In triangle ABC, O is the orthocentre (the point where all three altitudes meet). If the measure of angle BOC is 100°, what is the measure of angle BAC?

Introduction / Context:
This question tests a standard but very important property of the orthocentre of a triangle. The orthocentre is the common point of intersection of the three altitudes of a triangle. There is a special relationship between the angles at the orthocentre and the angles at the vertices of the triangle. Many geometry questions in exams use this relationship, so recognizing the formula will save time and reduce calculations.

Given Data / Assumptions:

• ABC is a triangle.
• O is the orthocentre of triangle ABC.
• The measure of angle BOC is 100°.
• We are asked to find the measure of angle BAC.
• The triangle is assumed to be acute-angled so that the orthocentre lies inside the triangle, which is the usual context for this standard relation.

Concept / Approach:
For an acute triangle, there is a well known relation between the angle at a vertex and the angle at the orthocentre opposite that vertex. In particular, the angle between the lines OB and OC at the orthocentre is related to the angle at vertex A by the formula: ∠BOC = 180° − ∠A where ∠A is the same as angle BAC. Using this formula, we can directly compute the missing vertex angle once ∠BOC is known, by simply rearranging the equation.

Step-by-Step Solution:
Step 1: Recall the relation for an acute triangle: ∠BOC = 180° − ∠A, where ∠A = ∠BAC. Step 2: The question gives ∠BOC = 100°. Substitute this into the formula: 100° = 180° − ∠BAC Step 3: Rearrange to find angle BAC: ∠BAC = 180° − 100° = 80° Step 4: Therefore, the measure of angle BAC is 80°.

Verification / Alternative check:
A quick verification can be done by constructing an approximate acute triangle and measuring angles. If we choose a triangle with angle A = 80°, then the relation predicts ∠BOC = 180° − 80° = 100°, which matches the given value. This consistency confirms that our formula has been applied correctly and the computed angle is accurate.

Why Other Options Are Wrong:
Option 2: 50° does not satisfy the relation 180° − ∠A = 100°. If ∠A were 50°, ∠BOC would be 130°, not 100°. Option 3: 100° is simply the value of ∠BOC. Using it again as angle BAC ignores the known geometric relation between these angles. Option 4: 60° would give ∠BOC = 120°, which contradicts the given data. Option 5: None of these is incorrect because 80° is a valid option and satisfies the exact formula.

Common Pitfalls:
Students sometimes confuse the orthocentre relation with the circumcentre relation. For the circumcentre of a triangle, the angle at the centre opposite a side is 2 times the angle at the circumference. For the orthocentre, the relation involves 180° minus the vertex angle instead. Another common pitfall is to assume the triangle is right angled or obtuse without checking context. In a right triangle, the orthocentre lies at the right angled vertex, and different relations apply. Here the use of the standard formula for an acute triangle is appropriate and leads directly to the correct result.

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