+25 1. In the figure below, line EF is the diameter of the circle. Arc AE = 35 and arc CF = 80. What is the measure of angle ABC, in degrees?
https://imgur.com/Fbgse5r
2. In the diagram, angle A = 30, arc DE = 170, and arc BC = 110. Find the measure of arc CE in degrees.
https://imgur.com/StUBt98
3. Angle SAP is 174 less than 4 times angle SOP. Find the degree measure of angle SAP.
https://imgur.com/Jfe3lFI
4. In the figure, if MR = MK, the measure of arc MK is 130, and the measure of arc MQ is 28, then what is angle RPK in degrees?
https://imgur.com/jnlDa4m
+129852 1. Angle ABE = Angle FBC = (35 + 80) / 2 = 57.5
Angle ABC = 180 - 57.5 = 122.5
2. Arc EC + DB = 360 - 170 - 110 = 80
So EC + DB = 80 (1)
Then
30 = (1/2)(EC -DB)
60 = EC - DB (2)
Using (1) , (2)
EC + DB = 80
EC - DB = 60 add these
2EC = 140
EC = 70
+129852 3. SOP and SAP are insribed angles intercepting the same arc....so, they are equal
So
4 SOP - 174 = SOP
3SOP = 174
SOP = 174 / 3 = 58 = SAP
4. Since MR = MK, then arc MR = 130
Since arc MQ = 28
So arc RQ = arc MR - arc MQ = 130 - 28 = 102
Then angle QKM = 14
Angle KPM = (1/2) (arc RQ + KM) = (1/2) ( 102 + 130) = 232 / 2 = 116
Then angle RPK = 180 - angle KPM = 180 - 116 = 64
