Similarity Criteria Formula

Similarity criteria is three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional.

The Formula

AA, SAS∼\sim, or SSS∼\sim ⇒△ABC∼△DEF\Rightarrow \triangle ABC \sim \triangle DEF

When to use: Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.

Quick Example

Triangle with angles 50°,60°,70°50°, 60°, 70° is similar to any other triangle with angles 50°,60°,70°50°, 60°, 70° by AA. 36=48=510=12  ⟹  SSS~\frac{3}{6} = \frac{4}{8} = \frac{5}{10} = \frac{1}{2} \implies \text{SSS\textasciitilde}

Notation

△ABC∼△DEF\triangle ABC \sim \triangle DEF means the triangles are similar with vertices corresponding in order.

What This Formula Means

Three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional sides with equal included angle), and SSS~ (all three pairs of sides in the same ratio).

Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.

Formal View

AA: (∠A=∠D,∠B=∠E)⇒△ABC∼△DEF(\angle A = \angle D, \angle B = \angle E) \Rightarrow \triangle ABC \sim \triangle DEF. SSS∼\sim: ∣AB∣∣DE∣=∣BC∣∣EF∣=∣AC∣∣DF∣⇒∼\frac{|AB|}{|DE|} = \frac{|BC|}{|EF|} = \frac{|AC|}{|DF|} \Rightarrow \sim. SAS∼\sim: ∣AB∣∣DE∣=∣BC∣∣EF∣\frac{|AB|}{|DE|} = \frac{|BC|}{|EF|} and ∠B=∠E⇒∼\angle B = \angle E \Rightarrow \sim

Worked Examples

Example 1

easy
Triangle ABCABC has angles 40°40°, 60°60°, 80°80°. Triangle DEFDEF has angles 40°40°, 60°60°, 80°80°. Are the triangles similar? Which criterion applies?

Answer

△ABC∼△DEF\triangle ABC \sim \triangle DEF by AA.

First step

1
Step 1: List the angle pairs: ∠A=∠D=40°\angle A = \angle D = 40°, ∠B=∠E=60°\angle B = \angle E = 60°, ∠C=∠F=80°\angle C = \angle F = 80°.

Full solution

  1. 2
    Step 2: All three angles match. However, for similarity, we only need two angles — the third is determined since angles sum to 180°.
  2. 3
    Step 3: The AA (Angle-Angle) criterion states: if two angles of one triangle equal two angles of another, the triangles are similar.
  3. 4
    Step 4: Conclude: △ABC∼△DEF\triangle ABC \sim \triangle DEF by AA.
AA is the most commonly used similarity criterion. Because all angles in a triangle sum to 180°, knowing two angles determines the third. Two triangles with the same angle measures have the same shape (though possibly different sizes), making them similar. Their corresponding sides are proportional.

Example 2

medium
In △ABC\triangle ABC: AB=6AB = 6, BC=9BC = 9, AC=12AC = 12. In △DEF\triangle DEF: DE=4DE = 4, EF=6EF = 6, DF=8DF = 8. Are the triangles similar? State the criterion.

Example 3

medium
Triangle ABC has sides 55, 1212, 1313. Triangle DEF has sides 1010, 2424, 2626. Prove the triangles are similar and state the criterion used.

Common Mistakes

  • Demanding all three angle pairs — AA is enough because the third angle follows automatically.
  • Matching sides in the wrong order — pair sides opposite equal angles before forming ratios.
  • Concluding congruence from proportional sides — equal ratios prove similarity, not congruence.

Why This Formula Matters

Similarity powers indirect measurement (shadow heights, map scales) and is the foundation of trigonometry, where same-angle triangles share fixed side ratios. The key insight AA captures — two equal angles alone fix the shape — is why a photo and its enlargement are 'the same.' Recognizing it by "Do the triangles match by equal angles or proportional sides (not equal lengths)?" — rather than by familiar numbers — is what lets a student tell it apart from congruence criteria and similarity and proportions in a mixed problem set.

Frequently Asked Questions

What is the Similarity Criteria formula?

Three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional sides with equal included angle), and SSS~ (all three pairs of sides in the same ratio).

How do you use the Similarity Criteria formula?

Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.

What do the symbols mean in the Similarity Criteria formula?

△ABC∼△DEF\triangle ABC \sim \triangle DEF means the triangles are similar with vertices corresponding in order.

Why is the Similarity Criteria formula important in Math?

Similarity powers indirect measurement (shadow heights, map scales) and is the foundation of trigonometry, where same-angle triangles share fixed side ratios. The key insight AA captures — two equal angles alone fix the shape — is why a photo and its enlargement are 'the same.' Recognizing it by "Do the triangles match by equal angles or proportional sides (not equal lengths)?" — rather than by familiar numbers — is what lets a student tell it apart from congruence criteria and similarity and proportions in a mixed problem set.

What do students get wrong about Similarity Criteria?

The procedure for similarity criteria is the easy part; the trap is demanding all three angle pairs. Asking "Do the triangles match by equal angles or proportional sides (not equal lengths)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Similarity Criteria formula?

Before studying the Similarity Criteria formula, you should understand: similarity, triangles, proportions.

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