Homework Help! Inscribed Angles

[mathematricks]

I've been stumped on my homework for the past few hours. I'd appreciate the help.

We're suppose to find the angle measures of #1 - 12.
So far i've gotten 7x + 8x + 12x + 9x = 360. 36x = 360x. X = 10.
Also, I know that angle 4 = 6, and angle 5 = 7, I'm just having trouble knowing where to start off.

Thank you

 

[DrPhil]

I've been stumped on my homework for the past few hours. I'd appreciate the help.
View attachment 2859
We're suppose to find the angle measures of #1 - 12.
So far i've gotten 7x + 8x + 12x + 9x = 360. 36x = 360x. X = 10.
Also, I know that angle 4 = 6, and angle 5 = 7, I'm just having trouble knowing where to start off.

Thank you

Having solved for X=10°, you now know the measures of the four corresponding arcs. Those arc measures lead directly to finding angles !, 2, 3, and the unnumbered interior angle at G. Isn't there some theorem about secants and tangents that applies here?

Carry on!

 

[mathematricks]

Having solved for X=10°, you now know the measures of the four corresponding arcs. Those arc measures lead directly to finding angles !, 2, 3, and the unnumbered interior angle at G. Isn't there some theorem about secants and tangents that applies here?

Carry on!

Knowing x=10, I have calculated the arcs to be: m arc PG = 70, m arc GY = 80, m arc YS = 120, and m arc PS = 90.

I am still having trouble trying to find the numbered angles and knowing which angle to start with.

 

[DrPhil]

Knowing x=10, I have calculated the arcs to be: m arc PG = 70, m arc GY = 80, m arc YS = 120, and m arc PS = 90.

I am still having trouble trying to find the numbered angles and knowing which angle to start with.

For three of those arcs, there is a "inscribed angle" in the problem. For instance, angle #1 at point P intercepts the arc GY. Surely you have learned the theorem that relates inscribed angles to the intercepted arc length? Find angles #1, 2, and 3 that way - then angle #4 from sum of a triangle, then #6 = #4, etc.

OR use the theorem about arcs intercepted by chords that intersect inside the circle to find #4-7.

You also need the theorem about angles between chords (or tangents) that intersect outside the circle, to find the angles at F and at T