Congruent Triangles

Learning Focus

Use SSS, SAS, ASA, AAS, and HL to decide whether triangles are congruent.

Big idea

Triangles are congruent if these parts match:

• SSS: three pairs of sides.
• SAS: two sides and the included angle.
• ASA and AAS: two angles and a side.
• HL: hypotenuse and a leg (right triangles only).

AAA and SSA do not guarantee congruence.

Worked example

Two right triangles each have hypotenuse $13$ and one leg $5$. By HL they are congruent.

The other leg: $\sqrt{13^2-5^2}=\sqrt{144}=12$ in both.

If $\triangle ABC\cong\triangle DEF$ with $AB=2x+1$ and $DE=11$, then $2x+1=11$ and $x=5$.

Takeaways

• Mark corresponding parts.
• SAS needs the angle between the sides.
• AAA and SSA are not criteria.