Answer:
Part 11) [tex]m\angle VSX=75^o[/tex]
Part 12) [tex]m\angle DBE=32^o[/tex]
Part 13) [tex]x=42^o[/tex]
Part 14) [tex]m\angle XYZ=40^o[/tex]
Step-by-step explanation:
Part 11)
step 1
Find the measure of arc WX
we know that
[tex]arc\ XT+arc\ TU+arc\ UV+arc\ VW+arc\ WX=360^o[/tex] ---> by complete circle
substitute the given values
[tex]80^o+80^o+125^o+35^o+arc\ WX=360^o[/tex]
[tex]arc\ WX=360^o-320^o=40^o[/tex]
step 2
Find the measure of angle VSX
we know that
[tex]m\angle VSX=arc\ WX+arc\ WV[/tex] ----> by central angle
Remember that
Central angle is the angle that has its vertex in the center of the circumference and the sides are radii of it
substitute the given values
[tex]m\angle VSX=40^o+35^o=75^o[/tex]
Part 12)
step 1
Find the value of x
we know that
[tex]arc\ GC+arc\ CD+arc\ DE+arc\ EF+arc\ FG=360^o[/tex] ---> by complete circle
[tex]arc\ FG=75^o[/tex] ----> by central angle
substitute the given values
[tex]133^o+80^o+(5x-8)^o+(4x+8)^o+75^o=360^o[/tex]
solve for x
[tex]9x=360^o-288^o\\9x=72\\x=8[/tex]
step 2
we know that
[tex]m\angle DBE=arc\ DE[/tex] ----> by central angle
so
[tex]m\angle DBE=(5x-8)^o=5(8)-8=32^o[/tex]
Part 13)
step 1
Find the measure of arc LM
we know that
The inscribed angle is half that of the arc comprising
so
[tex]48^o=\frac{1}{2}[arc\ LM][/tex]
[tex]arc\ LM=2(48^o)=96^o[/tex]
step 2
Find the measure of arc LK
we know that
[tex]arc\ LK+arc\ LM=180^o[/tex] ---> because segment KM is a diameter (the diameter divide the circle into two equal parts)
substitute the given value
[tex]arc\ LK+96^o=180^o[/tex]
[tex]arc\ LK=180^o-96^o=84^o[/tex]
step 3
Find the measure of angle x
we know that
The inscribed angle is half that of the arc comprising
so
[tex]x^o=\frac{1}{2}[arc\ LK][/tex]
substitute the given value
[tex]x=\frac{1}{2}[84^o]=42^o[/tex]
Part 14)
step 1
Find the value of x
we know that
[tex](17x+10)^o+134^o+(10x)^o=360^o[/tex] ----> by complete circle
solve for x
[tex]27x+144^o=360^o\\27x=216\\x=8[/tex]
step 2
Find the measure of arc XZ
[tex]arc\ XZ=10x^o=10(8)=80^o[/tex]
step 3
Find the measure of angle XYZ
we know that
The inscribed angle is half that of the arc comprising
so
[tex]m\angle XYZ=\frac{1}{2}[arc\ XZ][/tex]
substitute the given value
[tex]m\angle XYZ=\frac{1}{2}[80^o]=40^o[/tex]
