One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in for angle , you should notice that even though there are two angles remaining to solve for within triangle ABC, those two angles each equal the same thing. So since the sum of all three has to be , and angle a already accounts for , has to equal the remaining . means that . And then you get to use the same logic all over again. Within triangle BCD, you know that and that the sum of the three angles must be . That means that , so .

One extremely helpful tool that you’ll often find in your geometry toolkit is the presence of isosceles triangles. Here, once you’ve filled in for angle , you should notice that even though there are two angles remaining to solve for within triangle ABC, those two angles each equal the same thing. So since the sum of all three has to be , and angle a already accounts for , has to equal the remaining . means that . And then you get to use the same logic all over again. Within triangle BCD, you know that and that the sum of the three angles must be . That means that , so .

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.

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.

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.

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 .

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 .

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.

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 .

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 .