1999 CEMC Pascal Problems/Problem 17

Problem

In the parallelogram, the value of $x$ is

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$\text{ (A) }\ 30 \qquad\text{ (B) }\ 50 \qquad\text{ (C) }\ 70 \qquad\text{ (D) }\ 80 \qquad\text{ (E) }\ 150$

We can name the vertices to make the problem easier to understand.

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Opposite angles in a parallelogram are equal, so we know $m\angle ABC = m\angle ADC = 80^{\circ}$ .

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We can notice that angles $\angle BFE$ and $\angle CFE$ form a straight angle, allowing us to set up an equation:

$m\angle BFE + m\angle CFE = 180$

$m\angle BFE + 150 = 180$

$m\angle BFE = 30^{\circ}$

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We can now use the fact that the angles in a triangle are always supplementary. This gives:

$x + m\angle BFE + m\angle ABC = 180$

$x + 30 + 80 = 180$

$x + 110 = 180$

$x = \boxed {\textbf {(C) } 70}$

~anabel.disher