Question

explore the properties of inscribed angles by following these steps.

1. move point c so the measure of arc ac is 50°. what is the measure of ∠abc?
2. make a conjecture. which measures will change if you move vertex b of the inscribed angle? angle abc arc ac both neither check

m∠abc = 52° mac = 104°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. That is, if $\angle ABC$ is an inscribed angle and $\overset{\frown}{AC}$ is the intercepted arc, then $m\angle ABC=\frac{1}{2}m\overset{\frown}{AC}$.

Step2: Analyze the effect of moving vertex B

The measure of the intercepted arc $\overset{\frown}{AC}$ is determined by the positions of points A and C on the circle. When we move vertex B of the inscribed angle $\angle ABC$, the measure of the arc $\overset{\frown}{AC}$ remains the same (since A and C are fixed in terms of the arc - defining positions). But the measure of $\angle ABC$ is related to the arc $\overset{\frown}{AC}$ by the formula $m\angle ABC = \frac{1}{2}m\overset{\frown}{AC}$. So, if we move vertex B, the measure of $\angle ABC$ will change.

Answer:

angle ABC