1. **State the problem:** We are given a circle with center $P$ and two diameters $\overline{BD}$ and $\overline{AC}$. We need to find the measure of the major arc $\overset{\frown}{DBC}$ in degrees. 2. **Analyze the given information:** - $\overline{BD}$ and $\overline{AC}$ are diameters, so they intersect at the center $P$ and divide the circle into four arcs. - $\angle APB = 19^\circ$ is the angle between radii $PA$ and $PB$. 3. **Understand the geometry:** - Since $BD$ is a diameter, points $B$ and $D$ lie on opposite ends of the circle. - Similarly, $AC$ is a diameter, so points $A$ and $C$ lie on opposite ends. - The angle $\angle APB = 19^\circ$ is the angle between radii $PA$ and $PB$. 4. **Find the measure of minor arc $\overset{\frown}{AB}$:** - The measure of the central angle $\angle APB$ equals the measure of the minor arc $\overset{\frown}{AB}$. - Therefore, $m\overset{\frown}{AB} = 19^\circ$. 5. **Find the measure of minor arc $\overset{\frown}{BC}$:** - Since $BD$ is a diameter, $\overset{\frown}{BC}$ is a quarter of the circle minus $\overset{\frown}{AB}$. - The circle is $360^\circ$, so each quarter arc is $90^\circ$. - $m\overset{\frown}{BC} = 90^\circ - 19^\circ = 71^\circ$. 6. **Find the measure of minor arc $\overset{\frown}{CD}$:** - $CD$ is the other half of the circle opposite $AB$ and $BC$. - Since $BD$ is a diameter, $\overset{\frown}{CD}$ is $180^\circ$. 7. **Calculate the measure of major arc $\overset{\frown}{DBC}$:** - Major arc $\overset{\frown}{DBC}$ consists of arcs $\overset{\frown}{DB}$ and $\overset{\frown}{BC}$. - Since $DB$ is the other half of the circle opposite $BD$, $m\overset{\frown}{DB} = 180^\circ$. - We already found $m\overset{\frown}{BC} = 71^\circ$. - Therefore, $m\overset{\frown}{DBC} = 180^\circ + 71^\circ = 251^\circ$. **Final answer:** $$m\overset{\frown}{DBC} = 251^\circ$$