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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

 Feb 14, 2024
 #1
avatar+128732 
+2

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

 

cool cool cool

 Feb 14, 2024
 #3
avatar+25 
+1

thank you so much!

chippers44  Feb 14, 2024
 #2
avatar+128732 
+2

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

 

 

cool cool cool

 Feb 14, 2024

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