Included Angle — Definition, Formula & Examples
An included angle is the angle that sits between two given sides of a triangle, formed at the vertex where those two sides meet.
Given two sides of a triangle, the included angle is the angle whose vertex is the common endpoint of those two sides and whose arms lie along those sides.
How It Works
To identify the included angle, look at which two sides you are given and find the vertex they share. The angle at that shared vertex is the included angle. For example, in triangle , if you are given sides and , the included angle is because is the endpoint common to both sides. This concept is essential for applying the SAS (Side-Angle-Side) congruence postulate, which requires the angle to be between the two known sides.
Worked Example
Problem: In triangle PQR, side PQ = 8 cm, side QR = 6 cm, and the angle at Q is 50°. Identify the included angle between sides PQ and QR, and state which congruence postulate you could use if another triangle has these same measurements.
Find the shared vertex: Sides PQ and QR both meet at vertex Q.
Identify the included angle: The angle formed at the shared vertex Q between the two given sides is the included angle.
Apply to congruence: Since you know two sides and the angle between them, you can use the SAS postulate to prove congruence with another triangle that has matching measurements.
Answer: The included angle is ∠Q = 50°, and SAS congruence applies.
Why It Matters
The included angle is a requirement for SAS triangle congruence and for the law of cosines when computing a third side. Misidentifying it — for instance, choosing a non-included angle — invalidates these methods and can lead to ambiguous or incorrect results in proofs and calculations.
Common Mistakes
Mistake: Using any angle in the triangle instead of the one between the two specified sides.
Correction: The included angle must be at the vertex shared by both given sides. If you use a different angle, you no longer have the SAS arrangement and cannot apply that postulate.