) If the legs of one right triangle are congruent to the legs of another right triangle, then the
two right triangles are congruent.
3.) If a leg in an acute angle of one right triangle are congruent to the corresponding parts of
another right triangle,then the two right triangles are congruent.
4.) If the hypothenese and an acute angle of a rihght triangle are congruent to the
hypothense and an acute angle of another right triangle, then the right triangles are
congruent.
5.) The base angles of an isosceles triangle are congruent.
6.) If a triangle is equilateral , then it is equiangular.
7.) The measure of each angle of any equilateral triangle is 60◦.
8.) If two angles of a triangle are congruent, then they are the base angles of an issoscels
triangle.
9.) Every equiangular triangle is equilateral.
3-D shapes
The conditions that the two cubes above are similar are:



Corresponding angles are congruent.
Corresponding sides are in proportional to each other.
The ratios of their measures are equal of the two corresponding sides.
The shapes above are called ‘similar’ as their four angles are corresponding to each
other. Also, their sides are proportionally corresponding.they are called similar because
all four angles are equal 90◦; thus all cubes have the same angle.the main concept is
that all cubes add upto 360◦.
GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY
Writing
skill
3
creativit
y flow
neatness
Drawing
skill
2
pictures
labeling
0
1
Formatting
skill
3
page
paragraphs
math
equations
1
Polygon
2
alwayssimilar
polygon
s appln
2
Triangle
2
theorem
s
applns
3D
shapes
3
perimete
r
vol,
SA
1
1
Questions
3
3D
shapes
congruenc
y variety
0
Example
s
2
ex+ans
0
Total
20
6
Hiral, you have not taken this work seriously… most of the work is copy pasted from some other
source which is not the intension of giving such assignment.. you wouldn’t have learnt anything
from this work… be serious!