Question

Let PQR be a triangle such that angle P = angle Q + angle R. Among all three exterior angles of triangle PQR, what is the smallest exterior angle, in degrees?

Answer 1

Answer:

The smaller exterior angle is the exterior angle in the vertex P, and it measures 90°

Step-by-step explanation:

The sum of the internal angles of a triangle needs to be 180 degrees, so we have two equations:

$mP = mQ + mR$

$mP + mQ + mR = 180\°$

Substituting $mQ + mR$ by $mP$ in the second equation, we have:

$mP + mP = 180\°$

$mP = 90\°$

The other two angles need to be lesser than mP, and the exterior angle is the supplement of the internal angle, so the bigger the internal angle, the smaller the exterior angle.

So if the bigger internal angle in this triangle is mP, the smaller exterior angle is also the angle in the vertex P:

$exteriorP = 180\° - mP$

$exteriorP = 90\°$