Determine the measure of angle BAC.

Given:

• A circle with center O.
• Points A, B, and C are on the circle.
• Angle ACB measures 36 degrees.

Analysis:

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

In this case, angle AOB is the angle subtended by arc AB at the center, and angle ACB is the angle subtended by arc AB at the circumference.

Therefore, the measure of angle AOB is twice the measure of angle ACB.

\[ \text{Angle AOB} = 2 \times \text{Angle ACB} \]

\[ \text{Angle AOB} = 2 \times 36^{\circ} = 72^{\circ} \]

Since OA and OB are radii of the circle, triangle AOB is an isosceles triangle.

In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, angle OAB = angle OBA.

The sum of angles in a triangle is 180 degrees.

\[ \text{Angle OAB} + \text{Angle OBA} + \text{Angle AOB} = 180^{\circ} \]

\[ \text{Angle OAB} + \text{Angle OBA} + 72^{\circ} = 180^{\circ} \]

\[ \text{Angle OAB} + \text{Angle OBA} = 180^{\circ} - 72^{\circ} = 108^{\circ} \]

Since angle OAB = angle OBA:

\[ 2 \times \text{Angle OAB} = 108^{\circ} \]

\[ \text{Angle OAB} = \frac{108^{\circ}}{2} = 54^{\circ} \]

Angle BAC is the same as angle OAB in this context.

Answer:

Angle BAC = 54°