ASA criterion

When two angles and the side between them match, the triangles are congruent.

Idea

ASA (Angle-Side-Angle) criterion. If two angles of one triangle and the side included between them are equal to the corresponding parts of another triangle, the triangles are congruent.

In symbols: if $\angle B = \angle E$, $BC = EF$, $\angle C = \angle F$, then $\triangle ABC \cong \triangle DEF$.

Why it works. Once you fix a side and the two angles at its ends, the triangle's two other sides are forced to meet at a single point , the apex. So the whole shape is locked.

Corollary: AAS criterion. Two angles and a non-included side also work. The reason: the third angle is determined (since angles sum to $180^\circ$), so AAS effectively becomes ASA after computing the third angle.

So practically you have four working criteria for general triangles: SSS, SAS, ASA, AAS , and a special one for right triangles (RHS) coming next.

Note: AAA alone fails. Three matching angles only give similar triangles (same shape), not necessarily congruent (same size). A small equilateral and a big equilateral both have $60^\circ$–$60^\circ$–$60^\circ$, yet are not congruent.

Worked examples

Example 1. $\triangle ABC$: $\angle A = 50^\circ, AB = 5, \angle B = 70^\circ$. $\triangle PQR$: $\angle P = 50^\circ, PQ = 5, \angle Q = 70^\circ$. Congruent?

By ASA, yes. (The side $AB$ is between $\angle A$ and $\angle B$.)

Example 2. AAS example. $\triangle ABC$: $\angle A = 40^\circ, \angle B = 60^\circ, BC = 7$. $\triangle DEF$: $\angle D = 40^\circ, \angle E = 60^\circ, EF = 7$. Congruent?

The third angle of each is $80^\circ$. Now $\angle B = 60^\circ, BC = 7, \angle C = 80^\circ$ in both , ASA. So yes.

Example 3. Two equilateral triangles, sides $5$ and $10$. Same three angles ($60^\circ$ each). Congruent?

No , same shape, different size. AAA does not give congruence.

Example 4. $\triangle ABC$: $\angle B = 90^\circ, BC = 3, \angle C = 45^\circ$. Determine the triangle.

Third angle $= 180-90-45 = 45^\circ$. So an isosceles right triangle with legs $3$ each, hypotenuse $3\sqrt{2}$. Uniquely determined by ASA.

Try it yourself

1. State ASA in symbols.
2. Two angles $60^\circ, 70^\circ$ and included side $8$ cm , congruent triangles?
3. AAA , does it guarantee congruence? Why not?
4. Two angles $45^\circ, 60^\circ$ in each, and a non-included side of $6$ cm each. Congruent?
5. If two pairs of angles match, what about the third pair?
6. State the difference between ASA and AAS.
7. Sketch a triangle from $\angle B = 50^\circ, BC = 7, \angle C = 60^\circ$.
8. Two right triangles with one acute angle $30^\circ$ each and corresponding side equal. ASA or AAS?

Activity

Make a protractor by folding paper. Draw a base line; from each end, draw rays at given angles. They meet at the apex of the triangle, which is uniquely determined. Convince yourself ASA gives uniqueness.