This problem hinges on your ability to recognize two important themes: one, that triangle ABC is a special right triangle, a 6-8-10 side ratio, allowing you to plug in 8 for side AB. And secondly, triangles ABC and CDE are similar triangles. You know this because each triangle is marked as a right triangle and angles ACB and ECD are vertical angles, meaning that they’re congruent. Since all angles in a triangle must sum to 180, if two angles are the same then the third has to be, too, so you’ve got similar triangles here.

With that knowledge, you know that triangle ECD follows a 3-4-5 ratio (the simplified version of 6-8-10), so if the side opposite angle C in ABC is 8 and in CDE is 12, then you know you have a 9-12-15 triangle. With the knowledge that side CE measures 15, you can add that to side BC which is 10, and you have the answer of 25.

An important point of recognition on this problem is that triangles JXZ and KYZ are similar. Each has a right angle and each shares the angle at point Z, so the third angles (XJZ and YKZ, each in the upper left corner of its triangle) must be the same, too.

Given that, if you know that JX measures 16 and KY measures 8, you know that each side of the larger triangle measures twice the length of its counterpart in the smaller triangle.

You also have enough information to solve for side XZ, since you’re given the area of triangle JXZ and a line, JX, that could serve as its height (remember, to use the  base x height equation for area of a triangle, you need base and height to be perpendicular; lines JX and XZ are perpendicular). Since , you can see that XZ must measure 10. And since XZ will be twice the length of YZ by the similarity ratio, YZ = 5, meaning that XY must also be 5.

The first important thing to note on this problem is that for each triangle, you’re given two angles: a right angle, and one other angle. Because all angles in a triangle must sum to 180 degrees, this means that you can solve for the missing angles.

In ABC, you have angles 36 and 90, meaning that to sum to 180 the missing angle ACB must be 54. And in XYZ, you have angles 90 and 54, meaning that the missing angle XZY must be 36.

Next, you can note that both triangles have the same angles: 36, 54, and 90.  This means that the triangles are similar, which also means that their side ratios will be the same. You just need to make sure that you’re matching up sides based on the angles that they’re across from.

You’re given the ratio of AC to BC, which in triangle ABC is the ratio of the side opposite the right angle (AC) to the side opposite the 54-degree angle (BC). In triangle XYZ, those sides are XZ and XY, so the ratio you’re looking for is .

Since lines x and y will add to a total of 180 degrees, you have two equations to work with:

x + y = 180

x = 3y

This means you can substitute 3y for x in order to solve for y:

3y + y = 180

4y = 180

y = 45

And since z will also sum with y to 180, then z must be 180 - 45 = 135 degrees.

If h is 121, then the angle immediately below h must be 59, as it is a supplementary angle formed by the diagonal line. Since g and k are parallel, this 59 degree angle must exactly match p as they are alternative interior angles.

Two angle rules are very important for this question:

1) The sum of the interior angles of a triangle is always 180. Here, since you have a 90-degree angle (CED) and a 35-degree angle (EDC) in the bottom triangle, you can then conclude that angle ECD must be 55.

2) Supplementary angles, angles that are adjacent to each other when two straight lines intersect, must sum to 180 degrees. If you know that ECD is 55, then ACE as a supplementary angle must form the other 125 degrees for those two angles to sum to 180. Therefore, the correct answer is 125.

An important thing to recognize in this problem is that you're dealing with two intersecting triangles that create external supplementary angles along the straight line on the bottom. To see this, consider the diagram below for which angles x and y have been added:

Angle y is an external supplementary angle to the triangle beside it so y = a + c. Why? Remember that y is supplementary to the angle beside it (x + 30) and (a + c) is supplementary to that same angle (the sum of interior angles of a triangle = 180.) Therefore y and (a + c) are identical. Anytime you have a straight line drawn off of a triangle you should recognize that the external supplementary angle equals the sum of the two opposite angles.

Using the same logic, you can see that x = b + d in the other intersecting triangle. Since the problem is asking for a + b + c + d, you should recognize that this question is really the same as what is x + y. Why? You can substitute x for b + d and y for a + c in the question stem. Since x + y = 180 - 30 on the straight line along the bottom, the correct answer is 150.

Note that another way to solve this problem involves seeing two large obtuse triangles: one with the angles a, c, and (x+30) and the other with the angles b, d, and (y+30). If you do that, you would have:

a+c+x+30=180, so a+c+x=150

b+d+y+30=180, so b+d+y=150

And you know that x+y+30=180 because x, 30, and y are all angles that make up the 180-degree straight line across the bottom of the figure. So x+y=150.

You can then sum the triangle equations:

a+c+x+b+d+y=150+150=300

And then plug in x+y = 150 and you're left with a+b+c+d=150.

This problem hinges on two important geometry rules:

1) The sum of all interior angles in a triangle is 180. Here you know that in the top triangle you have angles of 30 and 80, meaning that the angle at the point where lines intersect must be 70, since 30+80=110, and the last angle must sum to 180.

2) Vertical angles - angles opposite one another when two straight lines intersect - are congruent. Because you have identified that the angle at the bottom of the triangle at the top is 70, that also means that the top, unlabeled angle of the bottom triangle is 70. That then lets you add 70+50+ as the three angles in the bottom triangle, and since they must sum to 180 that means that .