Question

select the correct answer from each drop - down menu.

in the diagram, points d and e are marked by drawing arcs of equal size centered at b such that the arcs intersect \\( \overrightarrow{ba} \\) and \\( \overrightarrow{bc} \\). then, intersecting arcs of equal size are drawn centered at points d and e. point p is located at the intersection of these arcs.

based on this construction, \\( m\angle abp \\) is \\( \square \\) , and \\( m\angle abc \\) is \\( \square \\) .

Explanation:

Step1: Recall the angle - bisector construction

The construction shown is for an angle bisector. When we construct an angle bisector of \(\angle ABC\) (in this case, \(\overrightarrow{BP}\) is the angle bisector of \(\angle ABC\)). Given that \(m\angle PBC = 32^{\circ}\) (from the diagram, assume the value of the non - bisected part adjacent to the construction).
Since \(\overrightarrow{BP}\) bisects \(\angle ABC\), \(m\angle ABP=m\angle PBC\).

Step2: Calculate \(m\angle ABP\)

By the property of angle bisector, \(m\angle ABP = 32^{\circ}\)

Step3: Calculate \(m\angle ABC\)

We know that \(m\angle ABC=m\angle ABP + m\angle PBC\). Substituting \(m\angle ABP=m\angle PBC = 32^{\circ}\), we get \(m\angle ABC=32^{\circ}+32^{\circ}\)

Answer:

\(m\angle ABP = 32\), \(m\angle ABC = 64\)