Question Bank #2, Question 47

This question is particularly tricky due to the orientations of angles, which lack perfectly parallel or perpendicular lines to allow for an easy comparison. Furthermore, some of angles have one very short arm, making it difficult to visualize the angle. As a result, we will use a modification of the reference method to solve this question

Although you could use the laptop method to answer this question, the short arms could potentially misdirect you. Additionally, the angles are too large for the knife method to be reliable and too small to use a 90 degree angle for reference. Thus, we must make use of a different reference angle, the 45 degree angle.

Diagonally connecting the corners of a perfect square creates a perfect 45 degree angle. Thus, we can imagine rectangles that have an edge lined up with one arm of the angle and the other arm diagonally connecting the corners, which are highlighted in red (see above). The closer this rectangle is to a perfect square, the closer the angle to 45 degrees. The rectangles for angles 1 and 4 appear to be good approximations of a square, so we can conclude that these angles are around 45 degrees. However, angles 2 and 3 appear significantly distorted, so we can conclude that these angles are significantly smaller than 45 degrees. Moreover, because angle 2 appears to be the most distorted, we can conclude that it is the smallest angle.

However, to double check that angle 2 is the smallest, let's compare angles 2 and 3 to a 45 degree reference angle:

First, let's imagine our perfect squares and diagonally connect the corners to produce a 45 degree reference angle, which is highlighted in blue (see above). Next, we can compare angles 2 and 3 to the reference angle. Since angle 3 is closer to the reference angle, it is closer to 45 degrees. Thus, angle 3 is larger than angle 2.

To conclude, we have found that angle 2 is the smallest angle using a 45 degree reference angle. Only one of the option choices displays angle 2 as the smallest. Thus, Option C must be correct.