Answer:
![m\angle 1=45^o](https://tex.z-dn.net/?f=m%5Cangle%201%3D45%5Eo)
![m\angle 2=75^o](https://tex.z-dn.net/?f=m%5Cangle%202%3D75%5Eo)
![m\angle 3=35^o](https://tex.z-dn.net/?f=m%5Cangle%203%3D35%5Eo)
![m\angle 5=75^o](https://tex.z-dn.net/?f=m%5Cangle%205%3D75%5Eo)
![m\angle 6=55^o](https://tex.z-dn.net/?f=m%5Cangle%206%3D55%5Eo)
![m\angle 7=50^o](https://tex.z-dn.net/?f=m%5Cangle%207%3D50%5Eo)
![m\angle 8=25^o](https://tex.z-dn.net/?f=m%5Cangle%208%3D25%5Eo)
![m\angle 9=35^o](https://tex.z-dn.net/?f=m%5Cangle%209%3D35%5Eo)
![m\angle 10=70^o](https://tex.z-dn.net/?f=m%5Cangle%2010%3D70%5Eo)
![m\angle 11=50^o](https://tex.z-dn.net/?f=m%5Cangle%2011%3D50%5Eo)
![m\angle 12=25^o](https://tex.z-dn.net/?f=m%5Cangle%2012%3D25%5Eo)
![m\angle 13=70^o](https://tex.z-dn.net/?f=m%5Cangle%2013%3D70%5Eo)
![m\angle 14=50^o](https://tex.z-dn.net/?f=m%5Cangle%2014%3D50%5Eo)
![m\angle 15=60^o](https://tex.z-dn.net/?f=m%5Cangle%2015%3D60%5Eo)
![m\angle 16=85^o](https://tex.z-dn.net/?f=m%5Cangle%2016%3D85%5Eo)
![m\angle 17=95^o](https://tex.z-dn.net/?f=m%5Cangle%2017%3D95%5Eo)
![m\angle 18=85^o](https://tex.z-dn.net/?f=m%5Cangle%2018%3D85%5Eo)
![m\angle 19=95^o](https://tex.z-dn.net/?f=m%5Cangle%2019%3D95%5Eo)
Step-by-step explanation:
step 1
Find the measure of arc FG
we know that
---> by complete circle
substitute the given values
![50^o+100^o+140^o+arc\ FG=360^o](https://tex.z-dn.net/?f=50%5Eo%2B100%5Eo%2B140%5Eo%2Barc%5C%20FG%3D360%5Eo)
![arc\ FG=360^o-290^o=70^o](https://tex.z-dn.net/?f=arc%5C%20FG%3D360%5Eo-290%5Eo%3D70%5Eo)
step 2
Find the measure of angle 9
we know that
The inscribed angle is half that of the arc comprising
so
![m\angle 9=\frac{1}{2}[arc\ GF]](https://tex.z-dn.net/?f=m%5Cangle%209%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20GF%5D)
we have
![arc\ FG=70^o](https://tex.z-dn.net/?f=arc%5C%20FG%3D70%5Eo)
substitute
![m\angle 9=\frac{1}{2}[70^o]=35^o](https://tex.z-dn.net/?f=m%5Cangle%209%3D%5Cfrac%7B1%7D%7B2%7D%5B70%5Eo%5D%3D35%5Eo)
step 3
Find the measure of angle 11
we know that
The inscribed angle is half that of the arc comprising
so
![m\angle 11=\frac{1}{2}[arc\ BD]](https://tex.z-dn.net/?f=m%5Cangle%2011%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20BD%5D)
we have
![arc\ BD=100^o](https://tex.z-dn.net/?f=arc%5C%20BD%3D100%5Eo)
![m\angle 11=\frac{1}{2}[100^o]=50^o](https://tex.z-dn.net/?f=m%5Cangle%2011%3D%5Cfrac%7B1%7D%7B2%7D%5B100%5Eo%5D%3D50%5Eo)
step 4
Find the measure of angle 12
we know that
The inscribed angle is half that of the arc comprising
so
![m\angle 12=\frac{1}{2}[arc\ BG]](https://tex.z-dn.net/?f=m%5Cangle%2012%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20BG%5D)
we have
![arc\ BG=50^o](https://tex.z-dn.net/?f=arc%5C%20BG%3D50%5Eo)
![m\angle 12=\frac{1}{2}[50^o]=25^o](https://tex.z-dn.net/?f=m%5Cangle%2012%3D%5Cfrac%7B1%7D%7B2%7D%5B50%5Eo%5D%3D25%5Eo)
step 5
Find the measure of angle 13
![m\angle 13=\frac{1}{2}[arc\ DF]](https://tex.z-dn.net/?f=m%5Cangle%2013%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20DF%5D)
we have
![arc\ DF=140^o](https://tex.z-dn.net/?f=arc%5C%20DF%3D140%5Eo)
![m\angle 13=\frac{1}{2}[140^o]=70^o](https://tex.z-dn.net/?f=m%5Cangle%2013%3D%5Cfrac%7B1%7D%7B2%7D%5B140%5Eo%5D%3D70%5Eo)
step 6
Find the measure of angle 4
we know that
---> the inscribed angle has the same arc comprising DF
therefore
![m\angle 4=70^o](https://tex.z-dn.net/?f=m%5Cangle%204%3D70%5Eo)
step 7
Find the measure of angle 14
we know that
---> the inscribed angle has the same arc comprising BD
therefore
![m\angle 14=50^o](https://tex.z-dn.net/?f=m%5Cangle%2014%3D50%5Eo)
step 8
Find the measure of angle 15
we know that
---> by form a straight line
substitute the given values
