The sum of the measures of $\angle{A}$ and $\angle{B}$ is $90^\circ$. The sum of the measures of $\angle{A}$ and $\angle{C}$ is $180^\circ$. The sum of the measures of $\angle{B}$ and $\angle{D}$ is $180^\circ$. What is the sum of the measures of $\angle{C}$ and $\angle{D}$ ?

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We'll write each statement in equation form:

$$m\angle{A} + m\angle{B} = 90^\circ$$ $$m\angle{A} + m\angle{C} = 180^\circ$$ $$m\angle{B} + m\angle{D} = 180^\circ$$

Since we wish to sum get the sum of $\angle{C}$ and $\angle{D}$, we can sum the last two equations:

$$m\angle{C} + m\angle{D} = ?$$ $$m\angle{A} + m\angle{C} + m\angle{B} + m\angle{D} = 180^\circ + 180^\circ$$

Rearrange. We can substitute the first equation:

$$m\angle{A} + m\angle{B} + m\angle{C}+ m\angle{D} = 360^\circ$$ $$90^\circ + m\angle{C}+ m\angle{D} = 360^\circ$$ $$m\angle{C}+ m\angle{D} = 270^\circ$$