Question

which statements are true? check all that apply. \\( m\overarc{cb} = 120^\circ \\) \\( m\overarc{cb} = 60^\circ \\) \\( m\angle cob = 2(m\angle cab) \\) \\( m\angle cob = 120^\circ \\) \\( m\angle cob = \frac{1}{2} (m\angle cab) \\) (there is a circle with center o, points a, b, c on the circle. triangle cab has angle at a of 60 degrees, and triangle cob with center o.)

Explanation:

Step 1: Recall the Inscribed Angle Theorem

The Inscribed Angle Theorem states that an inscribed angle is half the measure of its intercepted arc. Also, the central angle is equal to the measure of its intercepted arc. For an inscribed angle \(\angle CAB\) and central angle \(\angle COB\) intercepting arc \(CB\), we have \(m\angle COB = 2(m\angle CAB)\). Given \(m\angle CAB=60^{\circ}\), so \(m\angle COB = 2\times60^{\circ}=120^{\circ}\). And the measure of arc \(CB\) (since central angle \(\angle COB\) intercepts arc \(CB\)) is equal to the measure of \(\angle COB\), so \(m\widehat{CB}=120^{\circ}\).

Step 2: Analyze each statement

• For \(m\widehat{CB} = 120^{\circ}\): Since \(m\angle COB = 120^{\circ}\) (central angle equals its arc), this is true.
• For \(m\widehat{CB}=60^{\circ}\): We found \(m\widehat{CB}=120^{\circ}\), so this is false.
• For \(m\angle COB = 2(m\angle CAB)\): By Inscribed Angle Theorem, this is true (as \(m\angle CAB = 60^{\circ}\), \(2\times60^{\circ}=120^{\circ}=m\angle COB\)).
• For \(m\angle COB = 120^{\circ}\): Calculated from \(m\angle CAB = 60^{\circ}\) and \(m\angle COB = 2(m\angle CAB)\), so this is true.
• For \(m\angle COB=\frac{1}{2}(m\angle CAB)\): This is the reverse of the Inscribed Angle Theorem, so false.

Answer:

• \(m\widehat{CB} = 120^{\circ}\) (True)
• \(m\widehat{CB}=60^{\circ}\) (False)
• \(m\angle COB = 2(m\angle CAB)\) (True)
• \(m\angle COB = 120^{\circ}\) (True)
• \(m\angle COB=\frac{1}{2}(m\angle CAB)\) (False)

So the true statements are \(m\widehat{CB} = 120^{\circ}\), \(m\angle COB = 2(m\angle CAB)\), \(m\angle COB = 120^{\circ}\)