2001 CEMC Pascal Problems/Problem 5

Problem

In the diagram, the value of $x$ is

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$\text{ (A) }\ 100 \qquad\text{ (B) }\ 65 \qquad\text{ (C) }\ 80 \qquad\text{ (D) }\ 70 \qquad\text{ (E) }\ 50$

We can see that angles $\angle ACB$ and $\angle BCD$ form a straight angle from the diagram, allowing us to create an equation:

$m\angle ACB + m\angle BCD = 180$

$m\angle ACB + 130 = 180$

$m\angle ACB = 50^{\circ}$

We also notice $m\angle CAB = m\angle ACB = 50^{\circ}$ because $AB = BC$ .

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The sum of the measures of the interior angles in a triangle is always $180^{\circ}$ . Using triangle $ABC$ , we have:

$x + m\angle CAB + m\angle ACB = 180$

$x + 50 + 50 = 180$

$x + 100 = 180$

$x = \boxed {\textbf {(C) } 80}$

~anabel.disher