 This video is called Solving Problems with Similar Triangles, video number one. In this video, our job is to find angle A, B, and C, and side length D, side length E. So let's take a look at angles first. We're given that the triangle MNP is similar to triangle QRS. And that word similar means, in terms of triangles at least, that angles are congruent and sides are proportional. So all of these angles are congruent. I see M is the first angle in MNP, Q is the first angle in QRS. And so that means that angle M is congruent to angle Q. So therefore, right off the bat, we know angle A is equal to 41. Continuing on with the angles, angle N must be congruent to angle R. And the reason that N and R must be congruent is N is the second letter of the triangle, and likewise R is the second letter of the triangle. However, we don't know anything about angle N and angle R. So let's take a look at the last angles named in the triangle. Angle P must be congruent to angle S. And since 82 is angle S, I know that B is equal to 82. And now using the triangle sum theorem, let's just look at triangle MNP. We know 41 degrees is angle M. Angle P is 82 degrees. And that means C must be equal to 57 degrees. And the reason why is the triangle sum theorem. The angles in a triangle must add up to 180. So here we have A, B, and C. Now let's take a look at side lengths D and E. In order to find D, I need to figure out how the side length NP is related to the other triangle. So NP. I see in the name of the triangle that NP is the second and third letters of triangle MNP. So second and third corresponds with second and third. And so now we have one ratio. In order to solve with a proportion, we need another ratio. And in particular, we need a known ratio. By known ratio, I mean we need to know something about both pairs of sides in the triangles. So MP is a known side length. And MP corresponds with QS. And QS is also a known side length. And so we can set up a proportion using that known ratio, MP and QS. And we can set it equal to the ratio that we're looking for. Now let's substitute in those values that we know NP. NP again refers to this length. NP is unknown. That's D units divided by RS, which is 12. And that equals MP over QS. And so now we can solve for D by cross multiplying. D times 24 is equal to 12 times 56, which is 648, divided by 24 on both sides. And we get that D is equal to 27. So now we found D. Now the next job is to find E. So length E refers to the side length QR. So let's set up our ratios as QRS, triangle QRS, divided by triangle MP. So length QR, QR correlates with MN. So QR divided by MN is one ratio. And in particular, it's one of the unknown ratios because length E is unknown. So that's the unknown ratio. And again, it's unknown because QR is unknown. And we want to set that ratio equal to a known ratio. The only known ratio in this pair of triangles is 56 and 24. In other words, it's length QS. So first and third. And then the corresponding side to QS. First and third is MP. So now we have two ratios set equal to each other. We have our proportion. And our job is to solve that proportion. So QR again is unknown. It's E units. MN is known. That's 42. And that's equal to QS, which is 24 over MP, which is 56. So let's cross multiply. And then if we divide by 56 on both sides, we get that E equals 18.