Question

what is m\\(\stackrel{\frown}{bc}\\)? (not drawn to scale) a 76 b 284 c 38 d 152 m\\(\angle abc=\frac{1}{2}m\stackrel{\frown}{ac}\\) m\\(\angle apb = \frac{1}{2}(m\stackrel{\frown}{ab}+m\stackrel{\frown}{cd})\\)

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Here, \(\angle BAC = 76^{\circ}\) is an inscribed angle that intercepts arc \(BC\). Let \(m\widehat{BC}=x\).

Step2: Apply the formula

We know that the measure of an inscribed angle \(\theta\) and its intercepted arc \(s\) are related by \(\theta=\frac{1}{2}s\). In our case, \(\angle BAC = 76^{\circ}\), and the arc it intercepts is \(m\widehat{BC}\). So, \(m\angle BAC=\frac{1}{2}m\widehat{BC}\).

Step3: Solve for \(m\widehat{BC}\)

If \(m\angle BAC = 76^{\circ}\) and \(m\angle BAC=\frac{1}{2}m\widehat{BC}\), then we can solve for \(m\widehat{BC}\) by multiplying both sides of the equation by 2. So, \(m\widehat{BC}=2\times m\angle BAC\). Substituting \(m\angle BAC = 76^{\circ}\), we get \(m\widehat{BC}=2\times76 = 152^{\circ}\).

Answer:

D. 152