SAS (Side-Angle-Side) Congruence
Two sides and the included angle (the angle between those sides) are enough to determine a unique triangle — and prove congruence.
Introduction
Sometimes you know two sides and an angle. If that angle is betweenthe two known sides (the “included” angle), SAS guarantees congruence. The included angle acts like a hinge — once you fix the angle and two arm lengths, the triangle is locked.
Past Knowledge
SSS Congruence (5.2.2). Rigid motions (Ch 4). Congruence basics (5.2.1).
Today's Goal
State, prove, and apply the SAS Congruence Postulate.
Future Success
ASA (5.2.4), AAS (5.2.5), HL (5.2.6), and CPCTC proofs (5.2.7).
Key Concepts
SAS Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
If , then .
⚠️ The Angle MUST Be Included
SSA (Side-Side-Angle where the angle is not between the two sides) does NOT guarantee congruence. This is sometimes called the “ambiguous case” — two different triangles can have the same SSA measurements.
Theorem & Proof
Two-Column Proof: SAS Congruence (Rigid Motion Argument)
Given: and with , ,
Prove:
Align one side, the included angle forces the ray, equal length forces the vertex.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Translate and rotate so and aligns with | Rigid motions exist to align any ray onto any other (translation + rotation) |
| 3 | maps to | and the rays are aligned, so lands exactly on |
| 4 | maps onto | and rigid motions preserve angles, so the other ray from aligns with the other ray from |
| 5 | maps to | and lies on at distance from , which is the same position as on |
| 6 | The rigid motion maps ; congruence by definition |
∎ Two sides and their included angle determine a unique triangle.
Worked Examples
Identifying SAS
In and : , , . Is this SAS?
The angle is between sides and — it's the included angle. ✓
Similarly, is between and .
Yes, by SAS.
SAS with a Shared Side
bisects and . Prove .
Side 1: (given)
Angle: (definition of angle bisector)
Side 2: (Reflexive Property)
The angle is included between the two sides at vertex .
by SAS.
Recognizing SSA (Not Valid!)
In and : , , . Can you prove congruence?
The known angle is not between the known sides and — it's opposite .
This is SSA, which is ambiguous. Two different triangles could satisfy these conditions.
No. SSA does NOT prove congruence. The angle must be included between the two sides.
Common Pitfalls
Confusing SAS with SSA
In SAS, the angle is between (included by) the two sides. In SSA, the angle is across from one of the sides. Only SAS works. Remember: the “A” must be in the middle of “SAS.”
Using SAS When Sides Aren't Adjacent to the Angle
Both sides must form the angle. If you have , the two sides must be and — the sides emanating from vertex .
Real-Life Applications
Surveying — Distance Across a River
Surveyors use SAS to find distances they can't measure directly. By measuring two accessible distances and the angle between them with a theodolite, they construct a congruent triangle on paper to determine the inaccessible distance.
Robotics — Arm Positioning
A robot arm with two segments of known length and a joint angle uses SAS geometry. The joint angle and arm lengths uniquely determine where the end effector lands — the same principle as the SAS postulate.
Practice Quiz
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