Angles Inscribed within Circles?

[wizkidz]

Confused about what the rules are regarding angles as they relate to arcs/angles within the circle. For reference, Question #10 on DAT Math Destroyer 2016: Given triangle ABC inscribed within a circle whose diameter AC forms one of the sides of the triangle. If Arc BC on the circle subtends an angle of 40, find the measure of angle BCA within the triangle.

Can someone explain the general rules for me? Thank you!

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[CU2times]

Confused about what the rules are regarding angles as they relate to arcs/angles within the circle. For reference, Question #10 on DAT Math Destroyer 2016: Given triangle ABC inscribed within a circle whose diameter AC forms one of the sides of the triangle. If Arc BC on the circle subtends an angle of 40, find the measure of angle BCA within the triangle.

Can someone explain the general rules for me? Thank you!

1. For an angle inscribed WITHIN the circle, the angle is equal to half of the arc that it intercepts. 2. For an angle outside the circle, the angle is equal to 1/2 * the difference of the arcs it intercepts.
For this problem, you need rule #1 to deduce that angle CAB is 20 degrees. then you also have to realize that the triangle will make a 90 degree angle at angle ABC according to Thales' theorem (since one side is the diameter, this is the general rule). Since 90 + 20 + angle BCA = 180degrees , angle BCA = 70 degrees.

A little tricky to describe this without pictures, but Hope this helps.

 

[wizkidz]

Thank you! For the second rule, what do you mean by 1/2* the difference of the arc it intercepts? what difference are you referring to?

 

[CU2times]

Thank you! For the second rule, what do you mean by 1/2* the difference of the arc it intercepts? what difference are you referring to?

Sorry for the late reply but what I mean is : 1/2 multiplied by the difference of intercepted arcs. When an angle is inscribed outside the circle, it's legs will be tangent to the circle, and where they intersect creates two arcs. It's very difficult to explain this without a picture but maybe you can google and get a better visual? Sorry I can't be more helpful!