Project in MathCongruent of Triangles

S.Y 2014 – 2015Submitted by:

Congruent Triangles

Triangles are congruent when they have exactly the same three sides and exactly the same three angles.

What is "Congruent" ... ? It means that one shape can become

another using Turns, Flips and/or Slides:

Rotation Turn!

Reflection Flip!

Translation Slide!

Congruent Triangles

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 SidesWhen 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.

How To Find if Triangles 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.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

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.

SAS CONGRUENCE-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)

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.

ASA CONGRUENCE-A

B C

D

E FNow 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.

AAS CONGRUENCE-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.

SSS CONGRUENCE-

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.

RHS CONGRUENCE-

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

INEQUALITIES IN A TRIANGLE

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>108+10>910+9>8

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