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Grade 10

Similarity in triangles


As the name suggests, similar triangles share common features but are not exactly the same. In other words, a triangle will either be enlarged or shrunk but the angles will always be the same.  


Features of similar triangles

In similar triangles:

  • all corresponding angles are equal
  • all corresponding sides are in the same ratio


Just like congruent triangles, similar triangles may be transformed, so take your time working out whether triangles are similar. 


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


Proofs for Similar Triangles

There are three distinct methods for proving two triangles are similar. You only need to use one of these proofs to show two triangles are similar, then all the features of similar triangles can be applied. 


AAA (Angle, Angle, ANGLE)

This proof is also sometimes called equiangular, which means all corresponding angles in similar triangles are equal. However, we only have to prove two pairs of corresponding angles are equal because the third angle must be equal since the angle sum of a triangle is always $180^\circ$180°.

The triangles below show that all three corresponding pairs of angles are equal. Therefore, we can say that $\triangle ABC$