Question

s1 p.7 central angles and inscribed angles #4 what is the measure, in degrees, of arc jki ? 4) m\overarc{jki} (with a circle diagram showing center, points j, i, k, arc ji is 80°)

Explanation:

Step1: Recall circle's total degrees

A circle has \( 360^\circ \), but arc \( JKI \) can also be calculated by summing arcs. First, note the given arc \( JI = 80^\circ \), and the diameter (straight line \( IK \)) creates a semicircle? Wait, no, \( IK \) is a diameter? Wait, the center is the dot, so \( IK \) is a diameter (180°), and \( J \) is a point. Wait, arc \( JKI \): let's see the arcs. The circle's total is 360, but arc \( JKI \) is the major arc? Wait, no, let's check the diagram. The arc \( JI \) is 80°, and the arc from \( J \) to \( K \) to \( I \). Wait, \( IK \) is a diameter, so arc \( IK \) is 180°? Wait, no, the center is the middle, so \( IK \) is a straight line through the center, so arc \( IK \) is 180°. Then arc \( JK \): since arc \( JI \) is 80°, and the angle from \( J \) to center to \( I \) is 80°, so the angle from \( J \) to center to \( K \) is \( 180^\circ - 80^\circ = 100^\circ \)? Wait, no, maybe better: the total circumference is 360°, so arc \( JKI \) is arc \( JK \) + arc \( KI \). Wait, arc \( KI \) is a semicircle? Wait, \( IK \) is a diameter, so arc \( KI \) (the lower arc) is 180°? Wait, no, the diagram: \( J \) is above, \( I \) is left, \( K \) is right. So arc \( JKI \) goes from \( J \) to \( K \) (right) to \( I \) (left). So arc \( JK \): the angle at center for \( JK \): since \( JI \) is 80°, and \( IK \) is 180° (diameter), so the angle from \( J \) to center to \( K \) is \( 180^\circ + 80^\circ \)? Wait, no, let's think again. The measure of a circle is 360°. The minor arc \( JI \) is 80°, so the major arc \( JKI \) would be \( 360^\circ - 80^\circ = 280^\circ \)? Wait, no, wait: arc \( JKI \) is the arc from \( J \) to \( K \) to \( I \). Let's see the arcs: from \( J \) to \( K \) is 180° + 80°? No, wait, the center: the line from center to \( J \), center to \( I \) is 80°, center to \( K \) is opposite to \( I \), so center to \( K \) is 180° from center to \( I \). So the angle between center \( O \), \( J \), \( I \) is 80°, so angle \( JOI = 80^\circ \). Then angle \( JOK \): since \( K \) is opposite \( I \), \( angle JOK = 180^\circ - 80^\circ = 100^\circ \)? No, that can't be. Wait, maybe the arc \( JKI \) is the sum of arc \( JK \) and arc \( KI \). Arc \( JK \): from \( J \) to \( K \), which is \( 180^\circ + 80^\circ \)? No, I think I made a mistake. Let's recall: the total degrees in a circle is 360°. The arc \( JI \) is 80°, so the arc from \( J \) to \( I \) the other way (through \( K \)) is \( 360^\circ - 80^\circ = 280^\circ \). Wait, that makes sense. Because the minor arc \( JI \) is 80°, so the major arc \( JKI \) (going through \( K \)) is \( 360 - 80 = 280 \). Let's check: arc \( JKI \) is \( J \) to \( K \) (180°) plus \( K \) to \( I \) (100°)? No, 180 + 100 = 280. Yes, because \( K \) to \( I \) is 180° - 80° = 100°? Wait, no, \( K \) to \( I \) is a semicircle? No, \( IK \) is a diameter, so arc \( IK \) is 180°, but from \( K \) to \( I \) through the bottom is 180°, but from \( K \) to \( I \) through \( J \) is 180°? No, I'm confused. Wait, the key is: the measure of arc \( JKI \) is the sum of arc \( JK \) and arc \( KI \). Arc \( JK \): from \( J \) to \( K \), which is a semicircle? No, \( J \) is at 80° from \( I \), so \( J \) is at 80° above the left (I), so \( K \) is at right (180° from I). So the angle from \( J \) to \( K \) is 180° + 80° = 260°? No, this is wrong. Wait, let's use the circle's total: 360°. The minor arc \( JI \) is 80°, so the major arc \( JKI \) is \( 360 - 80 = 280 \). Yes, that's the formula: major arc = 360 - minor arc. So since arc \( JI \) is 80° (minor), arc \( JKI \) (major) is \( 360 - 80 = 280 \).

Step2: Calculate the measure

So \( m\overarc{JKI} = 360^\circ - 80^\circ = 280^\circ \).

Answer:

280