Triangles are congruent when they have

exactly the same three sides and exactly

the same three angles.

It means that one shape can become another using Turns, Flips and/or Slides:

Rotation Turn!

Reflection Flip!

Translation Slide!

When two triangles are congruent they

will have exactly the same three

sides and exactly the same three angles.

The equal sides and angles may not be

in the same position (if there is a turn or a

flip), but they are there.

Same Sides

When the sides are the same then the

triangles are congruent.

For example:

is congruent to: and

because they all have exactly the same

sides.

But:

is NOT congruent to:

because the two triangles do not

have exactly the same sides.

Same Angles

Does this also work with angles? Not

always!

Two triangles with the same angles might

be congruent:

is congruent to:

only because they are the same size

But they might NOT be congruent because of different sizes:

is NOT congruent to:

because, even though all angles match, one is larger than the other.

So just having the same angles is no guarantee they are congruent.

Two triangles are congruent if they have:

exactly the same three sides and

exactly the same three angles.

But we don't have to know all three sides

and all three angles ...usually three out of

the six is enough.

SAS(side-angle-side) congruence

› Two triangles are congruent if two

sides and the included angle of one

triangle are equal to the two sides

and the included angle of other

triangle.

A

B C

P

Q R

S(1) AC = PQ

A(2) ∠C = ∠R

S(3) BC = QR

Now If,

Then ∆ABC ≅ ∆PQR (by SAS

congruence)

› 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.

A

B C

D

E F

Now If, A(1) ∠BAC = ∠EDF

S(2) AC = DF

A(3) ∠ACB = ∠DFE

Then ∆ABC ≅ ∆DEF (by ASA

congruence)

ASA(angle-side-angle) congruence

•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.

A

B C P

Q

R

Now If, A(1) ∠BAC = ∠QPR

A(2) ∠CBA = ∠RQP

S(3) BC = QR

Then ∆ABC ≅ ∆PQR (by AAS

congruence)

SSS(side-side-side) congruence

•If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

Now If, S(1) AB = PQ

S(2) BC = QR

S(3) CA = RP

A

B C

P

Q R

Then ∆ABC ≅ ∆PQR (by SSS

congruence)

RHS(right angle-hypotenuse-side) congruence

•If in two right-angled triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

Now If, R(1) ∠ABC = ∠DEF = 90°

H(2) AC = DF

S(3) BC = EF

A

B C

D

E F

Then ∆ABC ≅ ∆DEF (by RHS

congruence)

PROPERTIES OF TRIANGLE

A

B C

A Triangle in which two sides are equal in length is called

ISOSCELES TRIANGLE. So, ∆ABC is a isosceles triangle with

AB = BC.

Angles opposite to equal sides of an isosceles triangle are

equal.

B C

A

Here, ∠ABC = ∠ ACB

The sides opposite to equal angles of a triangle

are equal.

CB

A

Here, AB = AC

Theorem on inequalities in a triangle

If two sides of a triangle are unequal, the angle opposite to the

longer side is larger ( or greater)

10

8

9

Here, by comparing we will get that-

Angle opposite to the longer side(10) is greater(i.e. 90°)

In any triangle, the side opposite to the longer angle is

longer.

10

8

9

Here, by comparing we will get that-

Side(i.e. 10) opposite to longer angle (90°) is

longer.

The sum of any two side of a triangle is greater than

the third side.

10

8

9

Here by comparing we get-

9+8>10

8+10>9

10+9>8

So, sum of any two sides is greater than the third side.