A rectangular strip of paper, white on one side and gray on the other, was folded as shown in the figure. What is the measure of the angle $\alpha$?

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-14.jpg?height=317&width=437&top_left_y=238&top_left_x=912)
A) $110^{\circ}$
B) $115^{\circ}$
C) $120^{\circ}$
D) $125^{\circ}$
E) $130^{\circ}$

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Solution

## ALTERNATIVE C

Consider the triangle $ABC$ in the figure below. It is a right triangle with $AB=1 \text{ cm}$ and $BC=2 \text{ cm}$, that is, one leg is half the hypotenuse. It follows that $D \hat{C} B = A \hat{C} B = 30^{\circ}$, and similarly, $C \hat{B} D = 30^{\circ}$. Since the sum of the interior angles of a triangle is $180^{\circ}$, it follows that $B \hat{D} C = 180^{\circ} - 30^{\circ} - 30^{\circ} = 120^{\circ}$. Since $B \hat{D} C$ and $\alpha$ are vertical angles, we conclude that $\alpha = 120^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-43.jpg?height=365&width=540&top_left_y=1094&top_left_x=672)