This document discusses congruent and similar shapes. It defines that congruent shapes have equal sides and equal angles, while similar shapes have the same shape but not necessarily the same size. It provides examples of figures that are congruent and similar. It also discusses properties of different types of triangles, such as isosceles triangles that have two equal sides, equilateral triangles that have three equal sides, and right triangles that have one 90 degree angle. The document concludes with methods to prove triangles congruent, such as using side-side-side properties, side-angle-side properties, or angle-side-angle properties.

1. Congruent and Similar
Triangles

2. Introduction
Recognizing and using congruent and
similar shapes can make calculations and
design work easier. For instance, in the
design at the corner, only two different
shapes were actually drawn. The design
was put together by copying and
manipulating these shapes to produce
versions of them of different sizes and in
different positions.
In this chapter, we will look in a little more
depth at the mathematical meaning of the terms
similar and congruent, which describe the relation
between shapes like those in design.

3. Similar and Congruent Figures
• Congruent polygons have all sides
congruent and all angles congruent.
• Similar polygons have the same shape;
they may or may not have the same size.
Worksheet :
Exercise 1 : Which of the following pairs are congruent
and which are similar?

4. Examples
These figures are
similar and congruent.
They’re the same shape
and size.
These figures are similar
but not congruent.
They’re the same shape,
but not the same size.

5. Another Example
These figures are neither
similar nor congruent.
They’re not the same
shape or the same size.
Even though they’re both
triangles, they’re not
similar because they’re
not the same shape
triangle.
Note: Two figures can be similar but not
congruent, but they can’t be congruent but not
similar. Think about why!

6. Congruent Figures
When 2 figures are congruent, i.e.
2 figures have the same shape
and size,
Corresponding angles are equal
Corresponding sides are equal
Symbol : ≡

7. Congruent Triangles
• AB = XY, BC = YZ, CA = ZX
∀∠A =∠X , ∠B =∠Y, ∠C =∠Z
ABC XYZ≡V VA
C
B
X
Z
Y
Note : Corresponding vertices are named in order.

8. THE ANGLE MEASURES OF A TRIANGLE AND
CONGRUENT TRIANGLES
• The sum of the angle measures of a triangle is 180o
Example
30o
65o
?
? = 85o
• Congruent triangles
90o
60o
5 cm
?
?
Example
Congruent triangles are triangles with the same shape and size
Angle = 60o
; side = 5cm

9. Isosceles triangles
• An isosceles triangle is the triangle which
has at least two sides with the same length
• In an isosceles triangle, angles that
are opposite the equal-length sides have
the same measure
The side = 82 cm, the angle = 76o
82cm
52o
?
?Example

10. Equilateral triangles
• An equilateral triangle has three sides of equal
length
• In an equilateral triangle, the measure of
each angle is 60o
Example 60o
100cm
?
?
Angle = 60o
, side = 100 cm

11. Right triangles and Pythagorean theorem
• A right triangle is the triangle with one right angle
• Pythagorean theorem
c2
= a2
+ b2
Example
Leg
a
Leg
b
Hypotenuse
c
c2
= 42
+ 32
= 25
60o
?
3 cm
4 cm
?
C = 5

12. Ex 10A Page 47
• Q2 a
• By comparing,
x = 4.8,
y = 42
• Q2 d
• By comparing,
x = 22,
y = 39 – 22
= 17
• Q2 b
• By comparing,
x = 16,
y = 30
( 180°- 75°- 75°)

13. Tests for Congruency
Ways to prove triangles congruent :
• SSS ( Side – Side – Side )
• SAS ( Side – Angle – Side )
• ASA ( Angle – Side – Angle ) or AAS
( Angle –Angle – Side )
• RHS ( Right angle – Hypotenuse – Side )

14. Thank You
MADE BY:
UMANG LAKHERA
CLASS: 10