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STATEMENT - 2 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

Find the correct option from the following:

A) Statement â€“ I is True, Statement-2 is True; Statement â€“ 2 is a correct explanation for the Statement â€“ I.

B) Statement â€“ I is True, Statement â€“ 2 is True; Statement â€“ 2 is not a correct explanation for Statement â€“ I.

C) Statement â€“ I is True, Statement â€“ 2 is False.

D) Statement â€“ I is False, Statement â€“ 2 is True.

Answer
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We have statement 1 as If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.

First, we will try to find out if it is true or not.

Letâ€™s consider two triangles as follows:

Without loss of generality, let us suppose that $\angle A = \angle D$ and $\angle B = \angle E$ â€¦â€¦â€¦(1)

We also know that the sum of all the angles of a triangle results in 180$^ \circ $.

So, \[\angle A{\text{ }} + \angle \;B{\text{ }} + \angle C{\text{ }} = {180^ \circ }\] â€¦â€¦.(2) and similarly \[\angle D{\text{ }} + \angle E{\text{ }} + \angle F = {180^ \circ }\]â€¦â€¦.(3)

Comparing (2) and (3), we have:-

\[\angle A{\text{ }} + \angle \;B{\text{ }} + \angle C{\text{ }} = \angle D{\text{ }} + \angle E{\text{ }} + \angle F\]

This implies \[\angle A{\text{ }} - \angle D{\text{ }} + \angle \;B{\text{ }} - \angle E{\text{ }} + \angle C{\text{ }} = \angle F\] â€¦â€¦.(4)

Using (1) in (4):-

$\angle C = \angle F$

Hence, all the corresponding angles are equal and thus, we have:-

$\vartriangle ABC \sim \vartriangle DEF$

This is because of the AAA similarity rule.

Now, in statement two the reason is different.

Hence, option (A) is discarded and since statement 1 is true, so option (D) is discarded as well.

Now, let us check statement 2.

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

This states that in the given triangles $\dfrac{{AB}}{{DE}} = \dfrac{{BC}}{{EF}}$.

This is a fact related to similar triangles which we can prove by constructing PQ parallel to EF and then using the theorem that: If a line is drawn parallel to one side of the triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Hence, it is true as well.

Hence, the correct option is (B).

We might make the mistake to infer statement 2 as the correct reason because Statement 1 implies statement 2.

We must remember the facts that: sum of all the angles of a triangle results in ${180^ \circ }$ and If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

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