**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 :**

Determine whether each pair of triangles is similar. Justify your answer.

**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 = 54^{0}

∠BCA = ∠DEF

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

Hence the triangles ABC and DEF are similar.

**Example 2 :**

Determine whether each pair of triangles is similar. Justify your answer.

**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 :**

Determine whether each pair of triangles is similar. Justify your answer.

**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.

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