2011 AMC 10B Problems/Problem 17
Problem 17
In the given circle, the diameter is parallel to
, and
is parallel to
. The angles
and
are in the ratio
. What is the degree measure of angle
?
Solution
![[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); pair[] ps={B,E,0}; dot(ps); label("$B$",B,NE); label("$E$",E,NW); label("$$",O,N); [/asy]](http://latex.artofproblemsolving.com/9/4/1/941cbc310056c64a2976b4088f6cfea36236f8a3.png)
We can let be
and
be
because they are in the ratio
. When an inscribed angle contains the diameter, the inscribed angle is a right angle. Therefore by triangle sum theorem,
and
.
because they are alternate interior angles and
. Opposite angles in a cyclic quadrilateral are supplementary, so
. Use substitution to get