1)Given angle APS = 77 degrees and arc AP = 123 degrees, find the degree measure of arc PS.

If angle APS = 77°, then arc AS is twice this = 154°

So...arc PS = (360 - 123 - 154) = 83°

2)In the figure below, EF is a diameter of the circle. Arc AE = 35 degrees and arc CF = 80 degrees. What is the measure of angle ABC, in degrees?

Angle AFE = (1/2)(35°) = 17.5°

Angle CEF = (1/2)(80°) = 40°

So ....angle EBF = angle ABC = (180 - 17.5 - 40) = 122.5°

3)In the diagram, angle A = 30 degrees, arc DE = 170 degrees, and arc BC = 110 degrees. Find the measure of arc CE, in degrees.

The sum of arcs CE + DB = (360 - 110 - 170) = 80”

And angle A = (1/2) (arc CE - arc DB).....so....

30 = (1/2)(arc CE - arc DB)

60 = CE - DB

So

CE + DB = 80

CE - DB = 60 add these

2CE = 140 divide both sides by 2

arc CE = 70°

4)In circle O, AP is a diameter of length 17 and PS is chord of length 8. What is the length of chord AS?

AS will form a leg of a right triangle with AP the hypotenuse and PS the other leg....so

AS = √ [ 17^2 - 8^2 ] = √ [ 289 - 64 ] = √ [ 225] = 15 units

5)Grogg draws the following figure. It so happens that angle SAP is 174 degrees less than 4 times angle SOP. Find the degree measure of angle SAP.

SAP + 174 = 4*SOP

But SAP and SOP intercept the same arc PS...so they are equal

So...substituting

SAP + 174 = 4*SAP subtract SAP from both sides

174 = 3*SAP dvide both sides by 3

58° = SAP

6)In the diagram, minor arc AB : minor arc BC : major arc CA = 1 : 3 : 5. What is angle ABC in degrees?

There are 1 + 3 + 5 = 9 equal arcs in the circle....and angle ABC spans 5/9 of these

So major arc CA = (5/9) (360) = (360/9) * 5 = 40 * 5 = 200°

And since angle ABC is an inscribed angle intercepting this arc, it is 1/2 of this = 100°

7)In the figure, if MR = MK, the measure of arc MK is 130 degrees, and measure of arc MQ is 28 degrees, then what is angle RPK, in degrees?

Since MR = MK, then arc MR = arc MK,,,,and the sum of these arcs is 260”

So minor arc RK = 360 - 260 = 100°

So...since angle RMK intercepts this arc it has 1/2 of its measure = 50°

And since arc MQ is 28, then angle QKM = 14

So in triangle PKM angle KPM = 180 - 14 - 50 = 116°.....and angle RPK is supplemental to this = 180 - 116 = 64°