in the figure below, $\\angle ape$ and $\\angle epd$ are congruent.\n\nwhat is the arc measure of minor arc…

Explanation:

Step1: Sum all central angles

The sum of all central angles in a circle is $360^{\circ}$. $$m\angle APB + m\angle BPC + m\angle CPD + m\angle DPE + m\angle EPA = 360^{\circ}$$

Step2: Substitute known angle values

Substitute $136^{\circ}$, $74^{\circ}$, and $42^{\circ}$ into the equation. $$136^{\circ} + 74^{\circ} + 42^{\circ} + m\angle DPE + m\angle EPA = 360^{\circ}$$

Step3: Simplify the equation

Combine the constant values. $$252^{\circ} + m\angle DPE + m\angle EPA = 360^{\circ}$$

Step4: Solve for congruent angles

Since $\angle APE \cong \angle EPD$, let $m\angle APE = m\angle EPD = x$. $$252^{\circ} + 2x = 360^{\circ}$$

Step5: Calculate the value of x

Subtract $252^{\circ}$ and divide by $2$. $$2x = 108^{\circ} \implies x = 54^{\circ}$$

Step6: Determine measure of arc AC

The arc measure equals the sum of central angles $\angle APB$ and $\angle BPC$. $$m\widehat{AC} = m\angle APB + m\angle BPC$$

Step7: Calculate final arc measure

Add the specific angle values. $$m\widehat{AC} = 136^{\circ} + 74^{\circ} = 210^{\circ}$$

Step8: Identify the minor arc

The minor arc is the shorter path, $360^{\circ} - 210^{\circ}$. $$m\widehat{AC}_{\text{minor}} = 360^{\circ} - 210^{\circ} = 150^{\circ}$$

Answer:

150^{\circ}