1. **State the problem:** We need to find the arc measure of the major arc BÂD on circle P, given the central angles \(\angle APB = 136^\circ\), \(\angle BPC = 74^\circ\), and \(\angle CPD = 42^\circ\). It is also given that \(\angle APE\) and \(\angle EPD\) are congruent. 2. **Understand the circle and arcs:** Points A, B, C, D, and E lie on the circle with center P. The arcs correspond to central angles at P. 3. **Calculate \(\angle APD\):** The central angles given cover arcs from A to B, B to C, and C to D. So, $$\angle APD = \angle APB + \angle BPC + \angle CPD = 136^\circ + 74^\circ + 42^\circ = 252^\circ.$$ This is the measure of the arc from A to D passing through B and C. 4. **Use the congruence of \(\angle APE\) and \(\angle EPD\):** Since \(\angle APE = \angle EPD\), point E divides the arc A to D into two arcs of equal central angles. Let \(\angle APE = \angle EPD = x\). Then, $$\angle APD = \angle APE + \angle EPD = x + x = 2x,$$ so $$x = \frac{\angle APD}{2} = \frac{252^\circ}{2} = 126^\circ.$$ 5. **Find arc BÂD major:** The circle total is \(360^\circ\). Arc BÂD major is the larger arc from B to D passing through A. Since \(\angle APB = 136^\circ\) is part of the smaller arc BÂD minor, arc BÂD major is $$360^\circ - 136^\circ = 224^\circ.$$ **Final answer: the arc measure of major arc BÂD is 224 degrees.