ASA (Angle-Side-Angle) Congruence
Two angles and the included side (the side between those angles) determine a unique triangle. If these three parts match, the triangles are congruent.
Introduction
Imagine standing at one end of a hallway. You know the hallway's length and the angles at both ends. That completely determines the triangle formed — you know exactly where the third point must be. That's ASA in action.
Past Knowledge
SSS (5.2.2), SAS (5.2.3). Rigid motions. Angle Sum Theorem (5.1.2).
Today's Goal
State, prove, and apply the ASA Congruence Theorem.
Future Success
AAS (5.2.5), HL (5.2.6), CPCTC (5.2.7).
Key Concepts
ASA Congruence Theorem
If two angles and the included side of one triangle are congruent to two angles and the included side of another, then the triangles are congruent.
If , then .
Included Side
The included side is the side that is a part of both angles — it connects the two vertices where the known angles are located. In and , the included side is .
Theorem & Proof
Two-Column Proof: ASA Congruence (Rigid Motion Argument)
Given: and with , ,
Prove:
Align the included side, equal angles force rays to match, unique intersection gives the third vertex.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Apply rigid motions to map and | , so a translation + rotation aligns onto |
| 3 | maps onto and maps onto | and ; rigid motions preserve angles, so rays from and align with rays from and |
| 4 | maps to | is the intersection of and ; is the intersection of and . Two non-parallel lines intersect at exactly one point. |
| 5 | Rigid motions map all three vertices: ; congruence by definition |
∎ Two angles and the included side determine a unique triangle.
Worked Examples
Direct ASA
In and : , , . Are they congruent?
and are at the endpoints of , so is the included side. ✓
by ASA.
Proof with Parallel Lines
Given , and and intersect at with . Prove .
Angle 1: (alternate interior angles, )
Side: (given)
Angle 2: (vertical angles)
by ASA.
Using the Third Angles Theorem
Given , , , , , . Can you use ASA?
Find the missing angles: and
Now check: , , and .
Side is included between and ; side is included between and .
by ASA (after using the Angle Sum Theorem to find the missing angles).
Common Pitfalls
Confusing ASA with AAS
In ASA, the side is between the two angles. In AAS (next lesson), the side is not between them. Both work, but they are different configurations — make sure you identify which one applies.
Forgetting to Match the Correct Side
The side must be the one connecting the vertices of the two known angles. If you know and , the included side is , not or .
Real-Life Applications
Navigation — Triangulation
Sailors and hikers use triangulation: from two known landmarks (a measured baseline distance apart), they measure the angle to a target from each endpoint. Two angles + included side (baseline) = ASA, giving the exact location of the target.
Astronomy — Stellar Parallax
Astronomers measure star distances using ASA: Earth's orbital diameter is the baseline (known side), and the tiny apparent shift angles at two points six months apart give the two angles. The resulting ASA triangle yields the star's distance.
Practice Quiz
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