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Congruence Tests for Triangles

Congruence Tests for Triangles

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

 

 

See also

Similarity tests for triangles

Worked Example

Problem: Triangle ABC has AB = 5 cm, angle B = 60°, and BC = 8 cm. Triangle DEF has DE = 5 cm, angle E = 60°, and EF = 8 cm. Are the two triangles congruent?
Step 1: Identify what information is given for each triangle. In both triangles, you know two sides and the angle between them (the included angle).
AB=DE=5 cm,∠B=∠E=60°,BC=EF=8 cmAB = DE = 5 \text{ cm}, \quad \angle B = \angle E = 60°, \quad BC = EF = 8 \text{ cm}
Step 2: Determine which congruence test applies. The known angle is between the two known sides, so this is the Side-Angle-Side (SAS) pattern.
Step 3: Apply SAS: since two sides and the included angle of triangle ABC are congruent to the corresponding two sides and included angle of triangle DEF, the triangles are congruent by SAS.
△ABC≅△DEF(SAS)\triangle ABC \cong \triangle DEF \quad (\text{SAS})
Answer: Yes, triangle ABC is congruent to triangle DEF by the SAS congruence test.

Another Example

Problem: In triangle PQR, PQ = 7, QR = 10, and PR = 12. In triangle XYZ, XY = 7, YZ = 10, and XZ = 12. Are the triangles congruent?
Step 1: List the three side lengths of each triangle and match corresponding sides.
PQ=XY=7,QR=YZ=10,PR=XZ=12PQ = XY = 7, \quad QR = YZ = 10, \quad PR = XZ = 12
Step 2: All three pairs of sides are congruent. This matches the Side-Side-Side (SSS) pattern — no angle information is needed.
Step 3: By the SSS congruence test, the two triangles are congruent.
△PQR≅△XYZ(SSS)\triangle PQR \cong \triangle XYZ \quad (\text{SSS})
Answer: Yes, the triangles are congruent by the SSS congruence test.

Frequently Asked Questions

Why doesn't SSA (Side-Side-Angle) work as a congruence test?
When you know two sides and a non-included angle, there can be two different triangles that satisfy those measurements — this is called the ambiguous case. Because the given information doesn't determine a unique triangle, SSA is not a valid congruence test. The one exception is the HL test, which works only for right triangles because the right angle constrains the geometry enough to eliminate ambiguity.
Why doesn't AAA (Angle-Angle-Angle) prove congruence?
Three matching angles guarantee the triangles have the same shape (they are similar), but say nothing about size. You could have two triangles with identical angles where one is much larger than the other. Without at least one matching side length, congruence cannot be established.

Congruence Tests for Triangles vs. Similarity Tests for Triangles

Congruence tests (SSS, SAS, ASA, AAS, HL) prove two triangles are exactly the same shape and size — all corresponding sides and angles are equal. Similarity tests (AA, SAS similarity, SSS similarity) prove two triangles have the same shape but possibly different sizes — corresponding angles are equal and corresponding sides are proportional. Congruence requires equal side lengths; similarity only requires equal ratios of side lengths.

Why It Matters

Congruence tests are essential tools in geometric proofs — they let you conclude that two triangles are identical without measuring every side and angle. Engineers and architects rely on triangle congruence to ensure that structural components are precisely the same shape and size. These tests also underpin more advanced results like proving properties of parallelograms, establishing angle bisector theorems, and justifying constructions with compass and straightedge.

Common Mistakes

Mistake: Using SSA (two sides and a non-included angle) as a valid congruence test.
Correction: SSA can produce two different triangles (the ambiguous case), so it does not prove congruence. The only exception is HL, which applies exclusively to right triangles where the right angle removes the ambiguity.
Mistake: Matching sides and angles in the wrong order (e.g., claiming SAS when the angle is not between the two given sides).
Correction: For SAS, the congruent angle must be the included angle — the one formed by the two congruent sides. Always check that corresponding parts are in the correct positions before citing a test.

Related Terms