Problem: In the adjoining figure, CD is the diameter of a semi-circle with center O. Point A lies on the extension of DC past C; point E lies on the semi-circle, and B is the point of intersection (distinct from E ) of line segment AI : with the semi-circle. If length AB equals length OD, and the measure of ∠EOD is 45∘, then the measure of ∠BAO is
Answer Choices:
A. 10∘
B. 15∘
C. 20∘
D. 25∘
E. 30∘
Solution:
Draw line segment BO, and let x and y denote the ineasures of ∠EOD and ∠BAO, respectively. Observe that AB=OD=OE=OA, and apply the theorem on exlerior angles of triangles to △ABO and △AEO to obtain ∠EBO=∠BEO=2y and
x=3y.
Thus 45∘=3y
15∘=y
OR
Since the measure of an angle formed by two secants is half the difference of the intercepled arcs.
y=21​(x−y),y=3x​=345∘​=15∘.
