In a circle with centre O, AB is a diameter and XY is a tangent at point C. If angle ACX = 35 degrees, what is the measure of angle CAB in degrees?

Introduction / Context:
This is a geometry question involving a circle, a diameter, and a tangent. It tests your understanding of the relationship between the angle formed by a tangent and a chord and the angles in the corresponding segment of the circle. It also uses the fact that an angle subtended by a diameter is a right angle.

Given Data / Assumptions:

• AB is a diameter of a circle with centre O.
• XY is a tangent at point C on the circle.
• Angle ACX = 35 degrees, where AC is a chord and CX is part of the tangent.
• Required: the angle CAB in degrees.

Concept / Approach:
Two key circle theorems are used. First, the angle between a tangent and a chord at the point of contact is equal to the angle in the opposite arc, that is, the angle made by that chord at a point on the circle on the opposite side. Second, an angle in a semicircle is a right angle, so any angle subtended by a diameter at the circumference is 90 degrees. Using these results, we can find the remaining angle of triangle ABC.

Step-by-Step Solution:
Step 1: Note that AC is a chord and CX is a tangent at C, so angle ACX is the angle between a chord and a tangent. Step 2: By the tangent chord theorem, angle ACX equals the angle in the opposite arc that is subtended by chord AC. Step 3: The chord AC subtends an angle at point B on the circle, so angle ABC = angle ACX = 35 degrees. Step 4: Since AB is a diameter, angle ACB is an angle in a semicircle and must be 90 degrees. Step 5: In triangle ABC, the sum of the interior angles is 180 degrees, so angle CAB + angle ABC + angle ACB = 180 degrees. Step 6: Substitute the known values: angle CAB + 35 + 90 = 180. Step 7: Simplify to get angle CAB = 180 - 125 = 55 degrees.

Verification / Alternative check:
You can verify the logic by sketching a circle with a horizontal diameter AB, a point C on the circle above AB, and a tangent at C. When AB is a diameter, triangle ABC is right angled at C. Using the known angle at B and the right angle at C, the third angle at A must be 55 degrees to keep the sum at 180 degrees. This geometric consistency confirms the answer.

Why Other Options Are Wrong:
45 degrees would reduce the total sum of angles in triangle ABC to 170 degrees, which is impossible. 35 degrees is simply the given angle ACX and belongs to a different location in the figure. 65 degrees makes the total 190 degrees, which is also impossible. 50 degrees similarly does not fit the required sum and does not follow from the tangent chord theorem.

Common Pitfalls:
A common mistake is to confuse the angle between the tangent and chord with the angle at the same point in the triangle instead of the angle at the opposite point on the circle. Another pitfall is forgetting that a diameter always subtends a right angle at the circumference. Careful reading of the geometry and drawing a neat diagram usually prevents such errors.

Final Answer:
The measure of angle CAB is 55 degrees.