Respuesta :

calculista

Answer:

Part 1) [tex]m\angle POQ=arc\ PQ[/tex]  see the procedure

Part 2)  [tex]m\angle PNQ=\frac{1}{2}[arc\ PQ][/tex]

Part 3) The angles are not congruent. The measure of angle PNQ is half the measure of angle POQ

Step-by-step explanation:

Part 1) Describe the relationship between angle POQ and arc PQ

we know that

Central angle is the angle that has its vertex in the center of the circumference and the sides are radii of it

we have that

[tex]m\angle POQ=arc\ PQ[/tex] ----> by central angle

Part 2) Describe the relationship between angle PNQ and arc PQ

we know that

The inscribed angle is half that of the arc it comprises.

In this problem, the angle PNQ is an inscribed angle

so

[tex]m\angle PNQ=\frac{1}{2}[arc\ PQ][/tex]

Part 3) Are the angle POQ and PNQ congruent?

The angles are not congruent, because

[tex]m\angle POQ=arc\ PQ[/tex]

[tex]m\angle PNQ=\frac{1}{2}[arc\ PQ][/tex]

substitute

[tex]m\angle PNQ=\frac{1}{2}[m\angle POQ][/tex]

therefore

The measure of angle PNQ is half the measure of angle POQ

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