In triangle PQR, the internal angle bisectors of ∠Q and ∠R meet at point O. If ∠P = 70°, what is the measure (in degrees) of ∠QOR at the incenter?

Introduction / Context:
This question involves the incenter of a triangle, which is the intersection point of the internal angle bisectors. A classic result gives the measure of the angle formed at the incenter between the bisectors corresponding to two vertices of the triangle. Recognising and using this formula is essential for solving such geometry problems quickly.

Given Data / Assumptions:

• Triangle PQR is any triangle with vertices P, Q and R.
• The internal angle bisectors of ∠Q and ∠R meet at point O, the incenter.
• ∠P is given as 70°.
• We are asked to find ∠QOR.
• All angles are internal and triangle is non degenerate.

Concept / Approach:
There is a standard theorem for a triangle with incenter O: the angle at the incenter between the bisectors of two angles of the triangle equals 90° plus half of the third angle. More precisely, ∠QOR = 90° + (∠P / 2). This comes from properties of angle bisectors and the fact that the incenter is equidistant from all three sides of the triangle. Applying this directly with the given value of ∠P produces the answer without complicated constructions.

Step-by-Step Solution:
Step 1: Recall the incenter angle formula: for triangle PQR with incenter O, ∠QOR = 90° + (∠P / 2). Step 2: The question gives ∠P = 70°. Step 3: Compute half of ∠P: ∠P / 2 = 70° / 2 = 35°. Step 4: Substitute into the formula: ∠QOR = 90° + 35°. Step 5: Therefore ∠QOR = 125°.

Verification / Alternative check:
We can support this result by a quick reasoning sketch. The three angles at the incenter formed by the bisectors sum to 360°. Each incenter angle is 90° plus half of the opposite angle. For example, ∠QOR corresponds to vertex P, and ∠ROP corresponds to vertex Q, and so on. Adding these expressions gives the total as 360° because the three vertex angles sum to 180°. This confirms that the formula ∠QOR = 90° + ∠P / 2 is consistent and that the numerical value 125° for ∠P = 70° is correct.

Why Other Options Are Wrong:
If we mistakenly took 90° minus half of ∠P, we would get 55°, which is not among the options. The options 110° or 115° correspond to incorrect manipulation of the formula, such as using 90° plus one third of ∠P or miscomputing half of 70°. The option 135° would require ∠P = 90°, not 70°. Only 125° matches the specific incenter relation with ∠P = 70°.

Common Pitfalls:
A common mistake is to confuse incenter formulas with those for the circumcenter or other triangle centers. Another error is to think that the incenter angles are simply half of the vertex angles or sum to 180°. Remember that incenter related angles between bisectors follow the 90° plus half opposite angle rule. Writing down the formula before substituting values helps avoid confusion and ensures accurate computation.

Final Answer:
The measure of ∠QOR is 125°.