![70^o+50^o+m\angle 15=180^o](https://tex.z-dn.net/?f=70%5Eo%2B50%5Eo%2Bm%5Cangle%2015%3D180%5Eo)
![m\angle 15=180^o-120^o=60^o](https://tex.z-dn.net/?f=m%5Cangle%2015%3D180%5Eo-120%5Eo%3D60%5Eo)
step 9
Find the measure of angle 16
Remember that the sum of the interior angles in any triangle must be equal to 180 degrees
so
![m\angle 12+m\angle 13+m\angle 16=180^o](https://tex.z-dn.net/?f=m%5Cangle%2012%2Bm%5Cangle%2013%2Bm%5Cangle%2016%3D180%5Eo)
substitute given values
![25^o+70^o+m\angle 16=180^o](https://tex.z-dn.net/?f=25%5Eo%2B70%5Eo%2Bm%5Cangle%2016%3D180%5Eo)
![m\angle 16=180^o-95^o=85^o](https://tex.z-dn.net/?f=m%5Cangle%2016%3D180%5Eo-95%5Eo%3D85%5Eo)
step 10
Find the measure of angle 17
we know that
---> by form a straight line
substitute the given value
![85^o+m\angle 17=180^o](https://tex.z-dn.net/?f=85%5Eo%2Bm%5Cangle%2017%3D180%5Eo)
![m\angle 17=180^o-85^o=95^o](https://tex.z-dn.net/?f=m%5Cangle%2017%3D180%5Eo-85%5Eo%3D95%5Eo)
step 11
Find the measure of angle 18
we know that
---> by vertical angles
therefore
![m\angle 18=85^o](https://tex.z-dn.net/?f=m%5Cangle%2018%3D85%5Eo)
step 12
Find the measure of angle 19
we know that
---> by vertical angles
therefore
![m\angle 19=95^o](https://tex.z-dn.net/?f=m%5Cangle%2019%3D95%5Eo)
step 13
Find the measure of angle 1
we know that
The measurement of the external angle is the half-difference of the arches that comprise
![m\angle 1=\frac{1}{2}[arc\ DF-arc\ GB]](https://tex.z-dn.net/?f=m%5Cangle%201%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20DF-arc%5C%20GB%5D)
substitute
![m\angle 1=\frac{1}{2}[140^o-50^o]=45^o](https://tex.z-dn.net/?f=m%5Cangle%201%3D%5Cfrac%7B1%7D%7B2%7D%5B140%5Eo-50%5Eo%5D%3D45%5Eo)
step 14
Find the measure of angle 3
we know that
---> the inscribed angle has the same arc comprising BD
therefore
![m\angle 3=35^o](https://tex.z-dn.net/?f=m%5Cangle%203%3D35%5Eo)
step 15
Find the measure of angle 2
we know that
---> by form a straight line
substitute the given values
![m\angle 2+35^o+70^o=180^o](https://tex.z-dn.net/?f=m%5Cangle%202%2B35%5Eo%2B70%5Eo%3D180%5Eo)
![m\angle 2=180^o-105^o=75^o](https://tex.z-dn.net/?f=m%5Cangle%202%3D180%5Eo-105%5Eo%3D75%5Eo)
step 16
Find the measure of angle 5
we know that
---> by vertical angles
therefore
![m\angle 5=75^o](https://tex.z-dn.net/?f=m%5Cangle%205%3D75%5Eo)
step 17
Find the measure of angle 6
we know that
The measurement of the external angle is the half-difference of the arches that comprise
![m\angle 6=\frac{1}{2}[arc\ DFG-arc\ BD]](https://tex.z-dn.net/?f=m%5Cangle%206%3D%5Cfrac%7B1%7D%7B2%7D%5Barc%5C%20DFG-arc%5C%20BD%5D)
substitute
![m\angle 6=\frac{1}{2}[210^o-100^o]=55^o](https://tex.z-dn.net/?f=m%5Cangle%206%3D%5Cfrac%7B1%7D%7B2%7D%5B210%5Eo-100%5Eo%5D%3D55%5Eo)
step 18
Find the measure of angle 8
we know that
---> the inscribed angle has the same arc comprising GB
therefore
![m\angle 8=25^o](https://tex.z-dn.net/?f=m%5Cangle%208%3D25%5Eo)
step 19
Find the measure of angle 7
Remember that the sum of the interior angles in any triangle must be equal to 180 degrees
so
![m\angle 5+m\angle 6+m\angle 7=180^o](https://tex.z-dn.net/?f=m%5Cangle%205%2Bm%5Cangle%206%2Bm%5Cangle%207%3D180%5Eo)
substitute
![75^o+55^o+m\angle 7=180^o](https://tex.z-dn.net/?f=75%5Eo%2B55%5Eo%2Bm%5Cangle%207%3D180%5Eo)
![m\angle 7=180^o-130^o=50^o](https://tex.z-dn.net/?f=m%5Cangle%207%3D180%5Eo-130%5Eo%3D50%5Eo)
step 20
Find the measure of angle 10
we know that
---> by form a straight line
substitute
![50^o+25^o+35^o+m\angle 10=180^o](https://tex.z-dn.net/?f=50%5Eo%2B25%5Eo%2B35%5Eo%2Bm%5Cangle%2010%3D180%5Eo)
![m\angle 10=180^o-110^o=70^o](https://tex.z-dn.net/?f=m%5Cangle%2010%3D180%5Eo-110%5Eo%3D70%5Eo)