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# Circle

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4  I've tried to solve these but I don't understand it at all. Please show how to solve this because I'm not good at these types of questions and I need to understand this. Thank you.

Sep 9, 2019

#1
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First one.....

Note that central angle CHD  is right  = 90°

And the measure of  arc  CD  has the same measure  = 90°

Angle DHE  also = 90°  [ since it is supplemental to angle CHD]

So arc  DE has the same measure  = 90°

And arc  FE = 34°

So...arc  FD  [I'm asssuming arc FED ]  is the sum of these two arcs  =  124°

The measure of arc DCF  =  360  -  measure of arc FD   =  360 - 124  = 236°

The measure of arc GDF   = measure of arc GCD  +  measure of arc FED  =  180 + 124  = 304°   Sep 9, 2019
#2
+1

Second one  [ at the top right ]

Notice that angle QUR  is a vertical angle to angle TUS.....so its measure is also  = 19°

And the measure of angle  PUR = 90°

So the measure of  arc PQ  =  measure of angle  PUQ  =  measure of  angle PUR - measure of angle QUR  =

90  - 19   =   71°

Measure of arc SR   =   180  - 19   =  161°

Measure of arc QRT  =  measure of arcs SR + QR =  180 + 19   =  199°

Measure of arc  PSR  =  measure of arc PT  + measure of arc TSR = 90  + 180  = 270°

Measure of  arc PS   =   measure of arc PT  + measur of arc TS  = 90  + 19    =  109°   Sep 9, 2019
#3
+1

Third one

Note that arc JK + arc KL  =180

Therefore

(5x + 23)  + (17x - 41)  = 180      simplify

22x  - 18  = 180         add 18 to both sides

22x  = 198       divide both sides by 22

x  = 9

arc JK  =  5x + 23   =   5(9) + 23  =  45 + 23  =  78°

Since angle MVJ  = angle LVK,  arc MJ  =  arc KL  =  17x - 41  = 17(9) - 41  =  112°

arc LMK  =  arc LMJ + arc JK   =  180 + 78  = 258°   Sep 9, 2019
#4
+1

Last one

Note that the sum of arcs   YX, XW and WV  =   360  - arc  VY  =  360  - 90    = 270°

So  we have this

(10x + 9)  + ( 21x - 12) + ( 8x)  = 270        simplify

39x - 3  =  270      add 3 to both sides

39x  = 273        divide both sides by 39

x  =  7

arc WX  =  21x - 12   =  21(7)  - 12  =  148 - 12   =   136°

arc YW  =   hard to tell which  arc is being referred to, here  arc YVW  or arc YXW ???

arc YX  = 10x + 9  =   10(7) + 9  =  70 + 9  = 79°

arc VXW  =  360  - arc VW  =  360  - 8x  =  360 - 8(7)  =  360 - 56   =   204°   Sep 9, 2019