2014 CEMC Gauss (Grade 8) Problems/Problem 12

Problem

If two straight lines intersect as shown, then $x - y$ is

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$\text{ (A) }\ 0 \qquad\text{ (B) }\ 40 \qquad\text{ (C) }\ 80 \qquad\text{ (D) }\ 60 \qquad\text{ (E) }\ 100$

Solution 1

Since the lines are straight, we know that $x$ and $40^{\circ}$ form a straight angle. Thus, we have:

$x^{\circ} + 40^{\circ} = 180^{\circ}$

$x = 140$

Using the same logic, $x$ and $y$ also form a straight angle, giving:

$x^{\circ} + y^{\circ} = 180^{\circ}$

$140^{\circ} + y^{\circ} = 180^{\circ}$

$y = 40$

Going back to the original problem, $x - y = 140 - 40 = \boxed {\textbf {(E) } 100}$ .

~anabel.disher

Solution 2

Like solution 1, we can use the fact that $x$ and $40^{\circ}$ form a straight angle. However, we can notice that $y^{\circ}$ and $40^{\circ}$ are vertical angles.

Thus, $y = 40$ , giving the same answer of $\boxed {\textbf {(E) } 100}$ .

~anabel.disher

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2014 CEMC Gauss (Grade 8) Problems/Problem 12

Problem

Solution 1

Solution 2