Circle centered at O has two radii, OA and OB, and two chords, AD and BD. Secants BC and AC are drawn that intersects each other outside the circle at C. Central angle BOA measures 250 degrees.

Question

What is the measure of angle BDA?
What is the measure of angle BCA?

Answer 1

1. The measure of angle BDA can be found using the Inscribed Angle Theorem, which states that an inscribed angle is half the measure of the central angle that intercepts the same arc. Since central angle BOA measures 250 degrees, angle BDA will be half of that, so angle BDA = 250/2 = 125 degrees.

2. The measure of angle BCA can be found using the Exterior Angle Theorem. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. In this case, angle BCA is an exterior angle of triangle ABC, so BCA = angle B + angle A. From the given information, we know that central angle BOA = 250 degrees, so angle B = 250/2 = 125 degrees. Since angles BDA and BAC are vertical angles, angle A = 125 degrees as well. Therefore, angle BCA = 125 + 125 = 250 degrees.