In triangle PQR, the sides PQ and PR are produced beyond Q and R to points A and B respectively. The bisectors of exterior angles AQR and BRQ intersect at point O. If angle QOR = 50 degrees, what is the measure of angle QPR?

Introduction / Context:
This is an advanced geometry question involving excentres of a triangle. The lines described are external angle bisectors at two vertices of a triangle, and their intersection defines an excentre. The angle at this excentre has a known relation to the corresponding interior angle of the triangle.

Given Data / Assumptions:
• Triangle PQR is given. • Side PQ is extended beyond Q to point A. • Side PR is extended beyond R to point B. • Angle bisectors of exterior angles AQR and BRQ meet at point O. • Angle QOR at point O is 50 degrees. • We are asked to find interior angle QPR of triangle PQR.

Concept / Approach:
The point O, where the bisectors of exterior angles at Q and R meet, is the excentre opposite vertex P. A key property of excentres is that, in triangle ABC with excentre I opposite A, angle BIC = 90 degrees - (A / 2). Translating this to triangle PQR with excentre O opposite P, angle QOR corresponds to angle BIC and angle at P corresponds to angle A. Therefore, angle QOR = 90 degrees - (angle QPR / 2). We can use this relation to solve for angle QPR.

Step-by-Step Solution:
1. Recognise that O is the excentre opposite vertex P because it lies at the intersection of the external angle bisectors at Q and R. 2. For a triangle, the angle at the excentre opposite vertex P satisfies angle QOR = 90 degrees - (angle QPR / 2). 3. We are given angle QOR = 50 degrees. 4. Set up the equation: 50 = 90 - (angle QPR / 2). 5. Rearrange: angle QPR / 2 = 90 - 50 = 40 degrees. 6. Multiply both sides by 2: angle QPR = 2 * 40 = 80 degrees.

Verification / Alternative check:
To check the formula, recall the well known incenter relation: in triangle ABC, if I is the incenter, angle BIC = 90 degrees + (A / 2). For the excentre opposite A, the sign changes and we have 90 degrees - (A / 2). The reasoning mirrors that of the incenter but uses external angles. Applying the excentre formula here leads directly to angle QPR = 80 degrees, confirming our calculation.

Why Other Options Are Wrong:
• 50°: This would give angle QOR = 90 - (50 / 2) = 65 degrees, which contradicts the given 50 degrees. • 60°: This would imply angle QOR = 90 - 30 = 60 degrees, again not equal to 50 degrees. • 100°: This would give angle QOR = 90 - 50 = 40 degrees, different from the given 50 degrees. • 70°: This would give angle QOR = 90 - 35 = 55 degrees, still inconsistent.

Common Pitfalls:
A common pitfall is to confuse the excentre with the incenter and mistakenly use the formula 90 degrees + (A / 2) instead of 90 degrees - (A / 2). Students also sometimes misidentify which vertex corresponds to the angle opposite the excentre and thus choose the wrong angle when applying the formula. Careful reading and a clear diagram help avoid these issues.

Final Answer:
The measure of angle QPR is 80°.