Question

Line segment XY is tangent to circle Z at point U.

If the measure of UV is 84º, what is the measure of ZYUV?

Answer 1

42 degrees

Explanation:

Just took the quiz

Answer 2

`m\angle YUV=42\°`

Explanation:

The picture of the question in the attached figure

we have that

`arc\ UV=84\°` -----> given problem

so

`m\angle UZV=84\°`------> by central angle

we know that

The triangle UZV is an isosceles triangle

because

ZU=ZV=radius

so

`m\angle ZUV=m\angle ZVU` -----> bases angle of the isosceles triangle

Remember that

The sum of the internal angles of any triangle must be equal to 180 degrees

so

`m\angle ZUV+m\angle ZVU+m\angle UZV=180\°\\2m\angle ZUV+m\angle UZV=180\°`

substitute and solve for m<ZUV

`2m\angle ZUV+84\°=180\°\\m\angle ZUV=(180\°-84\°)/2=48\°`

Find the measure of angle YUV

we have that

`m\angle ZUV+m\angle YUV=90\°` ------> by complementary angles

substitute the given value and solve for m<YUV

`48\°+m\angle YUV=90\°\\m\angle YUV=90\°-48\°=42\°`