Question

in circle k, mab = 58°, mbc = 94°, and mcd = 80°. explain why 232° cannot be a possible measure of da. select all the statements that support mda ≠ 232°. show hints the non - shared endpoints of an inscribed angle and all the points on the circle between them form an intercepted arc. minor arcs are named by their two endpoints, and their measure is equal to the measure of the central angle. the major arc dca measures 232°. the measure of an inscribed angle is equal to half the measure of its intercepted arc. major arcs are named by their two endpoints and a point on the arc, and their measure is equal to 360° minus the measure of the central angle. the only possible measure for the minor arc da is 128°.

Explanation:

Step1: Recall arc - measure relationships

The sum of the measures of the major and minor arcs of a circle is 360°. Given \(m\overparen{AB}=58^{\circ}\), \(m\overparen{BC}=94^{\circ}\), and \(m\overparen{CD}=80^{\circ}\).

Step2: Calculate the sum of the known arcs

\(m\overparen{AB}+m\overparen{BC}+m\overparen{CD}=58^{\circ}+94^{\circ}+80^{\circ}=232^{\circ}\)

Step3: Calculate the measure of \(\overparen{DA}\)

Let \(m\overparen{DA}=x\). Then, since the sum of the measures of the arcs of a circle is 360°, we have \(x + 232^{\circ}=360^{\circ}\), so \(x=m\overparen{DA}=360^{\circ}- 232^{\circ}=128^{\circ}\). A major arc has a measure greater than 180° and a minor arc has a measure less than 180°. Since \(m\overparen{DA}=128^{\circ}<180^{\circ}\), \(\overparen{DA}\) is a minor arc and cannot have a measure of 232°.

• The measure of an inscribed - angle is half the measure of its intercepted arc is not relevant to this problem about the non - equality of the measure of \(\overparen{DA}\) to 232°.
• Major arcs are usually named with three points (not just two endpoints in most cases) and their measure is greater than 180°, which also supports that \(\overparen{DA}

eq232^{\circ}\) as \(\overparen{DA}\) is a minor arc.

Answer:

• The major arc \(\overparen{DCA}\) measures 232°. (This is correct as \(m\overparen{AB}+m\overparen{BC}+m\overparen{CD}=232^{\circ}\) and \(\overparen{DCA}\) is a major arc).
• Minor arcs are named by their two endpoints, and their measure is equal to the measure of the central angle. (This is a general property but not directly relevant to showing \(\overparen{DA}

eq232^{\circ}\)).

• The only possible measure for the minor arc \(\overparen{DA}\) is 128°. (Correct as calculated above). So the statements that support \(m\overparen{DA}

eq232^{\circ}\) are: "The only possible measure for the minor arc \(\overparen{DA}\) is 128°." and "The major arc \(\overparen{DCA}\) measures 232°."