# SIMILAR TRIANGLES

How to prove two triangles are similar ?

If two triangles are similar, then the measures of their corresponding sides are proportional, and the measures of their corresponding angles are equal.

If ΔABC  ΔDEF,

then AB/DE  =  BC/EF  =  AC/DF Let us see some example problems to understand how to prove two triangles are similar.

Example 1 : Solution :

The angles BAC and DFE  are congruent.To prove the above triangles are similar, we need to prove one more pairs of angles are equal.

To check whether the angles BCA and DEF are equal, let us find the measure of angle BCA from triangle ABC.

∠BAC + ∠ABC + ∠BCA  =  180

21 + 105 +  ∠BCA  =  180

126 + ∠BCA  =  180

∠BCA  =  180 - 126

∠BCA  =  540

∠BCA  =  ∠DEF

 In triangle ABC∠BAC∠BCA In triangle DEF∠DFE∠DEF

Hence the triangles ABC and DEF are similar.

Example 2 : Solution :

The angles ACB and FDE are congruent.To prove the above triangles are similar, we need to prove one more pairs of angles are equal.

To check whether the angles ABC and DEF are equal, let us find the measure of angle ABC from triangle ABC.

∠ABC + ∠BAC + ∠ACB  =  180

ABC + 79 +  60  =  180

∠ABC + 139  =  180

∠ABC  =  180 - 139

∠ABC  =  41

∠ABC    ∠DEF

Hence the above triangles ABC and DEF are not similar.

Example 3 : Solution :

To check whether the above triangles are similar, we need to find the missing angles of triangle ABC.

∠ABC + ∠BAC + ∠ACB  =  180

84 +  ∠BAC + ∠ACB  =  180

2∠BAC  =  180 - 84

2∠BAC  =  96

∠BAC  =  96/2

∠BAC  =  48  =  ∠ACB

The corresponding angles of BAC  and DEF are not same. Hence the above triangles are not similar. After having gone through the stuff given above, we hope that the students would have understood "How to prove two triangles similar".