In triangle ABC, the incentre is I. If angle ABC = 90° and angle ACB = 70°, what is the measure of angle BIC (in degrees)?

Introduction / Context:
This problem uses a standard result about the angles formed at the incentre of a triangle. The incentre is the point where the internal angle bisectors meet and is the centre of the inscribed circle. Knowing how angles at the incentre relate to the angles of the triangle is useful in many geometry questions.

Given Data / Assumptions:

• I is the incentre of triangle ABC.
• Angle ABC = 90°.
• Angle ACB = 70°.
• We need to find the measure of angle BIC.

Concept / Approach:
In any triangle, if I is the incentre, the angle between the angle bisectors at vertices B and C (that is, angle BIC) is given by the formula:
angle BIC = 90° + (A / 2),
where A is the measure of angle at vertex A. This result comes from the fact that the sum of the angles around point I is 360° and each angle at the incentre is formed by halves of the original triangle angles. So our first step is to find angle A using the triangle angle sum, then apply the formula.

Step-by-Step Solution:
Step 1: Use the angle sum property of a triangle: A + B + C = 180°. Step 2: Substitute B = 90° and C = 70°: A + 90° + 70° = 180°. Step 3: Simplify: A + 160° = 180°, so A = 20°. Step 4: Use the incentre formula for angle BIC: angle BIC = 90° + (A / 2). Step 5: Substitute A = 20°: angle BIC = 90° + 20° / 2 = 90° + 10° = 100°.

Verification / Alternative Check:
We can check other incentre angle formulas to ensure consistency. For example, angle CIA = 90° + (B / 2) and angle AIB = 90° + (C / 2). Using B = 90° and C = 70°, we get angle CIA = 90° + 45° = 135° and angle AIB = 90° + 35° = 125°. The three angles at the incentre must sum to 360°, and indeed 100° + 135° + 125° = 360°, confirming that the formula and calculation are correct.

Why Other Options Are Wrong:
115°, 110°, or 105° do not match the formula 90° + (A / 2) once A is correctly found to be 20°. Using any of those values would make the sum of angles at the incentre differ from 360°, contradicting the geometric structure of the triangle and its incentre.

Common Pitfalls:
A frequent mistake is to use the wrong vertex angle in the formula and write angle BIC = 90° + (B / 2), which would be incorrect. Another pitfall is forgetting to first compute angle A from the angle sum rule. Some students also confuse the incentre with the circumcentre or orthocentre, each of which has different angle relations. Carefully recalling that angle BIC uses the angle at vertex A helps avoid these errors.

Final Answer:
The measure of angle BIC is 100°.