Answer:
[tex]m\angle YUV=42\°[/tex]
Step-by-step explanation:
The picture of the question in the attached figure
we have that
[tex]arc\ UV=84\°[/tex] -----> given problem
so
[tex]m\angle UZV=84\°[/tex]------> by central angle
we know that
The triangle UZV is an isosceles triangle
because
ZU=ZV=radius
so
[tex]m\angle ZUV=m\angle ZVU[/tex] -----> bases angle of the isosceles triangle
Remember that
The sum of the internal angles of any triangle must be equal to 180 degrees
so
[tex]m\angle ZUV+m\angle ZVU+m\angle UZV=180\°\\2m\angle ZUV+m\angle UZV=180\°[/tex]
substitute and solve for m<ZUV
[tex]2m\angle ZUV+84\°=180\°\\m\angle ZUV=(180\°-84\°)/2=48\°[/tex]
Find the measure of angle YUV
we have that
[tex]m\angle ZUV+m\angle YUV=90\°[/tex] ------> by complementary angles
substitute the given value and solve for m<YUV
[tex]48\°+m\angle YUV=90\°\\m\angle YUV=90\°-48\°=42\°[/tex]
