Seth is using the figure shown below to prove the Pythagorean Theorem using triangle similarity: In the given triangle DEF, angle D is 90° and segment DG is perpendicular to segment EF.

The figure shows triangle DEF with right angle at D and segment DG. Point G is on side EF.

Part A: Identify a pair of similar triangles. (2 points)

Part B: Explain how you know the triangles from Part A are similar. (4 points)

Part C: If EG = 2 and EF = 8, find the length of segment ED. Show your work. (4 points)

Part A: DEG and DGF are similar. Parts B and C: Below

Step-by-step explanation:

First, let's cover what we know. We know that angle D is 90 degrees, and we know that EGD and FGD are, too. We know that from the definition of perpendicular lines, as it's stated that segment DG is perpendicular to EF. We also know that EDG and FDG are each 45 degrees based on the position of DG in relation to angle D. DEG and DFG are each 45 degrees based on the Triangle Sum Theorem (all the angles add up to 180 degrees). We can say that Triangles DGE and DFG are similar based on the AA Similarity Postulate, because we know that multiple angles are congruent.

As for part C, Let's cover what we know. We know that DEG and DGF are isosceles triangles. Although we usually associate isosceles triangles with side lengths, a triangle is isosceles if there are two congruent angles. Please message or respond if you need me to clarify why that is. Anyways, We know that the length of DG will have to be either 2 or 8. I'll assume that you know what a hypothenuse and a leg are, so I'll work under that basis. In this case, our legs would be DG and either EG or GF. (I don't see an attached diagram, so I can't confirm which it is.) The hypothenuse would either be DE or DF. Since this triangle is isosceles, we know that the legs will match up (It can't be the hypothenuse, because the hypothenuse is the longest side.) You can visually determine whether DG is 2 or 8 units long by comparing its length to EG and GF, and you can kind of fill in the blanks for the showing your work part. I'd also be happy to clear anything up for anyone who responds to this answer!