## Tag Archives: IITJEE Foundation Maths

### Proof of SAS Congruency Test of two triangles

Theorem: If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are equal in all respects.

Construction: Let ABC, DEF be two triangles in which $AB=DE$ and

$AC=DF$ and the included angle $\angle BAC$ is equal to included angle $\angle EDF$.

It is required to prove that the

$\Delta ABC = \Delta DEF$ in all respects.

Proof: Apply the $\Delta ABC$ to the $\Delta DEF$ so that the point A falls on the point D; and the side AB along the side DE. Then, because $AB=DE$ so the point B must coincide with the point E. And, because AB falls along DE, and the

$\angle BAC=\angle EDF$, so AC must fall along DF. And, because

$AC=DF$, the point C must coincide with the point F. Then, since B coincides with E, and C with F, hence, the side BC must coincide with the side EF. Hence, the $\Delta ABC$ coincides with the $\Delta DEF$, and is therefore equal to it in all respects. QED.

Nalin Pithwa

### Congruency Tests of two triangles — for PreRMO, RMO, IITJEE Foundation Math

Uptil now, we have discussed quite a bit of algebra, number theory and calculus for RMO, IITJEE Main and Advanced Mathematics. Let us switch back to plane geometry for a while.

In school, we all learn about tests of congruency of two triangles. But, have you wondered what is the exact meaning of “two plane figures are congruent”? Try to Google and find the answer for yourself. You can also post your answers as comments to this blog article.

The reasons I discuss these are two fold: (1) do you know the proofs of these congruency tests? In schools, students are just taught the meaning of these and how to apply them. Well, I will give you proofs of these in my blog at some later time. (2) There are some nuances to  these tests; when they cannot be applied, or rather, which *tests* fail to apply.

Here we go…

Two triangles are equal in all respects when the following three parts in each are severally equal:

1)Two sides, and the included angle. (the SAS Test for Congruency of 2 triangles).

2) The three sides. (the SSS Test for congruency of two triangles.

3) Two angles and one side, the side given in one triangle CORRESPONDING to that given in the other (ASA and AAS Test for Congruency of two triangles).

The two triangles are not, however, necessarily equal in all respects, when any three parts of one are equal to the corresponding parts of the other.

For example:

1) When the three angles of one are are equal to  the three angles of the  other, each to each, the fig 1 (attached, please download) shows  that the triangles need not be equal in all respects. (If you recall, such triangles are called similar, not  congruent because congruent means exactly same; two twins are similar, but not same).

2) When two sides and one angle in one are equal to two sides and one angle of the other, the given angles being opposite to equal sides, the  fig 2 (attached, please download) shows that the triangles need not be equal in all respects.

For, if \$AB=DE\$ and \$AC=DF\$ and the \$\angle ABC = \angle DEF\$, it will be seen that the shorter of the given sides in the triangle DEF may lie in either of  the positions DF or DF’.

Important Note: From these data, it may be shown that the angle opposite to the equal sides AB, DE are either equal (as for instance, the \$\angle ACB\$ and

\$\angle DF’E\$) or supplementary as (the \$\angle ACB and \angle DFE\$); and, that in the former case the triangles are equal in all respects. This is called the ambiguous case in the congruence of triangles.

Note: ambiguous means that which has some uncertainty. In other words, there could be more than one possibilities.

If the given angles at B and E are right angles, the ambiguity disappears.

In  the next few blog articles, we will discuss the proofs of congruency tests of two triangles.

More later,

Nalin PIthwa

### Math Basics A Fallacy in Geometry

You might think that crazy things like 1=2 can be *proved* if we make basic blunders in Algebra. But, in geometry also you can *prove* wild things like a right angle is an obtuse angle — errr…hmm…provided you make flaws in some fundamental assumption/axiom/theorem/property. So, check how strong are your basics in Euclidean/plane or high school geometry — point out my mistake in the proof “a right angle is an obtuse angle!”.

Please download the JPEG images of the two page *proof* 🙂

More later,

Nalin