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Identify similar triangles

Similar triangles have the same shape: their corresponding angles are congruent and their corresponding side lengths are proportional by a common scale factor, even when the triangles differ in size or orientation. Identification relies on criteria such as AA, SAS, or SSS and requires matching corresponding vertices rather than comparing sides in arbitrary order; this understanding supports proportional reasoning, indirect measurement, and geometric proof without extending to higher-dimensional similarity or advanced trigonometric applications.

Detailed Explanation: Identify similar triangles

To identify similar triangles, check whether they have the same shape. Their corresponding angles must be equal, and their corresponding sides must have the same scale factor.

A useful shortcut is the AA criterion:

  • If two angles in one triangle match two angles in another triangle, the triangles are similar.
  • The third angles must also match because the angles in every triangle add to 180∘180^\circ.

Example

Suppose triangle ABCABC and triangle DEFDEF have these angle measures:

∠A=40∘,∠B=70∘\angle A=40^\circ,\qquad \angle B=70^\circ ∠D=70∘,∠E=40∘\angle D=70^\circ,\qquad \angle E=40^\circ

Determine whether the triangles are similar and name the corresponding vertices.

Step 1: Match equal angles.

  • ∠A=40∘\angle A=40^\circ matches ∠E=40∘\angle E=40^\circ, so A↔EA\leftrightarrow E.
  • ∠B=70∘\angle B=70^\circ matches ∠D=70∘\angle D=70^\circ, so B↔DB\leftrightarrow D.

Step 2: Match the remaining angles.

The third angle in triangle ABCABC is

∠C=180∘−40∘−70∘=70∘.\angle C=180^\circ-40^\circ-70^\circ=70^\circ.

The third angle in triangle DEFDEF is

∠F=180∘−70∘−40∘=70∘.\angle F=180^\circ-70^\circ-40^\circ=70^\circ.

Therefore, C↔FC\leftrightarrow F.

Step 3: Write the similarity statement in matching order.

Since

A↔E,B↔D,C↔F,A\leftrightarrow E,\qquad B\leftrightarrow D,\qquad C\leftrightarrow F,

the correct statement is

△ABC∼△EDF.\triangle ABC\sim\triangle EDF.

The order matters. It tells you that the corresponding sides are

AB↔ED,BC↔DF,AC↔EF.AB\leftrightarrow ED,\qquad BC\leftrightarrow DF,\qquad AC\leftrightarrow EF.

Thus, the triangles are similar by AA.

Learn by doing: Identify similar triangles

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    Similar Triangles - Separate (No Rotation) to Side

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