Congruence Criteria for Triangles

Why This Matters

Triangle congruence is the foundation of geometric proof—and you're being tested on your ability to choose the right criterion for a given situation, not just recite definitions. These criteria connect directly to broader algebraic geometry concepts: rigid motions, isometries, equivalence relations, and the logical structure of mathematical proof. When you understand why each criterion works, you unlock the ability to construct elegant proofs and solve complex problems involving geometric relationships.

Don't just memorize "SSS, SAS, ASA, AAS, HL" as a list. Know what information each criterion requires, when to apply each one, and how the logical properties of congruence allow you to chain arguments together. Exam questions will give you partial information about triangles and ask you to determine congruence—your job is to match the given data to the correct criterion efficiently.

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Side-Based Criteria

These criteria rely primarily on comparing side lengths. When you can measure or calculate sides directly, these are often your most straightforward path to proving congruence.

Side-Side-Side (SSS)

• Three pairs of equal sides guarantee congruence—no angle measurements needed, making this criterion powerful when side lengths are given or calculable
• Works for all triangle types: scalene, isosceles, and equilateral triangles all satisfy SSS when corresponding sides match
• Rigid determination principle—three fixed side lengths can only form one unique triangle shape, which is why this criterion is logically complete

Side-Angle-Side (SAS)

• Two sides and the included angle must match—the angle must be between the two sides being compared, not just any angle
• Included angle is critical: if the angle isn't between the measured sides, SAS doesn't apply (this is a common exam trap)
• More efficient than SSS when you have angle information, since you only need to verify two sides instead of three

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Angle-Based Criteria

These criteria emphasize angle measurements combined with strategic side information. The key insight: two angles of a triangle determine the third (since angles sum to $180°$), so angle-based criteria are often interchangeable.

Angle-Side-Angle (ASA)

• Two angles and the included side must match—the side lies between the two known angles
• Determines the third angle automatically since $\angle A + \angle B + \angle C = 180°$, giving you complete angle information
• Best when angles are easier to measure than sides, such as in surveying or construction problems

Angle-Angle-Side (AAS)

• Two angles and any corresponding side—the side does not need to be between the angles
• Logically equivalent to ASA because knowing two angles determines the third, effectively converting AAS to ASA
• More flexible than ASA in proof situations where the known side isn't conveniently located between measured angles

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Special Case: Right Triangles

Right triangles have additional structure that simplifies congruence testing. The presence of a $90°$ angle provides built-in information that other triangles lack.

Hypotenuse-Leg (HL)

• Hypotenuse and one leg determine a right triangle completely—this is a special case derived from the Pythagorean theorem
• Only applies to right triangles: the $90°$ angle must be established first before invoking HL
• Shortcut version of SAS—the right angle serves as the "included angle," so you only need to verify two sides

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Logical Properties of Congruence

These properties govern how congruence behaves as a mathematical relation. They transform congruence from a static comparison into a dynamic tool for building multi-step proofs.

Reflexive Property

• Every figure is congruent to itself—written as $\triangle ABC \cong \triangle ABC$
• Essential for shared elements in proofs where two triangles share a common side or angle
• Proof workhorse: when triangles overlap or share a side, cite reflexive property to establish that shared element as congruent to itself

Symmetric Property

• Congruence works both directions—if $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$
• Allows flexibility in proof structure so you can work from either triangle toward the other
• Notation matters: the order of vertices indicates corresponding parts, so symmetry lets you reverse the correspondence

Transitive Property

• Congruence chains together—if $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC \cong \triangle GHI$
• Powers multi-step proofs where you establish intermediate congruences to reach your final conclusion
• Connects distant triangles that don't share obvious relationships by routing through a common third triangle

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Quick Reference Table

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Self-Check Questions

1. You know two triangles have equal corresponding sides $AB = DE$ and $BC = EF$, plus equal angles $\angle B = \angle E$. Which criterion applies, and why does the angle's position matter?

2. Compare and contrast ASA and AAS: under what circumstances would you choose one over the other, and why are they logically equivalent?

3. A proof involves two triangles that share side $\overline{CD}$. Which property of congruence must you cite to use this shared side in your argument?

4. Why does the HL criterion only work for right triangles? What would go wrong if you tried to apply it to an obtuse triangle?

5. If you've proven $\triangle PQR \cong \triangle STU$ and separately proven $\triangle STU \cong \triangle XYZ$, what property allows you to conclude $\triangle PQR \cong \triangle XYZ$, and how might this appear in a multi-step FRQ?