AAS Congruence

High School

Definition

AAS is an abreviation of Angle-angle-side congurence and sometimes written as SAA congruence. This theorem states that two triangles are congruent when two pairs of corresponding angles and a pair of opposite sides are the congruent.

Worked examples

\(\triangle ABC \cong \triangle DEF\) when \(\angle A \cong \angle D\), \(\angle B \cong \angle E\), and \(\overline{AC} \cong \overline{DF}\)

Two angles and the non-included side (opposite one of the angles) match, so AAS confirms congruence.

Given \(\angle P = 50^\circ\), \(\angle Q = 60^\circ\), \(\overline{PR} = 8\) and \(\angle X = 50^\circ\), \(\angle Y = 60^\circ\), \(\overline{XZ} = 8\), then \(\triangle PQR \cong \triangle XYZ\)

The side PR is opposite angle Q, and XZ is opposite angle Y; two angles and opposite side match by AAS.

Common mistakes

Using AAS when the given side is between the two angles → That is ASA congruence, not AAS AAS requires the side to be opposite one of the angles, not included between them.
Matching two angles and any random side → The side must correspond correctly to the opposite side in the other triangle Correspondence matters: if side AB is opposite angle C, match it to the side opposite the corresponding angle.
Writing \(\triangle ABC \cong \triangle DEF\) when vertices do not correspond in order → List vertices so matching angles align: \(\angle A \leftrightarrow \angle D\), \(\angle B \leftrightarrow \angle E\), etc. Congruence notation requires corresponding vertices in matching positions.

Where you'll use it next

AAS congruence is essential for proving triangles congruent in geometry proofs, solving problems involving overlapping triangles, and later in trigonometry when analyzing triangle properties and relationships.

Found in 1 StudyPug lesson

Mastering Triangle Congruence: ASA and AAS Proofs

Geometry

Unlock the power of geometric reasoning with our in-depth guide to ASA and AAS proofs. Learn to confidently prove triangle congruence and apply these skills to advanced geometry problems.

10th Grade10th