In the diagram below, triangle ABC is inscribed in the circle and AC=AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD?
Recognize the key property of an inscribed angle: An inscribed angle is half the measure of the central angle that subtends the same arc.
Apply the property to angle BAC: Since angle BAC is inscribed in the circle and subtends arc BC, the measure of central angle BOC is twice the measure of angle BAC. Therefore, BOC = 2 * 42° = 84°.
Identify the tangent-radius relationship: A tangent to a circle from an external point creates a right angle with the radius drawn to the point of contact.
Apply the tangent-radius relationship to angle ACD: Angle DCE is a right angle because DE is tangent to the circle at C. Angle DCB is half of central angle BOC because D is the midpoint of arc BC.
Therefore, in triangle DCB, angles DCB and DCE are complementary, and angle ACD = 180° - (DCB + DCE) = 180° - (84° + 90°) = 56°.
Therefore, the measure of angle ACD is 56 degrees.