### Video Transcript

The diagram shows the path of a light ray through a triangular prism. Which of the following correctly relates the apex angle of the prism ๐ด to the angles of incidence and the angles of refraction of the incident and emergent rays shown? (A) ๐ one plus ๐ two equals ๐ด. (B) ฮฆ one plus ฮฆ two equals ๐ด. (C) ฮฆ one plus ๐ two equals ๐ด. (D) ๐ one plus ฮฆ two equals ๐ด. And (E) 180 degrees minus the quantity ๐ one plus ฮฆ two equals ๐ด.

In our diagram, we see our incident ray of light entering the triangular prism and then ultimately leaving it. The initial angle of incidence of this ray is ฮฆ one, the initial angle of refraction is ๐ one, and then the second angle of incidence is ฮฆ two and the second angle of refraction is ๐ two. We want to develop an expression for the apex angle ๐ด in terms of these values. Weโre going to be able to do this using a geometric approach.

Starting off, letโs clear some space on screen, and letโs note that the apex angle ๐ด is part of this quadrilateral in orange. If we think about the corners of this four-sided shape, we know that this corner here is a right angle. Thatโs because itโs defined by a line normal or perpendicular to the face of the prism on this side. And for the same reason, this angle here in the quadrilateral is also 90 degrees.

At this point, letโs recall that for any four-sided shape, if we add together the angles of the four corners of that shape, we always get a total of 360 degrees. We can apply this fact to our quadrilateral in orange. The angle at the top of this shape is ๐ด. Then, thereโs one 90-degree angle and a second 90-degree angle. And lastly, thereโs this angle at the bottom of the shape. Weโll leave that label blank for now. Our rule for four-sided objects says these four angles added together must equal 360 degrees.

Notice on the left we have 90 degrees plus 90 degrees; thatโs 180 degrees. And if we subtract this angle from both sides of the equation, then 180 degrees minus 180 degrees on the left goes to zero. And we have an expression that says that ๐ด plus our unknown angle is equal to 180 degrees. If we now subtract the angle ๐ด from both sides, canceling this angle on the left, we see that the unknown angle in our quadrilateral is 180 degrees minus ๐ด.

Knowing this, letโs now draw a zoomed-in version of this triangle that weโve highlighted in pink. The angle at the top left of this triangle is ๐ one. The angle at the top right is ฮฆ two. And we now know that the angle at the bottom is 180 degrees minus ๐ด. Similarly to what we found for our four-sided shape, if we consider a three-sided shape in general, a triangle, the sum of these three interior angles always equals 180 degrees. This means we can write that ๐ one plus ฮฆ two plus 180 degrees minus ๐ด equals 180 degrees. If we then subtract 180 degrees from both sides of this equation, that angle measure cancels out entirely.

Whatโs left is an expression that reads ๐ one plus ฮฆ two minus ๐ด equals zero. And if we then add ๐ด to both sides of the equation so that ๐ด cancels on the left, we find that ๐ one plus ฮฆ two equals ๐ด. Reviewing our answer options, this matches option (D). This is how the apex angle ๐ด can be expressed in terms of the angles of incidence and angles of refraction of this scenario. The initial angle of refraction, ๐ one, plus the second angle of incidence, ฮฆ two, equals the apex angle ๐ด.