ASA Congruence Rule: Definition, Proof, Examples

ASA Congruence Rule: Definition, Proof, Examples

The ASA congruence rule states that two triangles are congruent when two angles and the included side (the side between those two angles) of one match the corresponding parts of the other. This article gives the precise statement, a proof, six worked examples, and the mistakes that trip students up most.

How ASA Differs from AAS, and Why the Word "included" Decides Everything

A fair question students raise early: isn't ASA the same as AAS? Not quite, and the difference is the position of the side.

Why ASA Works At All

Once you fix two angles and the side between them, the third angle is forced (the three angles of a triangle add to 180°), and the two remaining sides are forced too — each must start at a fixed endpoint and leave at a fixed angle, so they meet at exactly one point. There is no freedom left. That is the intuition behind the proof below.

ASA Congruence Rule Proof

Statement to prove: In triangles ABC and DEF, if ∠B=∠E, BC=EF, and ∠C=∠F, then △ABC≅△DEF.

Proof (superposition):

Place △DEF onto △ABC so that side EF falls exactly along side BC.

This is possible because BC=EF, so E lands on B and F lands on C.

Since ∠E=∠B, ray ED falls along ray BA.

Since ∠F=∠C, ray FD falls along ray CA.

Point D lies on ray BA and on ray CA at once, so D must be the single point where those rays cross, which is A.

With D on A, E on B, and F on C, the triangles coincide completely.

The matching parts that follow, namely AB=DE and AC=DF, are guaranteed by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the tool you reach for once congruence is established.

Examples of ASA Congruence Rule

Example 1

In triangles PQR and XYZ, ∠Q=∠Y=50°, QR=YZ=6 cm, and ∠R=∠Z=70°. Are the triangles congruent?
The two angles are ∠Q, ∠R and the side QR lies between them.
The matching parts ∠Y, ∠Z and side YZ fit the same pattern.
By ASA, △PQR≅△XYZ.

Example 2

Two triangles share ∠A=∠D=40° and ∠C=∠F=60°, with AB=DE=5 cm. Can you conclude congruence by ASA?
A natural first move is to say "two angles and a side match, so it's ASA, done."
But the given side is AB, which is not between ∠A and ∠C.
It is, however, a valid AAS setup, and AAS does prove congruence.
Final answer: Congruent by AAS, not ASA. The label matters because the side is not included.

Example 3

In △ABC and △CDA sharing the common side AC, ∠BAC=∠DCA and ∠BCA=∠DAC. Prove △ABC≅△CDA.
The shared side is AC.
AC=CA (common side, included between the marked angles)
∠BAC=∠DCA (given)
∠BCA=∠DAC (given)
By ASA, △ABC≅△CDA.

Example 4

A triangle has ∠B=55° and ∠C=65° with BC=8 cm. A second triangle has ∠E=55°, ∠F=65°, EF=8 cm. Find the third angle in each, then state the congruence.
Third angle of the first: 180°−55°−65°=60°.
Third angle of the second: 180°−55°−65°=60°.
By ASA, the triangles are congruent.

Example 5

In the figure, O is the midpoint of AB, and ∠OAC=∠OBD with C and D on opposite sides. Show △AOC≅△BOD.
∠OAC=∠OBD (given)
AO=BO (O is the midpoint, included side)
∠AOC=∠BOD (given)
By ASA, △AOC≅△BOD.

Example 6

Two triangular plots are surveyed across a stream. From a baseline PQ=40 m, the angle to a marker M is measured as ∠QPM=72° from P and ∠PQM=48° from Q. A second survey gives a baseline RS=40 m with ∠SRT=72° and ∠RST=48°. Show the two triangles are congruent.
By ASA, △PQM≅△RST.

Where The Rule Earns Its Keep: Triangulation You Can Trust

ASA is not a textbook curiosity. It is the reason an entire family of measurement techniques works without anyone ever pacing out the distance in question.

Tripping Points To Avoid

Mistake 1: Using a side that is not included

Mistake 2: Writing the congruence statement in the wrong vertex order

Mistake 3: Confusing ASA with AAA

Key Takeaways