AAS (Angle-Angle-Side) Congruence
Two angles and a non-included side also prove congruence — because the third angle is automatically determined by the Angle Sum Theorem.
Introduction
AAS is closely related to ASA. If you know two angles, you automatically know the third (Angle Sum Theorem). So AAS gives you all three angles plus a side — which is enough information for ASA once you rearrange.
Past Knowledge
ASA (5.2.4). Angle Sum Theorem (5.1.2). Third Angles Theorem.
Today's Goal
Prove AAS using ASA + Third Angles Theorem and apply it in proofs.
Future Success
HL (5.2.6), CPCTC proofs (5.2.7), and future similarity work.
Key Concepts
AAS Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another, then the triangles are congruent.
If , then .
Why AAS Works
The key insight: if two angles of one triangle equal two angles of another, then by the Third Angles Theorem, the third angles are also equal. So you really have three angles plus a side — and the side is now included between two of those angles, giving you ASA.
Quick Summary of All Valid Shortcuts
✅ SSS, SAS, ASA, AAS
❌ AAA (similar only), SSA (ambiguous)
Theorem & Proof
Two-Column Proof: AAS Congruence Theorem
Given: and with , ,
Prove:
Strategy: Use the Third Angles Theorem to obtain a third angle, then apply ASA.
Two known angles → Third Angles Theorem gives the third → rearrange into ASA.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Third Angles Theorem: if two pairs of angles are congruent, the third pair must be also (angle sum = 180°) | |
| 3 | We now have: | Steps 1 and 2 (rearranging the known congruences) |
| 4 | ASA Congruence (, side , match , side , ) |
∎ AAS reduces to ASA via the Third Angles Theorem.
Worked Examples
Identifying AAS vs ASA
, , . Which method?
Side connects vertices and . We know but not — we know instead.
So the side is not between the two known angles → AAS, not ASA.
by AAS.
Proof with Perpendicular Lines
Given: , , and bisects . Prove .
Angle 1: (given)
Angle 2: (definition of angle bisector)
Side: (Reflexive Property)
The side is not between and → AAS configuration.
by AAS.
Choosing the Right Method
Given: , , . Which congruence method applies?
The side connects vertices and . Both and are at the endpoints of this side.
Wait — the side IS between the two known angles! This is actually ASA, not AAS.
Always check whether the side is included before claiming AAS.
by ASA (side is included between the two angles).
Common Pitfalls
Mislabeling AAS as ASA
Both work, but identify the correct one. If the known side connects the two known-angle vertices, it's ASA. If it does NOT connect them, it's AAS.
Thinking AAA Works
AAS requires a side. Knowing all three angles (AAA) only proves similarity, not congruence. Without a side, the triangles could be different sizes.
Real-Life Applications
Crime Scene Investigation
Forensic analysts use AAS when reconstructing trajectories. Knowing the angle of impact, the angle at a known reference point, and a measured distance lets them reconstruct the unique triangle of the projectile's path.
Cartography — Map Making
Cartographers historically triangulated large regions using AAS. From a known baseline and angle measurements to two distant landmarks, they could reconstruct the exact triangle and compute all distances, building accurate maps without direct measurement.
Practice Quiz
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