Congruence Criteria

Why This Matters

Triangle congruence is the foundation of geometric proof—it's how you establish that two figures are exactly the same in both shape and size. When you're asked to prove relationships in geometry, you're almost always working backward from a congruence criterion to unlock additional information about sides and angles. The criteria you'll learn here (SSS, SAS, ASA, AAS, and HL) aren't just random rules; they represent the minimum information needed to guarantee two triangles are identical.

You're being tested on your ability to select the right criterion for a given situation and apply it strategically in proofs. Don't just memorize the acronyms—understand what each one requires, why it works, and when to use it. The real skill is recognizing which criterion fits the information you're given, then leveraging CPCTC to prove whatever the problem actually asks for.

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Side-Based Criteria

These criteria prioritize side measurements as the primary evidence for congruence. When you know enough about the sides, the angles are forced to match.

Side-Side-Side (SSS) Congruence

• Three pairs of corresponding sides equal—if all three sides match, the triangles must be congruent regardless of angle measures
• No angle information required—this makes SSS ideal when you're given side lengths but no angle data
• Works for all triangle types—scalene, isosceles, and equilateral triangles all follow this criterion

Hypotenuse-Leg (HL) Congruence

• Requires a right angle—this criterion applies only to right triangles, making the right angle your first checkpoint
• Hypotenuse plus one leg—if these two measurements match between right triangles, congruence is guaranteed
• Shortcut version of SAS—the right angle is "built in," so you don't need to state it separately as your included angle

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Side-Angle Combination Criteria

These criteria use a mix of side and angle measurements. The key is whether the angle is "included" (sandwiched between the sides) or not.

Side-Angle-Side (SAS) Congruence

• Two sides and the included angle—the angle must be formed by the two sides you're comparing
• Included angle is critical—if the angle isn't between the two sides, SAS doesn't apply (this is a common exam trap)
• Frequently appears in proofs—especially when triangles share a common side or vertex

Angle-Side-Angle (ASA) Congruence

• Two angles and the included side—the side must be the one connecting the two angles
• Angle-heavy approach—useful when angle measures are easier to find than side lengths
• The third angle is automatic—since triangle angles sum to $180°$, knowing two angles determines the third

Angle-Angle-Side (AAS) Congruence

• Two angles and a non-included side—the side is not between the two angles
• Works because the third angle is fixed—two angles determine the third, effectively converting AAS to an ASA situation
• Watch the correspondence—the non-included side must correspond to the same position in both triangles

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Properties That Support Proofs

These aren't congruence criteria themselves—they're logical properties that help you build congruence arguments and extend your conclusions.

Reflexive Property of Congruence

• A figure is congruent to itself—written as $\overline{AB} \cong \overline{AB}$ or $\angle X \cong \angle X$
• Essential for shared elements—when two triangles share a side or angle, the reflexive property justifies counting it for both
• Appears in nearly every proof—if triangles overlap or share a boundary, expect to use this property

Symmetric Property of Congruence

• Congruence works both directions—if $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$
• Allows flexible reasoning—you can establish congruence in whichever order is convenient
• Useful for reordering statements—especially when working backward in a proof

Transitive Property of Congruence

• Chain congruence across figures—if $\triangle A \cong \triangle B$ and $\triangle B \cong \triangle C$, then $\triangle A \cong \triangle C$
• Connects multiple relationships—essential when a proof involves three or more triangles
• Think of it as a bridge—the middle figure links the first and third

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Using Congruence Results

Once you've established triangle congruence, these tools let you extract additional information.

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

• Unlocks all six corresponding parts—once triangles are congruent, every pair of corresponding sides and angles is also congruent
• Used after proving congruence—CPCTC is never your first step; it's the payoff for establishing congruence
• The real goal of most proofs—problems often ask you to prove a specific side or angle relationship, and CPCTC is how you get there

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Quick Reference Table

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Self-Check Questions

1. You know two triangles have two pairs of congruent sides and one pair of congruent angles. What additional information do you need to determine whether SAS or AAS applies?

2. Which two congruence criteria both require exactly two angles? How do they differ in what side information they need?

3. A proof involves two triangles that share side $\overline{BD}$. Which property justifies using this side as evidence for both triangles, and how would you write the congruence statement?

4. Compare and contrast SSS and HL: What do they have in common, and why does HL require fewer pieces of information?

5. In a two-column proof, you've just written "$\triangle ABC \cong \triangle XYZ$ by ASA." The problem asks you to prove $\overline{AC} \cong \overline{XZ}$. What is your next statement and justification?