Question

given that p is in the interior of ∠abc, m∠pbc = 30°, ∠dbc is a right - angle, and m∠abd = 26°, what are the possible measures of ∠abp? the possible measures of ∠abp are ° or °.

Explanation:

Step1: Consider two - case scenarios

There are two possible positions of ray $BP$ relative to $\angle ABD$ and $\angle DBC$.

Step2: First case: $\angle ABP=\angle DBC - \angle PBC-\angle ABD$

Since $\angle DBC = 90^{\circ}$, $\angle PBC = 30^{\circ}$, and $\angle ABD=26^{\circ}$, then $\angle ABP=90 - 30 - 26=34^{\circ}$.

Step3: Second case: $\angle ABP=\angle DBC+\angle ABD - \angle PBC$

$\angle ABP=90 + 26-30 = 86^{\circ}$.

Answer:

$34$; $86$