Triangle $PQR$ is isosceles and the measure of angle $R$ is $40^\circ$. The possible measures of angle $P$ are $x,y,z$. What is the value of the sum $x + y + z$?
Let me consider both cases and find the values of angle $P$. If angle $P$ is equal to angle $R$, then they are both $40^\circ$. That means angle $Q$ must be $180 - 40 - 40 = 100^\circ$. So, one possible value of angle $P$ is $40^\circ$. If angle $Q$ is equal to angle $R$, then they are both $40^\circ$. That means angle $P$ must be $180 - 40 - 40 = 100^\circ$. So, another possible value of angle $P$ is $100^\circ$. Then, angle $P$ and angle $Q$ must be $180 - 40 = 140^\circ$ divided by $2$, which is $70^\circ$. So, another possible value of angle $P$ is $70^\circ$. Those are the only three possible values of angle $P$, since the triangle is isosceles and the measure of angle $R$ is $40^\circ$. Therefore, the sum $x + y + z$ is $40 + 70 + 100 = 210^\circ$. 