Question

explore the properties of inscribed angles by following these steps. 1. move point c so the measure of the inscribed angle abc is 50°. what is the measure of arc ac? 100 ° 2. move point c so the measure of arc ac is 50°. what is the measure of ∠abc? ° (image: a circle with points a, b, c on it. m∠abc = 52°, m⌢ac = 104°)

Explanation:

Step1: Recall the inscribed angle theorem

The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. So, if \( \angle ABC \) is an inscribed angle intercepting arc \( AC \), then \( m\angle ABC=\frac{1}{2}m\widehat{AC} \), or equivalently, \( m\widehat{AC} = 2m\angle ABC \). For the reverse, to find the inscribed angle when the arc is known, we use \( m\angle ABC=\frac{1}{2}m\widehat{AC} \).

Step2: Solve for \( m\angle ABC \) when \( m\widehat{AC} = 50^\circ \)

Using the formula \( m\angle ABC=\frac{1}{2}m\widehat{AC} \), substitute \( m\widehat{AC}=50^\circ \) into the formula.

$$ m\angle ABC=\frac{1}{2}\times50^\circ = 25^\circ $$

Answer:

\( 25 \)