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Similarity Tests for Triangles

Similarity Tests for Triangles

AA, HL, SAS, and SSS similarity. These tests describe combinations of congruent angles and/or proportional sides that are used to determine if two triangles are similar.

 

 

See also

Congruence tests for triangles

Worked Example

Problem: Triangle ABC has sides AB = 6, BC = 9, and AC = 12. Triangle DEF has sides DE = 4, EF = 6, and DF = 8. Determine whether the two triangles are similar, and if so, identify which test applies.
Step 1: Check whether corresponding sides are proportional. Compare the ratios of the shortest, middle, and longest sides.
DEAB=46=23,EFBC=69=23,DFAC=812=23\frac{DE}{AB} = \frac{4}{6} = \frac{2}{3}, \quad \frac{EF}{BC} = \frac{6}{9} = \frac{2}{3}, \quad \frac{DF}{AC} = \frac{8}{12} = \frac{2}{3}
Step 2: All three ratios are equal, so every pair of corresponding sides is in the same proportion.
DEAB=EFBC=DFAC=23\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = \frac{2}{3}
Step 3: By the SSS Similarity Test, if all three pairs of corresponding sides are proportional, the triangles are similar.
△ABC∼△DEF\triangle ABC \sim \triangle DEF
Answer: The triangles are similar by SSS Similarity, with a scale factor of 2/3.

Another Example

Problem: In triangle PQR, angle P = 50° and angle Q = 65°. In triangle XYZ, angle X = 50° and angle Z = 65°. Are the triangles similar?
Step 1: Find the third angle in each triangle using the fact that angles in a triangle sum to 180°.
∠R=180°−50°−65°=65°,∠Y=180°−50°−65°=65°\angle R = 180° - 50° - 65° = 65°, \quad \angle Y = 180° - 50° - 65° = 65°
Step 2: Match corresponding angles: angle P = angle X = 50°, and angle Q = angle Z = 65°. Two pairs of corresponding angles are congruent.
∠P=∠X,∠Q=∠Z\angle P = \angle X, \quad \angle Q = \angle Z
Step 3: By the AA Similarity Test, two pairs of congruent angles are enough to guarantee similarity.
△PQR∼△XYZ\triangle PQR \sim \triangle XYZ
Answer: Yes, the triangles are similar by AA Similarity.

Frequently Asked Questions

What is the difference between similarity tests and congruence tests for triangles?
Similarity tests determine whether two triangles have the same shape (equal angles and proportional sides). Congruence tests determine whether two triangles have the same shape AND the same size (equal angles and equal sides). Every pair of congruent triangles is also similar, but similar triangles are not necessarily congruent.
How many similarity tests are there and which one is used most often?
There are four standard tests: AA (Angle-Angle), SSS (Side-Side-Side), SAS (Side-Angle-Side), and HL (Hypotenuse-Leg, for right triangles). AA is the most commonly used because you only need two pairs of equal angles, and the third angle is automatically determined since angles in a triangle sum to 180°.

Similarity Tests vs. Congruence Tests

Similarity tests check whether corresponding angles are equal and corresponding sides are proportional, meaning the triangles have the same shape but may differ in size. Congruence tests check whether corresponding angles and sides are exactly equal, meaning the triangles are identical in both shape and size. For similarity, you compare side ratios; for congruence, you compare side lengths directly.

Why It Matters

Similarity tests are essential in geometry whenever you need to relate two triangles that share the same shape but differ in size. They are the foundation of indirect measurement—for example, using shadows to calculate the height of a building. In trigonometry and coordinate geometry, similarity arguments underpin the definitions of trigonometric ratios and proofs about parallel lines cut by transversals.

Common Mistakes

Mistake: Using SSA (Side-Side-Angle) as a similarity test.
Correction: SSA is not a valid similarity test, just as it is not a valid congruence test. Two triangles can share two proportional sides and one non-included congruent angle yet still not be similar. Always use AA, SSS, SAS, or HL.
Mistake: Comparing sides for equality instead of proportionality when testing similarity.
Correction: Similarity requires that corresponding sides are in the same ratio, not that they are equal. Compute the ratio of each pair of corresponding sides and check that all ratios match.

Related Terms