Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent.

Just as there are specific methods for proving triangles congruent (SSS, ASA, SAS, AAS and HL), there are also specific methods that will prove triangles similar.

There are three accepted methods for proving triangles similar:

AA
To prove two triangles are similar, it is sufficient to show that two angles of one triangle are congruent to the two corresponding angles of the other triangle.
 If two angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar. (proof of this theorem is shown below)

 PROOF: * This will be a transformational proof. We will search for a sequence of transformations that will map ΔABC onto ΔDEF . 1. Since the triangles are different sizes, we will start by dilating ΔABC to a smaller size. The smaller size needs to match the side lengths of ΔDEF, so we match AB with DE by setting the scale factor to be . The new dilated triangle will be ΔA'B'C'. 2. After the dilation, we know that ΔA'B'C'∼ΔABC because a dilation is a similarity transformation. 3. After the dilation, ∠A ∠A' and ∠B ∠B' since dilations preserve angle measure. 4. After the dilation, A'B' = k•AB = DE, which gives us . 5. We are given that ∠A ∠D and ∠B ∠E, and from step 3 that ∠A ∠A' and ∠B ∠B'. Using the transitive property, we get ∠A' ∠D and ∠B' ∠E. 6. ΔA'B'C' ΔDEF by angle side angle (ASA) for congruent triangles. 7. Since ΔA'B'C' ΔDEF and ΔA'B'C' ∼ ΔABC, we have ΔDEF ∼ ΔABC.

CCSS and PARCC specifically mention AA in relation to similar triangles. Engage NY also mentions SSS and SAS methods.

The next two methods for proving similar triangles are NOT
the same theorems used to prove congruent triangles.

SSS
To prove two triangles are similar, it is sufficient to show that the three sets of corresponding sides are in proportion.
 If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.

SAS
To prove two triangles are similar, it is sufficient to show that two sets of corresponding sides are in proportion and the angles they include are congruent.
 If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.

Once the triangles are similar ...
the remaining sets of angles will be congruent and the remaining corresponding sides will be in proportion. Just be sure the triangles are similar before using the following theorem.
 The corresponding sides of similar triangles are in proportion.