Question

consider $odot b$ in the figure. what is $mangle abc$? $mangle abc=\text{select choice}$ degrees select choice 32 16

Explanation:

Step1: Use the property of angles in a circle

The central - angle theorem states that the measure of an inscribed angle is half the measure of the central angle subtended by the same arc. Here, \(\angle ADC\) is an inscribed angle and \(\angle ABC\) is a central angle subtended by the same arc \(\overset{\frown}{AC}\). Also, \(\angle DBC\) and \(\angle BDC\) are related in \(\triangle BDC\). In \(\triangle BDC\), \(BC = BD\) (radii of the same circle \(\odot B\)), so \(\angle BDC=\angle BCD = 2x - 8\). And \(\angle ABC\) is the exterior - angle of \(\triangle BDC\). By the exterior - angle property of a triangle, \(\angle ABC=\angle BDC+\angle BCD\). Since \(\angle ABC = 3x\) and \(\angle BDC=\angle BCD = 2x - 8\), we have the equation \(3x=(2x - 8)+(2x - 8)\).

Step2: Solve the equation for \(x\)

\[

$$\begin{align*} 3x&=2x - 8+2x - 8\\ 3x&=4x-16\\ 4x - 3x&=16\\ x&=16 \end{align*}$$

\]

Step3: Find the measure of \(\angle ABC\)

Substitute \(x = 16\) into the expression for \(\angle ABC\). Since \(\angle ABC = 3x\), then \(m\angle ABC=3\times16 = 48\) degrees.

Answer:

48