Triangle Congruence Theorems

Introduction to Congruence

• Concept of Congruence: Symbolized by an equal sign with a squiggly line.
• Example: Two running shoes are congruent if they are identical in size and shape, even if rotated or colored differently.
• Triangles: Triangles are congruent if they have the same size and shape, meaning their corresponding sides and angles are identical.

Triangle Congruence Theorems

There are five main theorems used to establish the congruence of triangles:

1. Side-Side-Side (SSS)

• Definition: If three pairs of corresponding sides in two triangles are congruent, the triangles are congruent.
• Implication: Same shape and size even with different orientations.

2. Side-Angle-Side (SAS)

• Definition: If two pairs of corresponding sides and the included angle are congruent, the triangles are congruent.
• Notation: The "A" (angle) is between the two "S" (sides).

3. Angle-Side-Angle (ASA)

• Definition: Two pairs of corresponding angles and the included side are congruent.
• Notation: The "S" (side) is between the two "A" (angles).

4. Angle-Angle-Side (AAS)

• Definition: Two pairs of corresponding angles and a non-included side are congruent.
• Difference from ASA: The congruent side is next to but not between the angles.

5. Hypotenuse-Leg (HL)

• Definition: Applicable only to right triangles.
• Criteria: The hypotenuse and one leg (non-hypotenuse side) are congruent.

Why Angle-Side-Side (ASS) Doesn’t Work

• Issue: Lacks information about the angle between the two sides.
• Example: Congruent angles and two sides can form different triangles (acute or obtuse).
• Conclusion: Different base lengths indicate non-congruence.

Conclusion

• Explore Further: Understanding why certain theorems work and others do not is crucial for proving triangle congruence.
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