The value of x is 16 degrees and the measure of the arc DFG for circles of homework 2 are 1 6.056 times radius of circle.
What is the property of a straight angle?
The property of straight angle says that, the angle of a straight line is equal to the 180 degrees.
It can also be define as the sum of all the angle made on the straight line is equal to the 180 degrees.
For the first problem, the line GE, which is the diameter of the circle has two angles over it (x-3) degree and (12x-25) degrees.
For the diameter GE the sum of these two angle will be equal to 180 degrees. Thus,
[tex](x-3)+(12x-25)=180\\x-3+12x-25=180\\13x-28=180\\13x=180+28\\x=\dfrac{208}{13}\\x=16^o[/tex]
The value of mDE will be equal to (12x-25) as it is verticle angle of mGF. Thus,
[tex]arcDE=(12x-25)\\arcDE=(12(16)-25)\\arcDE=167^o[/tex]
Similarly the value of mEF
[tex]arcEF=(x-3)\\arcEF=(16-3)\\arcEF=13^o[/tex]
Let the radius of the circle is R. Then the measure of arc DFG is equal to the sum of areDEF and arc GF. Thus,
[tex]arcDFG=\dfrac{2\pi R}{2}+\dfrac{2 \pi R \angle GHF}{360}\\arcDFG=\dfrac{2\pi R}{2}+\dfrac{2 \pi R (167^o)}{360}\\arcDFG=6.056R[/tex]
For the second problem, the angle MN and QO are equal due to the vertical angle property. Therefore,
[tex](10x-45)=(6x-1)\\10x-6x=-1+45\\4x=44\\x=11[/tex]
The value of mMN is,
[tex]MN=(10x-45)\\MN=(10(11)-45)\\MN=65^o[/tex]
The value of NP using the property of straight line for line segment MP can be given as,
[tex]NP=180-MN\\NP=180-65\\NP=115^o[/tex]
Thus, the value of x is 16 degrees and the measure of the arc DFG for circles of homework 2 are 1 6.056 times radius of circle.
Learn more about the a straight angle here;
https://brainly.com/question/594583
