Question

11
type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s).

points (a) and (b) are the endpoints of an arc of a circle. chords are drawn from the two endpoints to a third point, (c), on the circle.

given (\text{m} \widehat{ab} = 64^{\circ}) and (\angle abc = 73^{\circ}), (\text{m}\angle acb =) 32 ({}^{\circ}) and (\text{m} \widehat{ac} =) 146 ({}^{\circ}).

Explanation:

🆕 New Concept Discovered: Inscribed Angles in Circles
An angle whose vertex is on a circle and whose sides contain chords of the circle is half the measure of its intercepted arc.

Step 1: Find the measure of angle ACB

The angle \(\angle ACB\) is an inscribed angle that intercepts the arc \(\widehat{AB}\).

The measure of an inscribed angle is half the measure of its intercepted arc:

$$ m\angle ACB = \frac{1}{2} \cdot m\widehat{AB} $$

Given that \(m\widehat{AB} = 64^\circ\):

$$ m\angle ACB = \frac{1}{2} \cdot 64^\circ = 32^\circ $$

Step 2: Find the measure of arc AC

The angle \(\angle ABC\) is an inscribed angle that intercepts the arc \(\widehat{AC}\).

Using the inscribed angle relationship:

$$ m\angle ABC = \frac{1}{2} \cdot m\widehat{AC} $$

Multiply both sides by 2 to solve for the arc measure:

$$ m\widehat{AC} = 2 \cdot m\angle ABC $$

Given that \(m\angle ABC = 73^\circ\):

$$ m\widehat{AC} = 2 \cdot 73^\circ = 146^\circ $$

Answer:

• \(m\angle ACB =\) 32
• \(m\widehat{AC} =\) 146