What Is an ASA Triangle?
Consider triangle $\triangle ABC$ where the two known angles are $\angle B$ and $\angle C$, and the known included side between them is $BC = a$. This configuration is called an ASA triangle.
Properties of an ASA Triangle
A valid ASA triangle must satisfy the following conditions:- Two angles are known: for example, $\angle B$ and $\angle C$
- The included side between them is known: $a = BC$
- The sum of all three interior angles always equals $180°$:
How to Solve an ASA Triangle
Solving an ASA 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 is:$\angle A = 180° - \angle B - \angle C$
Step 2 — Find the Unknown Sides Using the Law of Sines
Once all three angles are known, apply the Law of Sines:$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
The two unknown sides are solved as:
$b = \frac{a \cdot \sin B}{\sin A}$
$c = \frac{a \cdot \sin C}{\sin A}$
ASA Triangle Congruence Theorem
Statement
The ASA Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.In formal notation: if $\angle ABC \cong \angle EFG$, $\overline{BC} \cong \overline{FG}$, and $\angle BCA \cong \angle FGE$, then:
$\triangle ABC \cong \triangle EFG$
Proof of the ASA Congruence Theorem
To prove: $\triangle ABC \cong \triangle DCB$Given that $AB \parallel CD$.
| Steps | Statements | Reasons |
|---|---|---|
| 1 | $AB \parallel CD$, $\angle ACB = \angle DBC$ | Given |
| 2 | $\angle ABC \cong \angle DCB$ | Alternate interior angles |
| 3 | $CB \cong CB$ | Reflexive property of congruence |
| 4 | $\triangle ABC \cong \triangle DCB$ | AAS postulate (Hence proved) |
Conclusion
An ASA triangle is uniquely determined when two angles and their included side are known. By applying the Angle Sum Rule to find the third angle $\angle A = 180° - \angle B - \angle C$, then using the Law of Sines $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$, all remaining sides can be found. The ASA Congruence Theorem further confirms that any two triangles sharing the same two angles and included side are necessarily congruent — a foundational principle in Euclidean geometry.
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