Triangles grade-09



Triangl is a three sided polygon. Triangle is made up with the parts of sides, angles and vertex.

congruence of triangle

If an object is Congruence with other  object, then the both object can be perfectly site one over another. So if two triangle are congruent then all of the parts of the two triangle should be same.

congruent triangle
congruent triangle

Consider the triangle i, ii, we can say both triangle are congruent because all the side and angle are same for bot triangle. But here we have to take care to say the name of both congruent triangle. Because in the figure i and ii,  the two triangles will be congruent  only if we keep the side AB of the one triangle over the side PQ of the another triangle. So the name of both congruent triangle should be in the congruent order.

here ΔABC≅ΔPQR. It can be write in many ways corect as well as wrong. so please take care and write the name of other congruent triangle in the fig. with the triangle ABC.

Important Notes. Corresponding parts of congruent triangle are same (CPCT).

In Order to find two triangles are congruent we have to check by keep both triangle one over another.  This is not possible in all time, so now we can study some criterias to check the congruence  of triangles.

SAS Congruence rule

if two side and one include angle are same for two triangle then the two triangle will be in congruence

This can’t be proven by previously known result. So it is accepted true as an axiom.


 


ASA congruence rule

If two angles and included side of two triangles are same then they are in congruent.

This can be proved so that it can be called as a theorem.

Theorem 7.1 (ASA) :

ASA CONGRUENT RULE
ASA CONGRUENT RULE

In the Above fig ∠B=∠E,  ∠C= ∠F and BC=EF (Given)

consider AB=DE Then we can say that ΔABC≅ΔDEF (SAS RULE)

now jus we have to prove that AB=DE

Suppose BA greater than DE, Then we can construct a line PC  as shown in the fig. 7.12

Then  PB=DE (by construction)

∠B=∠E (given)

BC=EF (given)

Ther for  ΔPCB≅ΔDEF (SAS)

As we know CPCT are same

∠PCB=∠DEF (CPCT)

∠ACB=∠DEF (given) there for ∠PCB=∠ACB and this is only possible when P is coincide with A

There for BA =ED and   ΔABC≅ΔDEF (SAS RULE) so ASA rule is proved for the congruence of triangles.



AAS CONGRUENCE RULE

This congruence rule says that if any two angle and any one side of two triangle is same then the triangle will be in congruence.

This is happening because, we know that the sum of all the angle of a triangle is 180º.  There for if two angle is same for two triangle then the third angle will also same. (same thing is subtracted from equal then the result will also same). If three angle and one side is same the these triangles will in congruence by ASA rule. And hence we can say that AAS rule is valid.



Some property of a triangle

As we know in an isosceles  triangle the two side are equal. By constructing many Isosceles triangle we can check the angle opposite to the same side.  then we can see that the opposite angles of the same sides are equal. This one can be prove by the congruence rule.

Theorem 7.2 : Angles opposite to equal side of an isosceles triangle are equal

isosceles triangle
isosceles triangle

Consider the triangle ABC where AB=AC , Now constructing the bisector of the angle A with the line AD. Now consider the two triangles BAD and CAD

AB=AC (Given)

∠BAD=∠CAD (By construction)

AD=AD (common line)

ther for ΔBAD≅ΔCAD

Ther for ∠B=∠C (CPCT)

So it is proved.

By the observation, trial and error we can say that converse  of this theorem is also true. That is if two angle of any triangle  equal then the opposite side will also equal. It can also proof as a theorem.

Theorem 7.3 : The side opposite to equal angle of a triangle are equal.

proof

isosceles triangle
isosceles triangle

Consider the triangle ABC. And draw the bisector of the ∠A  by the line AD, is also perpendicular to BC.

∠BAD=∠CAD (By construction)

AD=AD (Common Side)

∠ADB=CDA=90 So,

ΔABD≅ΔACD (ASA rule)  So,

AB=AC (CPCT)



SSS congruence rule

Theorem  7.4 : If three side of one triangle are equal to three side of other triangle then two triangle are congruent

As we know if three side of two triangle are equal then there three angle are also equal. so this theorem can proven by SAS SSA rules.

RHS congruence rule

Theorem 7.5 : In two right angle triangle if the hypotenuse and one side is same then two triangle are congruent.

Here RHS stands for Right angle, Hypotenuse, Side. But here this three can be any order.

Inequalities in triangle

By observing  we can understand that any one side of the triangle increased then the opposite angle also increase.

Theorem 7.6 : If two side of the triangle are unequal, the angle opposite to the larger side is larger.

triangle
triangle

From this fig. we can see that, when the one side of the triangle increased, the angle opposite to that side will also increase. thus the theorem is proved.

Theorem 7.7 We can also say that the side opposite to the biggest angle will be the logestside. 

Theorem 7.8 The sum of any two side of a triangle is greater than the third side.

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