Question

A circle is shown. Points Q, U, A, D are on the circle. Lines connect the points to form a quadrilateral. Angle Q U A is 111 degrees. Arc Q U is 88 degrees.
What is the measure of Arc A U?
44°
50°
64°
92°

Answer 1

The measure of Arc AU of the given Circle Geometry is; 50°

How to find the measure of arc angle?

The circle theorem we will apply to solve for the arc measure states that “Measure of angles subtended to any point on the circumference of the circle from the same arc is equal to half of the angle subtended at the center by the same arc.”

We can express the above theorem as;

Angle at the center = 2 × Angle at the circumference

From the attached image below  and from the given statement in the question, we are given that; ∠QUA = 111°

Therefore, applying the circle theorem earlier quoted, we can say that; m∠QDA = 2 × ∠QUA

m∠QDA = 2 × 111° = 222°

Which gives;

∠QOA =  = 360° - 222° = 138°

∠QOA = ∠QOU + ∠UOA (by angle addition property)

Thus;

∠QOA = 138° = 88° + ∠UOA

∠UOA = 138° - 88° = 50°

Thus, the measure of Arc AU is;

Arc AU = ∠UOA = 50°

Read more about arc angle at; brainly.com/question/27890907

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