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:

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