4.2 Similarity Criteria (AA, SAS, SSS)

Defining Similar Figures

Two figures are similar when one can be mapped onto the other by a similarity transformation, meaning a dilation followed by a rigid motion (a translation, reflection, and/or rotation). The dilation handles the size difference; the rigid motion handles position and orientation. This is the Next Generation definition behind standard G-SRT.A.2.

Because a dilation preserves angles and scales all lengths by the same factor k, two similar figures have:

• Corresponding angles congruent (equal measure), and
• Corresponding sides proportional (the same ratio k).

The symbol for similarity is ~. Writing triangle ABC ~ triangle DEF states the correspondence in order: A matches D, B matches E, C matches F. Vertex order is not decorative; it tells you which sides and angles pair up.

Congruent vs. Similar

Congruence is the special case of similarity where k = 1.

The Three Triangle Similarity Criteria

You do not need to check every angle and every side. For triangles, three shortcuts each guarantee similarity. These support G-SRT.A.3 and G-SRT.B.5.

AA (Angle-Angle). If two angles of one triangle are congruent to two angles of another, the triangles are similar. Two angles are enough because the third is forced: the angle sum of every triangle is 180 degrees. AA is by far the most common criterion on the Regents.

SAS~ (Side-Angle-Side similarity). If two pairs of corresponding sides are proportional and the included angles (the angles between those sides) are congruent, the triangles are similar.

SSS~ (Side-Side-Side similarity). If all three pairs of corresponding sides are proportional, the triangles are similar.

Note the parallel to congruence: AA replaces ASA/AAS, SAS~ replaces SAS, and SSS~ replaces SSS, but with proportional sides instead of equal ones. Just as with congruence, SSA and AAA give no new information for similarity beyond AA.

The Ratio of Similarity and Setting Up Proportions

The ratio of similarity (or scale factor) between two similar figures is the ratio of any pair of corresponding lengths. Once you know it, every other pair of corresponding sides obeys the same ratio.

Worked example 1. Two similar triangles have corresponding sides 6 and 10. A second side of the smaller triangle is 9. The scale factor from small to large is 10/6 = 5/3, so the matching larger side is 9 * (5/3) = 15.

Worked example 2. The smaller triangle has sides 4, 6, 8; the larger has corresponding sides 6, 9, x. The scale factor is 6/4 = 9/6 = 3/2, so x = 8 * (3/2) = 12.

Worked example 3. Triangle RST ~ triangle XYZ with RS = 8, XY = 12, and ST = 10. Corresponding sides give the proportion RS/XY = ST/YZ, that is 8/12 = 10/YZ. Cross-multiply: 8 * YZ = 120, so YZ = 15.

Set Up Proportions Consistently

The safest habit is to write each ratio the same direction, image over preimage (or large over small) on both sides. A cross-multiplication error usually comes from flipping one ratio. In RS/XY = ST/YZ, both numerators (RS, ST) belong to triangle RST and both denominators (XY, YZ) belong to triangle XYZ.

Perimeter, Area, and the k-Ladder

Similar figures obey the same scaling ladder as dilations. If the ratio of similarity is k, then:

• Perimeters are in ratio k,
• Areas are in ratio k^2, and
• Volumes (for similar solids) are in ratio k^3.

So two similar polygons with corresponding sides 5 and 8 have areas in the ratio 5^2 : 8^2 = 25 : 64, not 5 : 8. A model built at linear scale 4 has 4^2 = 16 times the surface area of the original. Forgetting to square (or cube) the ratio is one of the most common similarity mistakes on the exam.

Applying SAS~ and SSS~

AA covers most Regents items, but the side-based criteria are also tested. For SSS~, check that all three ratios reduce to the same number. Triangles with sides 3, 5, 7 and 9, 15, 21 are similar because 3/9 = 5/15 = 7/21 = 1/3. If even one ratio differs, the triangles are not similar.

For SAS~, verify that two sides are proportional and confirm the angle between them is congruent. An equal angle that is not included between the two sides does not qualify, which is the similarity version of the SSA failure.

Why AAA and SSA Add Nothing New

Three congruent angles (AAA) is just AA with a redundant third angle, so it is not a separate rule. SSA, two sides and a non-included angle, can describe two differently shaped triangles, so it never guarantees similarity. Keep every proof to AA, SAS~, or SSS~ and you stay on firm ground.

Correspondence Controls Everything

When a problem states triangle ABC ~ triangle DEF, the smallest angle in one triangle matches the smallest in the other, and the shortest side matches the shortest side. If a figure is rotated or reflected, relabel mentally so corresponding vertices line up before you write any proportion. Mismatched correspondence is the leading cause of wrong similarity answers even when the criterion itself was chosen correctly.