Question

practice relating inscribed, central, and circumscribed angles of a circle.
which statements are true? check all that apply.
$\boldsymbol{mwidehat{cb} = 120^circ}$
$\boldsymbol{mwidehat{cb} = 60^circ}$
$\boldsymbol{mangle cob = 2(mangle cab)}$
$\boldsymbol{mangle cob = 120^circ}$
$\boldsymbol{mangle cob = \frac{1}{2}(mangle cab)}$

Explanation:

Step1: Identify inscribed angle property

An inscribed angle is half the measure of its intercepted central angle and its intercepted arc. Here, $\angle CAB$ is an inscribed angle intercepting arc $\overparen{CB}$ and central angle $\angle COB$.

Step2: Calculate arc $\overparen{CB}$ measure

Given $m\angle CAB = 60^\circ$, the intercepted arc $\overparen{CB}$ has twice the measure of the inscribed angle:
$m\overparen{CB} = 2 \times 60^\circ = 120^\circ$

Step3: Calculate $\angle COB$ measure

The central angle $\angle COB$ is equal to the measure of its intercepted arc $\overparen{CB}$:
$m\angle COB = m\overparen{CB} = 120^\circ$

Step4: Verify angle relationship

From the inscribed angle theorem, $m\angle COB = 2(m\angle CAB)$, which holds as $120^\circ = 2 \times 60^\circ$.

Answer:

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