What Is an AAS Triangle?
Consider triangle $\triangle ABC$ where the two known angles are $\angle A$ and $\angle B$, and the known non-included side is $BC = a$. This configuration is called an AAS triangle.
The AAS condition is one of the standard triangle congruence criteria. If two triangles share the same two angles and a corresponding non-included side, the triangles are congruent.
Properties of an AAS Triangle
A valid AAS triangle must satisfy the following conditions:- Two angles are known: for example, $\angle A$ and $\angle B$
- One non-included side is known: for example, $a = BC$
- The sum of all three interior angles always equals $180°$:
How to Solve an AAS Triangle
Solving an AAS triangle means finding all unknown angles and sides. The process involves two steps.Step 1 — Find the Missing Angle Using the Angle Sum Rule
Since the sum of angles in any triangle is $180°$, the third angle can always be found:$\angle C = 180° - \angle A - \angle B$
Step 2 — Find the Unknown Sides Using the Law of Sines
Once all three angles are known, apply the Law of Sines to find the remaining sides:$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
From this, each unknown side is solved as:
$b = \frac{a \cdot \sin B}{\sin A}$
$c = \frac{a \cdot \sin C}{\sin A}$
Conclusion
An AAS triangle is straightforward to solve because having two angles and one non-included side is enough to fully determine the triangle. By applying the Angle Sum Rule to find $\angle C = 180° - \angle A - \angle B$, then using the Law of Sines $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$, all missing sides and angles can be calculated accurately. Mastering AAS triangles builds a strong foundation for tackling more complex trigonometry problems.
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