Question

using the image provided, find m∠abc. 84° 20° 62° 32

Explanation:

Step1: Consider angle - sum property of \(\triangle DBC\)

\((3r-32)+(r - 12)+2p = 180\)

Step2: Simplify the left - hand side

\(4r+2p=224\)

Step3: Further simplify

\(2r + p=112\)
Since we have two variables and only one equation and no other information about \(p\), we can't find \(r\) and thus can't find \(m\angle ABC\).

Answer:

We need more information to solve this problem as there are two variables \(r\) and \(p\) and not enough equations to find their values and thus \(m\angle ABC=(3r - 12)^{\circ}\). If we assume some relationship between the angles (for example, if \(\triangle DBC\) is a triangle and we use the angle - sum property of a triangle \(180^{\circ}\)), we can solve for \(r\). Let's assume \(\triangle DBC\) has the angle - sum property:
\[

$$\begin{align*} (3r-32)+(r - 12)+2p&=180\\ 4r+2p&=180 + 32+12\\ 4r+2p&=224\\ 2r + p&=112 \end{align*}$$

\]
But without more information about \(p\) we can't solve for \(r\). If we assume \(p\) is such that we can solve for \(r\), we first solve the equation for \(r\) and then substitute into \(m\angle ABC=(3r - 12)^{\circ}\). Since we don't have sufficient data, we can't give a numerical answer from the given options.