Geometry / Unit 5: Congruent Triangles
5.4 ASA, AAS and HL Congruence
Notes
Angles can also help prove triangles congruent.
- ASA: two angles and the included side.
- AAS: two angles and a nonincluded side.
- HL: in right triangles, congruent hypotenuses and one pair of congruent legs.
- AAA and SSA do not work; they don't guarantee congruence.
- Example: $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\overline{BC} \cong \overline{EF}$. $\overline{BC}$ is not between $\angle A$ and $\angle B$, so this is AAS.
Practice
$\triangle ABC$ and $\triangle DEF$ have $\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$ and $\angle B \cong \angle E$. Which postulate proves them congruent?
$\overline{AB}$ is between $\angle A$ and $\angle B$, so it is the included side.
Two right triangles have congruent hypotenuses and one pair of congruent legs. Which theorem proves them congruent?
HL is the special case for right triangles.
Which set of information does NOT prove two triangles congruent?
AAA only shows the triangles have the same shape; they could be different sizes.
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