Similar Triangles

We will discuss here about the similar triangles.

If two triangles are similar then their corresponding angles are equal and corresponding sides are proportional.

Here, the two triangles XYZ and PQR are similar.

So, ∠X = ∠P, ∠Y = ∠Q, ∠Z = ∠R and \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) = \(\frac{XZ}{PR}\).

∆XYZ is similar to ∆PQR.  We write ∆XYZ ∼ ∆PQR (the symbol ‘∼’ means ‘similar to‘.)

Corrosponding Sides:

Sides opposite to equal angles in similar triangles are known as corresponding sides and they are proportional.

Here from the given figures ∠X = ∠P, ∠Y = ∠Q and ∠Z = ∠R.

Therefore, XY and PQ are corresponding sides as they are opposite to ∠Z and ∠R respectively.

Similarly from the given figure, YZ and QR are a pair of corresponding sides. XZ and PR are also a pair of corresponding sides.

Thus, \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) = \(\frac{XZ}{PR}\), as corresponding sides of similar triangles are proportional.

Corrosponding Angles:

Angles opposite to proportional sides in similar triangles are known as corresponding angles.

If ∆XYZ ∼ ∆PQR and \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) = \(\frac{XZ}{PR}\) then ∠X = ∠P as they are opposite to corresponding sides YZ and QR respectively.

Similarly from the given figure, ∠Y = ∠Q and ∠Z = ∠R.

Congruency and Similarity of Triangles:

Congruency is a particular case of similarity. In both the cases, three angles of one triangle are equal to the three corresponding angles of the other triangle. But in similar triangles the corresponding sides are proportional, while in congruent triangles the corresponding sides are equal.

∆XYZ ∼ ∆TUV.

Therefore, \(\frac{XY}{TU}\) = \(\frac{YZ}{UV}\) = \(\frac{XZ}{TV}\) = k, where k is the constant of proportionality or the scale factor of size transformation.

∆XYZ ≅ ∆PQR.

Here, \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) = \(\frac{XZ}{PR}\) = 1.

Therefore, in congruent triangles the constant of proportionality between the corresponding sides is equal to one. Thus, congruent triangles have the same shape and size while similar triangles have the same shape but not necessarily the same size.

Congruent triangles are always similar, but similar triangles are not necessarily congruent.

Note: Triangles that are similar to the same triangle are similar to each other.

Here, ∆XYZ ∼ ∆PQR and ∆ABC ∼ ∆PQR.

Therefore, ∆XYZ ∼ ∆ABC.

9th Grade Math

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