This problem heavily leverages two rules:

1) The sum of the angles in a triangle is 180.

2) Supplementary angles - adjacent angles created when one line intersects another - must sum to 180.

Here you can first leverage the 140-degree angle to fill in that its adjacent neighbor - its supplementary partner - must then be 40. and that gives you two of the three angles in the uppermost triangle: 20 and 40. You can use that to determine that the third angle must then be 120.

From there you should see that the 120-degree angle is a vertical angle, meaning that its opposite will also be 120. And that gives you a second angle in the lower-right triangle. Knowing that you have angles of 15 and 120 means that the third angle of that triangle must be 45.  And since that angle is supplementary to angle x, x must then be 135.

This problem heavily leans on two important lines-and-angles rules:

1) The sum of the three interior angles of a triangle is always 180.

2) Supplementary angles - angles next to each other formed by two lines intersecting - must also sum to 180.

Here you can then determine that the angle next to the 95-degree angle is 85, and since that angle is the lower-right hand angle of the little triangle at the top, you can close out that triangle. With angles of 40 and 85, that means that the lower left hand angle must be 55.

From there, you can use the fact that parallel lines will lead to congruent angles. Since lines  and  are parallel, the angle next to  will be 55 degrees, meaning that  will then be 125.

This problem tests two important rules. For one, the angle measure of a straight line is 180. Here if you follow line  you can see that its angle is broken in to three segments:  and the blank angle between them. Those three angles must sum to 180, so if you already know that  and , then the unlabeled angle between them must equal  so that .

Next, know that when lines intersect to form angles at a particular point, opposite (vertical) angles are congruent. The angle of measure  is directly opposite the angle you just calculated to be  degrees, so  has to be  as well.

The third side of a triangle is always greater than the difference of the other two sides and less than the sum of the other two sides. This applies to every side of a triangle. In other words, you can arbitrarily pick any one side to be the “third side,” and then that side must be greater than the difference of the other two and less than the sum of those two.

Here that means that the third side must greater than the difference of  and . Since , that means that  is not an option. It also means that the third side must be less than the sum of  and . Since , that rules out  as an option. You know that the third side must be greater than  and less than : only , option II, fits.

Whether you use the Pythagorean Theorem or you quickly recognize that this is a  triangle, you can solve for the length of side .

You can prove that this is a  triangle because the hypotenuse is twice as long as one of the legs of the triangle. This then fits the side ratio , so you can fill in the middle side as .

Or you can use Pythagorean Theorem. Since you know side YZ is the hypotenuse, you can set it up as . This means that , so  and .

Once you've determined that , you can calculate the area. The area is half the base times the height, where the base and height form a -degree angle. This means that you can use  and . .

There is a rule about isosceles triangles that isn't obvious the first time you see it, but that the SAT likes to test: if an isosceles triangle includes a -degree angle, then it must be an equilateral triangle.

You can prove this by testing the cases: if you know that you have an isosceles triangle with a measure of  for one angle, then you can call your angles , , and . You know that  must sum to  (a rule of triangles), and that one of the following cases must be true:

 matches . if that's true, then your three angles are , , and . Since , that means that . Here, all three angles are .

 matches . The same as the above.

 matches . This then means that , so . If , then  so x = 60, which means that . Here, again, all angles are .

This problem blends two important rules related to triangles:

1) The sum of the interior angles of a triangle is .

2) In an isosceles triangle, two angles have the same measure.

Here, although you're not explicitly told which angles have the same measure, you can deduce that it must be angles XZY and YXZ - the two angles that do not measure  degrees. Note that if  were to be the "match," then you would already have  degrees assigned to those two angles, but that would violate the  rule.

Therefore, you know that your three angles are , , and  (where  represents the unknown, matching angle). You can then say that , so  and .

One of the most convenient things about isosceles right triangles is that you can use the two shorter sides as the base and the height to find the area, since they're connected by a right angle:

So if you know that  is the area in an isosceles right triangle, you can use  to solve for  as the length of each of the shorter sides. This means that , which you can simplify to:

And then solve for .

Because this is an isosceles right triangle, the sides will form the ratio , meaning that the hypotenuse will measure . If you sum the two shorter sides of  with the hypotenuse of , you reach .