Question

angles of a circle. which statements are true? check all that apply. ( moverarc{cb} = 120^circ ); ( moverarc{cb} = 60^circ ); ( mangle cob = 2(mangle cab) ); ( mangle cob = 120^circ ); ( mangle cob = \frac{1}{2}(mangle cab) )

Explanation:

Step1: Recall Inscribed Angle Theorem

The Inscribed Angle Theorem states that an inscribed angle is half the measure of its subtended central angle. Here, $\angle CAB$ is an inscribed angle subtended by arc $CB$, and $\angle COB$ is the central angle subtended by arc $CB$. Given $m\angle CAB = 60^\circ$.

Step2: Calculate $m\angle COB$

By the Inscribed Angle Theorem, $m\angle COB = 2\times m\angle CAB$. Substituting $m\angle CAB = 60^\circ$, we get $m\angle COB = 2\times60^\circ = 120^\circ$.

Step3: Determine Arc $CB$ Measure

The measure of a central angle is equal to the measure of its subtended arc. So, $m\widehat{CB}=m\angle COB = 120^\circ$.

Step4: Evaluate Each Statement

• $m\widehat{CB}=120^\circ$: True (from Step3).
• $m\widehat{CB}=60^\circ$: False (contradicts Step3).
• $m\angle COB = 2(m\angle CAB)$: True (from Step2, $120^\circ = 2\times60^\circ$).
• $m\angle COB = 120^\circ$: True (from Step2).
• $m\angle COB=\frac{1}{2}(m\angle CAB)$: False (since $120^\circ

eq\frac{1}{2}\times60^\circ$).

Answer:

• $m\widehat{CB}=120^\circ$
• $m\angle COB = 2(m\angle CAB)$
• $m\angle COB = 120^\circ$