Ray OC bisects the angle AOB, Ray OD bisects angle AOC, ray OE bisects angle AOD , Ray OF bisects angle AOE, adn Ray OG bisects angel FOC
a. if measure of angle BOF=120 then the measure of angle DOE=?
b. if the measure of angle COG=35 then the measure of angle EOG=?
Answers:
It helps to draw a picture of what they are telling you...
Draw angle AOB. and let "x" represent that angle of <AOB
(sorry... but I don't know how to do symbols either... so the "less than" symbol is to mean "angle" AOB)
Okay... Now bisect that angle... From vertex O, draw a ray and put a C at the end of the line you drew from O...
That means that < AOC and < COB = 1/2 < AOB = (1/2) x
Now if you split that in half again... that means that < AOD and < DOC = 1/2 < AOC = 1/4 <AOB = (1/4) x
Now if you split that in half again... that means that < AOE and < EOC = 1/2 < AOD = 1/4 < AOC = 1/8 < AOB = (1/8) x
Now if you split that in half again... that means that < AOF and < FOC = 1/2 < AOE = 1/4 < AOD = 1/8 < AOC = 1/16 < AOB = (1/16) x
Now that you have found all that... let's look at the first question...
A) If measure of < BOF=120, then the measure of < DOE=?
Okay... we just said that "x" is going to represent < AOB. and we found out that <AOF = (1/16) x , right?
What do we also know?
We know that < AOF + < FOB = < AOB . which then tells us that < AOB - < AOF = < FOB
Soo. x - (1/16) x = < FOB
That means that < FOB = (15/16) x
If < BOF = 120 degrees, what is x?
(15/16) x = 120 degrees
x = 120 (16/15) = 128 degrees . so < AOB = 128 degrees
... with me?
That means that < AOC = 1/2 < AOB = 1/2 (128 deg) = 64 deg
We also know that <AOF = < AOB - < FOB = 128 deg - 120 = 8 degrees
We also know that <AOF = 1/2 < AOE . so because <AOF = 8 degrees... that means that < AOE = 16 degrees
We know that < AOE = < DOE because the problem tells us that ray OE bisects < AOD.
< AOE = 16 degrees. so that means that < DOE = 16 degrees
< DOE = 16 degrees ** ANSWER **
____________________________
Now ... let's look at the second question...
B) If the measure of < COG=35, then the measure of < EOG=?
Okay... scratch the angles that we found in Part A) (you know, the bit about how we found that < AOB = 128 degrees from know that < FOB = 120 degrees), but "remember" all the relationships that we found before we started answering Part A)
Remember, we said that "x" is going to represent < AOB. and we know that < AOC = 1/2 < AOB = (1/2) x
We also found out that <AOF = (1/16) x , right?
What do we also know?
We know that < FOG = < GOC because the problem tells us that ray OG bisects < FOC.
We know that < AOF + < FOG + < GOC = < AOC .
When we substitute the relationships that we know in terms of "x"... and < COG = 35 degrees... which also means that < FOG = 35 degrees... we get...
(1/16) x + 35 degs + 35 degs = (1/2) x
** Now solve for "x".**
(1/16) x + 70 degs = (1/2) x
(1/2) x - (1/16) x = 70 degs
(8/16) x - (1/16) x = 70 degs
(7/16) x = 70 degs
x = (70 degs)(16/7) = 160 degrees
So because x = 160 degs... we know now that < AOB = 160 degs
Now we have to set up another equation so that we can find < EOG. so let's write one which shows what angles PLUS < EOG will make up < AOC. what do you see on your drawing?
You should see this...
We know that < EOG = < FOG - < FOE .
When we substitute the relationships that we know in terms of "x"... and we know that < FOG = < COG = 35 degrees... we get...
< EOG = 35 degs - (1/16) x
Remember we found that x = 160 degs in this Part B) of the question...?
So, since x = 160 degs... just plug it in for "x"... like this...
< EOG = 35 degs - (1/16)(160 degs) = 35 degs - 10 deg = 25 degs.
< EOG = 25 degrees ** ANSWER **
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