An octagon is shown.A dotted line extends from each of the points where one side meets another side, so that there are 8 dotted lines meeting at an asterisk in the center of the octagon. These dotted lines divide the octagon into 8 triangles. There is one additional dotted line pointing directly down from the center asterisk. A right angle is formed at the point where this line meets the base of the octagon, in the center of the bottom triangle. The right base angle of this triangle is labeled with a 2. The vertex angle of the triangle directly to the right of this triangle is labeled with a 1. The 2 and the 1 are on either sides of the same dotted line.

Question

An octagon is shown.A dotted line extends from each of the points where one side meets another side, so that there are 8 dotted lines meeting at an asterisk in the center of the octagon. These dotted lines divide the octagon into 8 triangles. There is one additional dotted line pointing directly down from the center asterisk. A right angle is formed at the point where this line meets the base of the octagon, in the center of the bottom triangle. The right base angle of this triangle is labeled with a 2. The vertex angle of the triangle directly to the right of this triangle is labeled with a 1. The 2 and the 1 are on either sides of the same dotted line.

The figure above is a regular octagon with radii and an apothem drawn. What is m<1?
A. 22.5°
B. 45°
C. 60°
D. 67.5°

Answer 1

We can start by using the fact that the sum of angles in a triangle is $180^\circ$. Let $x$ be the measure of angle 1. Then, angle 2 (the right angle) has measure $90^\circ$ and the other acute angle in triangle 2 has measure $180^\circ - x - 90^\circ = 90^\circ - x$.

Since the octagon is regular, angle 1 is congruent to angles 3, 5, and 7, which are the other angles in the four triangles adjacent to triangle 2. Therefore, the sum of angles 1, 3, 5, and 7 is $4x$. The sum of angles in a quadrilateral is $360^\circ$, so angles 2, 4, 6, and 8 each have measure $\frac{1}{4}(360^\circ - 90^\circ) = 67.5^\circ$.

Now we can use the fact that the sum of angles around a point is $360^\circ$. The eight angles around the center asterisk add up to $8\cdot 67.5^\circ = 540^\circ$. Adding in the additional angle pointing down, we get $540^\circ + 90^\circ = 630^\circ$.

Finally, we can use the fact that the apothem of a regular polygon divides each interior angle in half. Angle 1 is one of the interior angles of the octagon, so it is divided in half by the apothem. Therefore, $x = \frac{m<1}{2}$. Since the sum of the angles in the octagon is $180(8-2) = 1080^\circ$, we have $\frac{m<1}{2} + \sum_{k=1}^8 67.5^\circ = 1080^\circ$. Simplifying, we get $\frac{m<1}{2} + 540^\circ = 1080^\circ$, which gives $m<1 = \boxed{\textbf{(D) }67.5^\circ}$.