Similarity Criteria Formula

Q: What do the symbols mean in the Similarity Criteria formula?

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

The Formula

AA, SAS\sim, or SSS\sim \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° is similar to any other triangle with angles 50°, 60°, 70° by AA. \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = \frac{1}{2} \implies \text{SSS\textasciitilde}

Notation

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

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
Step 1: List the angle pairs: \angle A = \angle D = 40°, \angle B = \angle E = 60°, \angle C = \angle F = 80°.
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
Step 3: The AA (Angle-Angle) criterion states: if two angles of one triangle equal two angles of another, the triangles are similar.
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.

Common Mistakes

Forgetting that AA is sufficient (you don't need all three angles explicitly)
Setting up proportions with non-corresponding sides
Confusing similarity (\sim) with congruence (\cong)

Why This Formula Matters

The foundation for indirect measurement—you can find the height of a building by measuring its shadow and comparing to a known object.

Frequently Asked Questions

What is the Similarity Criteria formula?

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What do the symbols mean in the Similarity Criteria formula?