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Geometry Similarity

Triangle similarity (AA, SAS, SSS)

20 practice questions 4 video lessons Theory + worked examples

Triangle Similarity (AA, SAS, SSS)

Common Core Geometry • Standard G-SRT.3 • Similarity

Triangle Similarity (AA, SAS, SSS) is a topic in Similarity in the Common Core State Standards. It is aligned to Standard G-SRT.3, which requires students to use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

The triangle similarity criteria — AA, SAS, and SSS — establish that two triangles are similar from limited information.

Common Core Geometry › Similarity › Triangle Similarity (AA, SAS, SSS)  —  Standard G-SRT.3

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Watch 4 video(s)
  • Triangle Similarity - AA SSS SAS & AAA Postulates, Proving Similar Triangles, Two Column Proofs Watch
  • Scale Factors Finding Length, Area, Volume in Similar Figures Watch
  • Triangle Similarity Criteria - AA, SSS, SAS Watch
  • Similar Areas and Volumes - GCSE Higher Maths Watch
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Theory

Two triangles are similar (\(\sim\)) when they have the same shape: equal corresponding angles and proportional corresponding sides. Three criteria prove it:

  • AA — two pairs of equal angles (the third follows automatically).
  • SAS — two pairs of proportional sides with equal included angles.
  • SSS — all three pairs of sides proportional.
AA is the workhorse. Because angles sum to \(180^\circ\), two equal pairs force the third, so two angles are enough.
AA similarity If two angles of one triangle equal two angles of another, the triangles are similar. two equal angles ⇒ AA similarity
Two equal pairs of angles give AA similarity.
Similarity criteria Similarity criteria Similarity criteria AA: two pairs of equal angles SAS: proportional sides + equal angle SSS: all three sides proportional
The three similarity criteria.

Proportional corresponding sides:

\[\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF}=k\]
corresponding sides of similar triangles share a common ratio, the scale factor
Set corresponding sides in a proportion and cross-multiply to find a missing length.

How to work with similar triangles

  1. Prove similarity with AA, SAS, or SSS.
  2. Match corresponding sides (opposite equal angles).
  3. Set a proportion and cross-multiply to solve.
  4. Read the scale factor as the common ratio.
Example 1 — AA similarity
Two triangles each have angles \(40^\circ\) and \(65^\circ\). Are they similar?
Solution

Two pairs of equal angles is AA, so the triangles are similar.

\(\triangle_1\)\(\sim\)\(\triangle_2\)
yes, similar by AA
Example 2 — Find a missing side
\(\triangle ABC\sim\triangle DEF\) with \(AB=6\), \(DE=9\), \(BC=8\). Find \(EF\).
Solution

Corresponding sides are proportional; set up and solve.

\(\dfrac{AB}{DE}\)\(=\)\(\dfrac{BC}{EF}\)
\(\dfrac{6}{9}\)\(=\)\(\dfrac{8}{EF}\)
\(6\cdot EF\)\(=\)\(72\)
\(EF\)\(=\)\(12\)
EF equals 12
Example 3 — SSS similarity
Triangles have sides \(3,4,5\) and \(9,12,15\). Are they similar?
Solution

Check that all three ratios are equal.

\(\dfrac{9}{3}=\dfrac{12}{4}=\dfrac{15}{5}\)\(=\)\(3\)

All equal, so similar by SSS.

yes, similar by SSS with ratio 3
Example 4 — The scale factor
For the triangles in Example 3, what is the scale factor from the small to the large?
Solution

It is the common ratio of corresponding sides.

\(k\)\(=\)\(3\)
the scale factor is 3

Common pitfalls

Match corresponding sides carefully — each pairs with the side opposite the equal angle.
AA proves similarity, not congruence. Equal angles alone allow different sizes.
All three ratios must match for SSS. Two out of three is not enough.

Frequently asked questions

What are the triangle similarity criteria?

AA (two equal angles), SAS (proportional sides with equal included angle), and SSS (all sides proportional).

Why is AA enough to prove similarity?

Because a triangle's angles sum to \(180^\circ\), two equal pairs force the third pair equal, so the shapes match.

How do you find a missing side in similar triangles?

Set corresponding sides in a proportion and cross-multiply to solve.

What is the scale factor between similar triangles?

The common ratio of corresponding sides; it tells how much larger one triangle is than the other.

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