Geometry Proofs Congruent Triangles

Introduction - geometry proofs congruent triangles:

Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. For a positive integer n, two integers a and b is said to be congruent modulo n, written:

a`-=`b (mod n)

if their difference a-b is an integer multiple of n. The number n is called the modulus of the congruence.

Source: Wikipedia

Theorem- Geometry Proofs Congruent Triangles:

SAS (SIDE ANGLE SIDE) geometry proofs congruent triangles Axiom:

“Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other” it is called as similar congruent triangle.

Note: This axiom refers to two sides and the angle between them, so it is known as side angle or SAS congruence axiom or SAS criterion.

In the figure given below, if AB=ab and `angle`B=`angle`B, then ∆ABC`~=`∆abc (i.e., CA=ca, `angle`A=`angle`a, `angle`C=`angle`c would also be true).                               

Note.  In SAS criterion, equality of included angle is essential. Thus if in ∆ABC and ∆abc. AB=ab, BC=bc and `angle`A=`angle`A (or `angle`C=`angle`C), then ∆ABC will not be congruent to ∆abc because the included angle `angle`B and `angle`b are not given to be equal.

Theorem for geometry proofs congruent triangles:

Angles opposite to two equal sides of a triangle are equal.

Given:

In ∆ABC, AB=AC.

To prove:

`angle`C=`angle`B.

Construction:

We draw the bisector AW of `angle`A which meets BC in W.

Proof:

In ∆’s ABW and ACW,

AB=AC                                  given

`angle`BAW=`angle`CAW                           construction

AW=AW                                  common side

∆ABW`~=`∆ACW                            SAS congruence Axiom

`angle`B=`angle`C                              

ASA (ANGLE SIDE ANGLE) geometry proofs congruent triangles:

“Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.”

Given:

In ∆’s ABC and abc

`angle`B=`angle`B, `angle`C=`angle`C and BC=bc.

Result:

∆ABC `~=`∆abc and hence

`angle`A=`angle`a, AB=ab and AC=ac.

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Example- Geometry Proofs Congruent Triangles:

In two right triangles, one side and an acute angle of one triangle are equal to one side and the corresponding acute angle of the other triangle. Prove that the two triangles are congruent.

Solution:

In ∆VST and ∆vst                                  

`angle`S=`angle`s                                            each 90o given

And      `angle`T=`angle`t                                           given

`angle`V=`angle`v                                           the sum of three angles of a triangle is 180o

Now, in ∆VST and ∆vst    

`angle`V=`angle`v                                            from

`angle`S=`angle`s                                            each=90

VS=vs                                               given

∆VST`~=` ∆vst                                  ASA congruence Axiom

`angle`2=`angle`3

Now in ∆VTS,

`angle`2=`angle`3

`angle`ST=`angle`VT                                            sides opp. To equal angle are equal

VS=VT

Hence, ∆VST is an isosceles triangle.