Question

In two triangles, if the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal.
Prove this theorem in detail.

Answer 1

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Given:-

In two triangles, if the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. It also called SSS (side-side-side) criterion.

Two triangles ABC and DEF are drawn in such a way that their corresponding sides are proportional. It means:

AB/DE = AC/DF = BC/EF

Proof:

To prove: ∠A = ∠C, ∠B = E and ∠C = ∠F

Hence, triangle ABC ~ DEF

In Triangle DEF, draw a line PQ so that DP = AB and DQ = AC

Since the corresponding sides of the two triangles are equal.

This implies;

DP/PE = DQ/QF = PQ/EF

This also means that ∠P = ∠E and ∠Q = ∠F

We had taken, ∠A=∠D, ∠B=∠P and ∠C=∠Q

Hence,

∠A = ∠D, ∠B = ∠E and ∠C = ∠F

Therefore, from AAA criterion;

Triangle ABC ~ DEF.

Hence Proved.

Answer 2

In two triangles, if the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. It also called SSS (side-side-side) criterion.

Two triangles ABC and DEF are drawn in such a way that their corresponding sides are proportional. It means:

AB/DE = AC/DF = BC/EF

Proof:

To prove: ∠A = ∠C, ∠B = E and ∠C = ∠F

Hence, triangle ABC ~ DEF

In Triangle DEF, draw a line PQ so that DP = AB and DQ = AC

Since the corresponding sides of the two triangles are equal.

This implies;

DP/PE = DQ/QF = PQ/EF

This also means that ∠P = ∠E and ∠Q = ∠F

We had taken, ∠A=∠D, ∠B=∠P and ∠C=∠Q

Hence,

∠A = ∠D, ∠B = ∠E and ∠C = ∠F

Therefore, from AAA criterion;

Triangle ABC ~ DEF.

Proved.