KHow wrote:

Could someone provide more insight about the rules related to this problem? I am having trouble correctly solving this one. Thank you!

reto wrote:

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If O is the center of the circle shown above, what is the measure of x?

A. 30°

B. 40°

C. 50°

D. 80°

E. 100°

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KHow , sure.

One central angle and one inscribed angle intercept

the same arc, in my diagram,

arc ABCentral angle measure yields arc measure

Arc measure yields inscribed angle measure \(= x\)

A central angle is formed by two radii,

OA and

OB with its vertex at center O

• degree measure of central angle ∠ AOB

Straight line = 180°

Two angles that lie on a straight line sum to 180°

100° + ∠AOB = 180°

∠ AOB = 80°1) ∠ AOB is "subtended" by arc AB, or ∠ AOB intercepts arc AB

2) degree measure of intercepted arc AB ?

The radii of sector AOB intercept the circumference of the circle and create

arc AB.We cam also say that ∠ AOB "subtends" arc AB

Rule: The length of an arc subtended by a central angle has the same degree measure as the central angle∠ AOB = 80°

Because ∠ AOB is a central angle,

arc AB also

= 80°3) from degree measure of arc AB, find inscribed ∠ ACB measure = \(x\)

An inscribed angle is formed by two intersecting chords,

AC and

BC, whose vertex lies on the circumference of the circle

4) An inscribed angle has a degree measure that is \(\frac{1}{2}\) that of the arc it intercepts

Rule: the degree measure of an inscribed angle is half that of the arc that the angle intercepts5) arc AB = 80°

6) x =?

∠ ACB = \(x = \frac{1}{2}\) of arc AB

\(x = \frac{1}{2} * 80°\)

\(x = 40°\)

Answer B

Hope that helps.

This site, here, and

this site, here discuss the relationship between and among inscribed angles, central angles, and intercepted arcs

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