Similarity Criteria Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Similarity Criteria.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Similarity is about shape, not size. Two matching angles are enough to guarantee two triangles have the same shape.

Common stuck point: AA only requires two angle pairs because the third angle is automatically determined (angles sum to 180°).

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: List the angle pairs: \angle A = \angle D = 40°, \angle B = \angle E = 60°, \angle C = \angle F = 80°.
  2. 2
    Step 2: All three angles match. However, for similarity, we only need two angles — the third is determined since angles sum to 180°.
  3. 3
    Step 3: The AA (Angle-Angle) criterion states: if two angles of one triangle equal two angles of another, the triangles are similar.
  4. 4
    Step 4: Conclude: \triangle ABC \sim \triangle DEF by AA.

Answer

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

Example 3

medium
Triangle ABC has sides 5, 12, 13. Triangle DEF has sides 10, 24, 26. Prove the triangles are similar and state the criterion used.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A tree casts a 15 m shadow, and at the same time a 2 m stick casts a 3 m shadow. How tall is the tree? Which similarity criterion justifies this method?

Example 2

hard
In \triangle PQR and \triangle STU: PQ = 8, QR = 12, \angle Q = 50°; ST = 6, TU = 9, \angle T = 50°. Show the triangles are similar and find the scale factor.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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