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 mangle1?
A. 22.5°
B. 45°
C. 60°
D. 67.5°

Answer 1

Since the octagon is regular, all of its angles are equal. Let this angle be $x$. Since angle 2 is a right angle, triangle 2 must be a 45-45-90 triangle, meaning angle 2 is 45 degrees. Therefore, angle 1 is $180^\circ - 90^\circ - x = 90^\circ - x$.

We can use the fact that the dotted lines divide the octagon into triangles and use the formula for the sum of angles in a polygon to get $8\cdot (180^\circ - 2x) + 360^\circ = 6(180^\circ)$, since there are 8 triangles formed and one additional 360 degree angle at the center. Simplifying, we get $x = 135^\circ$.

Therefore, angle 1 is $90^\circ - 135^\circ = \boxed{\textbf{(A) } 22.5^\circ}$.

Answer 2

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Answer 3

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Answer 4

To find the measure of angle 1, we need to understand the properties of a regular octagon.

A regular octagon has 8 equal sides and 8 equal angles. Each angle of a regular octagon measures 360 degrees divided by the number of sides, which in this case is 8. So each angle of a regular octagon measures 45 degrees.

Now, let's look at the figure provided. The dotted lines divide the octagon into 8 equal triangles. We are given that the right base angle of the triangle at the bottom is labeled with a 2. From this information, we can conclude that the measure of angle 2 is 90 degrees.

Since the dotted line labeled with a 2 is the base of the triangle and forms a right angle, the remaining angle of this triangle must be 180 degrees - 90 degrees = 90 degrees.

Now, the dotted line labeled with a 1 is the hypotenuse of the triangle directly to the right of the triangle with angle 2. Since this is a triangle within a regular octagon, we know that this triangle has a right angle and two equal angles. Thus, the remaining angle of this triangle must be (180 degrees - 90 degrees) / 2 = 45 degrees.

Therefore, the measure of angle 1 is 45 degrees.

The answer is B. 45°