Question

if ∠dpb = 45 degrees, find the measure of all the other angles.
∠dpa = 135 degrees
∠apc = 45 degrees
∠cpb = 135 degrees

Explanation:

Step1: Identify vertical - opposite angles

Vertical - opposite angles are equal. $\angle DPB$ and $\angle APC$ are vertical - opposite angles, so $\angle APC=\angle DPB = 45^{\circ}$.

Step2: Identify linear - pair of angles

$\angle DPB$ and $\angle DPA$ form a linear - pair. The sum of angles in a linear - pair is $180^{\circ}$. So, $\angle DPA=180^{\circ}-\angle DPB$. Substituting $\angle DPB = 45^{\circ}$, we get $\angle DPA = 135^{\circ}$.

Step3: Identify another vertical - opposite pair

$\angle DPA$ and $\angle CPB$ are vertical - opposite angles. So, $\angle CPB=\angle DPA = 135^{\circ}$.

Answer:

$\angle APC = 45$ degrees, $\angle DPA=135$ degrees, $\angle CPB = 135$ degrees