In triangle ABC, O is the orthocentre. If the angle ∠BOC formed at the orthocentre by lines OB and OC is 110°, then what is the measure of angle ∠BAC (in degrees)?

Introduction / Context:
This geometry question tests your knowledge of special angle relationships associated with the orthocentre of a triangle. The orthocentre is the point where the three altitudes of a triangle intersect. For an acute triangle, there is a well known relationship between the angle at the orthocentre formed by two vertices and the opposite vertex angle in the original triangle. Understanding this relationship allows you to convert between an angle at the orthocentre and an angle at a vertex of the triangle without directly constructing all altitudes, which is very useful in time bound aptitude exams.

Given Data / Assumptions:
• Triangle ABC is a non degenerate triangle with orthocentre O.
• Angle ∠BOC is given as 110 degrees.
• O is the intersection point of the three altitudes of triangle ABC.
• The triangle is assumed to be acute so that the usual interior angle relationships with the orthocentre apply directly.
• We need to find angle ∠BAC, which is the angle at vertex A of the triangle.

Concept / Approach:
A key property of the orthocentre states that the angle at the orthocentre formed by joining it to any two vertices is supplementary to the angle at the remaining vertex. More precisely, in an acute triangle, the angle ∠BOC at the orthocentre is related to the angle at vertex A by the formula ∠BOC = 180° - ∠A. This is analogous to a similar relation for the circumcentre, where angles involving the circumcentre are often twice the vertex angles, but here for the orthocentre the relationship involves a supplement. Using this property, you can directly obtain the required angle at A from the given angle at O.

Step-by-Step Solution:
Step 1: Recall the orthocentre angle property: in triangle ABC with orthocentre O, angle ∠BOC = 180° - ∠A, where ∠A = ∠BAC. Step 2: Substitute the given value ∠BOC = 110° into the relation: 110° = 180° - ∠BAC. Step 3: Rearrange the equation to solve for ∠BAC: ∠BAC = 180° - 110°. Step 4: Compute the difference: 180° - 110° = 70°. Step 5: Therefore, the measure of angle ∠BAC is 70 degrees.

Verification / Alternative check:
If angle A is 70°, then angle ∠BOC for an acute triangle should be 180° - 70° = 110°, which matches the given information. This confirms that our use of the property is correct. Additionally, because 70° is an acute angle, it is consistent with the assumption that the triangle is acute, where the orthocentre lies inside the triangle and the standard relationships for orthocentre angles hold. There is no contradiction, so the answer is consistent and verified.

Why Other Options Are Wrong:
If ∠BAC were 110°, then ∠BOC would be 180° - 110° = 70°, which is the reverse of what is given. If ∠BAC were 100°, then ∠BOC should be 80°, not 110°. If ∠BAC were 90°, then ∠BOC should be 90°, which again does not match the given 110°. Thus all other options are inconsistent with the fundamental relation ∠BOC = 180° - ∠BAC, leaving 70° as the only correct choice.

Common Pitfalls:
A frequent mistake is to confuse the orthocentre with the circumcentre and incorrectly use a relation like ∠BOC = 2∠A, which actually belongs to the circumcentre in certain contexts, not to the orthocentre. Another common error is to think in terms of angle sums in the triangle alone and ignore the specific properties of the special point described in the question. Carefully noting that O is the orthocentre and recalling the correct identity ∠BOC = 180° - ∠A is crucial. Students should also be careful with subtraction, as reversing 180° - 110° can lead to incorrect values if done hastily.

Final Answer:
The measure of angle ∠BAC is 70°.