tangent-secant-theorem, chord-theorem, angle?

robmma82

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please see pictures. i applied the mentioned theorems but can't find angle PEQ=QRB=ABP. What am I missing?
 

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please see pictures. i applied the mentioned theorems but can't find angle PEQ=QRB=ABP. What am I missing?

The numbers you wrote in look good (at least those I could read easily).

What I did was to use the given angles to write equations relating the four arcs, which I called w, x, y, and z. You can write four equations (one for each given angle, and one saying they add up to 360 degrees), and then solve this system of equations.
 
you do not miss anything.

please see pictures. i applied the mentioned theorems but can't find angle PEQ=QRB=ABP. What am I missing?

The answer is obvious. In triangle ABR, you have 35+?+80+?=180. 2?=65. ?=32.5.
 
How could I be so blind...

Thanks to Dr. Peterson for the fast and elegant answer which I got noticed by my mail account just today... I used a similar way:
https://www.onlinemathe.de/forum/Tangenten-Sekanten-Satz-Sehnensatz-Winkel

And special thanks to yma16. How could I be so blind being so close. I even had a little question mark in my angle ABP (on my paper after I took the picture) but couldn't put 2 and 2 together... ****
 
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