Answer:
m∠AOC = 96°
Explanation:
Line DE is tangent to circle O at the point of tangency C.
Since the tangent of a circle is always perpendicular to the radius, and OC is the radius of circle O, m∠OCE = 90°. Therefore:
m∠OCA + m∠ACE = 90°
m∠OCA + 48° = 90°
m∠OCA = 90° - 48°
m∠OCA = 42°
OA and OC are the radii of circle O. Therefore, triangle AOC is an isosceles triangle with apex ∠AOC. This means that its base angles ∠OCA and ∠CAO are congruent, so m∠OCA = m∠CAO = 42°.
As the interior angles of a triangle sum to 180°, to find the measure of ∠AOC, we can subtract the measures of ∠OCA and ∠CAO from 180°:
m∠AOC = 180° - m∠OCA - m∠CAO
m∠AOC = 180° - 42° - 42°
m∠AOC = 138° - 42°
m∠AOC = 96°
Therefore, the measure of ∠AOC is 96°.