In triangle PQR, angle QPR is 45 degrees. The internal bisectors of angles PQR and PRQ intersect at point O. What is the measure of angle QOR (in degrees)?

Introduction / Context:
This question is about angles formed at the incenter of a triangle. In triangle PQR, the internal angle bisectors from vertices Q and R intersect at O, which is the incenter. We are given that angle QPR (the angle at P) is 45 degrees and asked to find angle QOR, the angle at the incenter between the bisectors from Q and R. This uses a standard property of incenters that relates the angle between two internal bisectors to the opposite vertex angle.

Given Data / Assumptions:
• Triangle PQR is any triangle with angle QPR = 45 degrees.
• The internal bisector of angle PQR and the internal bisector of angle PRQ intersect at O, the incenter of triangle PQR.
• We must determine the measure of angle QOR in degrees.
• All angles mentioned are internal angles in the triangle or at the incenter.

Concept / Approach:
In any triangle, the incenter is the intersection point of the three internal angle bisectors. A key property is that the angle at the incenter formed by the bisectors of two angles is equal to 90 degrees plus half of the remaining angle. Specifically, the angle between the bisectors of angles at Q and R is 90° + (angle P)/2. In this problem, angle P is given as 45 degrees, so we can directly apply this formula to find angle QOR.

Step-by-Step Solution:
Step 1: Note that the bisectors from Q and R meet at O, making O the incenter of triangle PQR. Step 2: Recall the standard result: angle QOR = 90° + (angle QPR)/2, where angle QPR is the remaining vertex angle at P. Step 3: Given angle QPR = 45°, substitute this value into the formula. Step 4: Compute half of angle QPR: (angle QPR)/2 = 45° / 2 = 22.5°. Step 5: Add 90° to this half angle: 90° + 22.5° = 112.5°. Step 6: Therefore, angle QOR is 112.5 degrees.

Verification / Alternative check:
We can verify this property by recalling the general relationships at the incenter. The three angles at the incenter formed by pairs of adjacent angle bisectors sum to 360 degrees. These angles are 90° + (A/2), 90° + (B/2), and 90° + (C/2) for a triangle with angles A, B, and C at the vertices. Their sum is 270° + (A + B + C)/2 = 270° + 180° / 2 = 270° + 90° = 360°, which is consistent. Applying this specifically with A = angle P, the angle between the bisectors at Q and R must be 90° + A/2. Substituting A = 45° gives 112.5° as found above.

Why Other Options Are Wrong:
107.5°, 117.5°, and 122.5° differ from 112.5° by 5° or 10°, and none of them equal 90° + 45° / 2.
135° would correspond to 90° + 45°, which would be the wrong application of the formula, using the full angle instead of half the angle at P. Therefore, these values do not match the incenter angle property for this triangle.

Common Pitfalls:
A common error is to use 90° + angle P instead of 90° + angle P/2, leading to 135° in this case. Others may confuse which vertex angle is used in the formula, sometimes taking half of Q or R instead of P. Another issue is forgetting that the bisectors intersect at the incenter, not some arbitrary point. Remembering the specific formula for angles at the incenter and carefully identifying the opposite vertex angle avoids these errors.

Final Answer:
The measure of angle QOR is 112.5 degrees.