a Let's label the vertices of the segment's endpoints that we see inside the circle.
Notice that BC and DC are both radii of the circle. Therefore these segments are congruent, which means △ BCD must be an isosceles triangle. According to the Base Angles Theorem we know that ∠B ≅ ∠D, which means m∠B= x.
Notice that the central angle ∠BCD has the same measure as its intercepted arc BD. Let's add that information to the diagram.
Since the two sides that are cut by transversals are parallel, we know by the Consecutive Interior Angles Theorem that they are supplementary. With this information we can write two equations.
x+67^(∘)&=180^(∘)
5y+(3y-16^(∘))&=180^(∘)
Let's solve them one at a time.
To find the length of y we must know the side's opposite angle. This will allow us to use the Law of Sines again. Since we know two angles in the triangle, we can use the Triangle Angle Sum Theorem to determine the remaining angle.
