Congruent Figures:
Congruent figures are the figures whose
shapes and sizes are same.
Congruent Triangles:
The triangles which can overlap each other
perfectly are called congruent triangles.
For congruent triangles corresponding
parts are equal.
If two triangles and are
congruent, we write as
![](https://tutormate.in/wp-content/uploads/2018/10/congruent-triangles.png)
And we will have
,
,
And
side side
side side
side side .
Criteria for congruency:
SAS criterion: Triangles are congruent if two sides and
included angle of one Triangle are equal to the
sides
and the included angle of other triangle.
![](https://tutormate.in/wp-content/uploads/2018/10/SAS-criterion.png)
ASA criterion: Two Triangles are congruent if two angles
and the included side of one triangle are equal
to
two angles and the included side of other triangle.
![](https://tutormate.in/wp-content/uploads/2018/10/ASA-criterion.png)
AAS criterion:Two Triangles are congruent if any two pairs
of angles and one pair of corresponding
sides
are equal.
![](https://tutormate.in/wp-content/uploads/2018/10/AAS-criterion.png)
SSS criterion: If three sides of one triangle are equal to
three sides of another triangle, then two
triangles
are congruent.
![](https://tutormate.in/wp-content/uploads/2018/10/SSS-criterion.png)
RHS theorem:
If
in two right triangles the hypotenuse and one side of one triangle are equal to
the hypotenuse and
one
side of other triangle, then the two triangles are congruent
![](https://tutormate.in/wp-content/uploads/2018/10/RHS-theorem.png)
Properties of triangle:
Theorem 1:
Angles opposite to equal sides of an
isosceles triangle are equal.
![](https://tutormate.in/wp-content/uploads/2018/10/theorem-1.png)
Given: is isosceles triangle with
To prove:
Construction: is
bisector of angle
Proof:
….( Given)
(By
construction)
common
(SAS
rule)
(CPCT)
Theorem 2:
The sides opposite to equal angles are
equal
![](https://tutormate.in/wp-content/uploads/2018/10/theorem-2.png)
Given: is triangle with
To prove:
Construction: is
bisector of angle Proof:
…(Given )
…. (By
construction)
….. (Common)
(AAS rule)
(CPCT)
Inequality in a triangle:
Theorem 3: If
two sides of a triangle are unequal, the longer side has greater angle opposite
to it.
Given:
To prove :
Construction:Take point D on
the side AC so that
Proof:
As
ABD is isosceles triangle.
….(By isosceles
triangle property)..…1
But is exterior
angle of
….. (Exterior
angle is greater than remote interior angle.)……………2
By and
But
…. (From the
figure)
Theorem 4:Converse of the theorem:
In a
triangle the greater angle has longer side opposite to it.
![](https://tutormate.in/wp-content/uploads/2018/10/theorem-4.png)
Given: In
To prove :
Proof:There are 3 cases
Case i:
Case ii:
Case iii: .
Case
i
(Angles
opposite to equal sides)
This is contradiction
Since
So
Case
2)
(Angle opposite to greater side)
contradicts to
So,
Theorem 5:
The
sum of any two sides of a triangle is greater than the third side.
![](https://tutormate.in/wp-content/uploads/2018/10/theorem5-construction.png)
Given:
To prove:
1)
2)
3)
Construction:
Produce to
such
that Join
.
![](https://tutormate.in/wp-content/uploads/2018/10/theorem-6.png)
Proof:
In , we have ..(By construction)
… (Angles
opposite to equal sides)
… (From the fig.)
……….. (Side opposite to larger angle)
.
Similarly, we can prove other two
inequalities.
Theorem6 :
In
the right angle triangle, the hypotenuse is the longest side.
![](https://tutormate.in/wp-content/uploads/2018/10/theorem-6.png)
Given:
, .
To prove: is largest side
i.e. and .
Proof:
(Angle sum
property of triangle)
.
and
are acute angles
,
.(Sides opposite to greater angles)
is largest side.