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ASA Congruence

ASA Congruence

Angle-side-angle congruence. When two triangles have corresponding angles and sides that are congruent as shown below, the triangles themselves are congruent.

 

Two congruent triangles ABC and DEF with tick marks on sides AC and DF, and arc marks on angles A, C, E, and F showing ASA...

 

 

See also

Congruence tests for triangles

Worked Example

Problem: Triangle ABC has angle A = 50°, side AB = 7 cm, and angle B = 60°. Triangle DEF has angle D = 50°, side DE = 7 cm, and angle E = 60°. Are the two triangles congruent by ASA?
Step 1: Identify the two angles and the included side in each triangle. In triangle ABC, the included side AB lies between angle A and angle B. In triangle DEF, the included side DE lies between angle D and angle E.
Step 2: Check whether the first pair of corresponding angles are equal.
∠A=∠D=50°\angle A = \angle D = 50°
Step 3: Check whether the included sides are equal.
AB=DE=7 cmAB = DE = 7 \text{ cm}
Step 4: Check whether the second pair of corresponding angles are equal.
∠B=∠E=60°\angle B = \angle E = 60°
Step 5: Since two angles and the included side of triangle ABC match two angles and the included side of triangle DEF, the ASA condition is satisfied.
△ABC≅△DEF(ASA)\triangle ABC \cong \triangle DEF \quad (\text{ASA})
Answer: Yes, triangle ABC is congruent to triangle DEF by ASA Congruence.

Another Example

Problem: In triangle PQR, angle P = 40°, angle Q = 75°, and side PQ = 10 cm. In triangle XYZ, angle X = 40°, angle Z = 75°, and side XZ = 10 cm. Can you use ASA to prove the triangles congruent?
Step 1: In triangle PQR, the side PQ is between angle P and angle Q. So the ASA pairing is: angle P = 40°, side PQ = 10 cm, angle Q = 75°.
Step 2: In triangle XYZ, the side XZ is between angle X and angle Z. So the ASA pairing is: angle X = 40°, side XZ = 10 cm, angle Z = 75°.
Step 3: Compare the corresponding parts: angle P corresponds to angle X (both 40°), side PQ corresponds to side XZ (both 10 cm), and angle Q corresponds to angle Z (both 75°). All three match.
△PQR≅△XYZ(ASA)\triangle PQR \cong \triangle XYZ \quad (\text{ASA})
Answer: Yes. Even though the letters are in different positions, the key is that the known side is included between the two known angles in both triangles, so ASA applies.

Frequently Asked Questions

What is the difference between ASA and AAS congruence?
In ASA, the known side is between the two known angles (it is the included side). In AAS (Angle-Angle-Side), the known side is not between the two known angles — it is adjacent to only one of them. Both rules prove congruence, but they use the side in a different position relative to the angles.
Why does ASA prove triangles are congruent?
If you fix two angles and the side between them, there is only one possible triangle you can draw. The third angle is already determined (since all angles sum to 180°), and the other two sides are forced into specific lengths. So any triangle with the same ASA measurements must be identical in shape and size.

ASA (Angle-Side-Angle) vs. SAS (Side-Angle-Side)

ASA uses two angles with the side between them, while SAS uses two sides with the angle between them. Both require the shared element (side in ASA, angle in SAS) to be the included one — meaning it sits between the other two known parts. Mixing up which element must be included is one of the most common errors students make.

Why It Matters

ASA is one of the foundational congruence criteria used in geometric proofs. Whenever you can identify two angles and an included side that match between triangles, you can immediately conclude the triangles are congruent without measuring every side and angle. This shortcut appears frequently in proofs involving parallel lines, bisectors, and many real-world applications like construction and surveying.

Common Mistakes

Mistake: Using a non-included side and calling it ASA.
Correction: The side must be between the two angles. If the known side is not between the two known angles, the correct rule is AAS, not ASA. Always check that the side connects the vertices of both given angles.
Mistake: Matching parts from the two triangles in the wrong order.
Correction: Corresponding parts must actually be in matching positions. Angle A should correspond to the angle in the other triangle that occupies the same relative position. Write out the congruence statement carefully so that corresponding vertices line up.

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