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| Quantity A | Quantity B |
| v | w |
Take a look at another example of a Geometry Quantitative Comparison in which both quantities are unknown values:
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| Quantity A | Quantity B |
| v | w |
As in the preceding example, there are no numbers given, so an exact value for any of these angles is impossible to determine. This does not, however, mean that the answer is necessarily (D).
What you need to know is the relative size of v and w. As this type of problem gets more difficult, it becomes more difficult to establish what you know. An important feature of this diagram is that the intersection of the three lines creates a triangle. Triangles, when they appear, are often very important parts of diagrams, because there are many rules related to triangles that test makers can make use of.
This question appears to be about angles. After all, the values in both quantities are angles. Create variables to represent the three angles of the triangle:
Part of the challenge is the fact that there are actually many relationships, and thus many equations you could create. For instance:
But not all of these relationships will help determine the relative size of v and w. You need to find relationships that will allow you to directly compare v and w.
The best bet for a link between v and w is the triangle in the center of the diagram. Try to express v and w in terms of x, y, and z.
Begin with v. Angles v and y are vertical angles, and thus equal. In Quantity A, replace v with y:
| Quantity A | Quantity B |
| v = y | w |
Now, if you can express w in terms of y, then you may be able to determine the relative size of v and w.
Based on the diagram, you know w + z = 180, so w = 180 − z.
| Quantity A | Quantity B |
| y | w = 180 − z |
You can’t directly compare y and (180 − z), so keep going. Try to find an equation that links y and z.
Remember, you also know that x + y + z = 180. Isolate z:
x + y + z = 180
z = 180 − x − y
Now, substitute (180 − x − y) for z in Quantity B:
| Quantity A | Quantity B |
| y | 180 − (180 − x − y) = x + y |
Now you can directly compare the two quantities. You know that x and y both represent angles, and so must be positive, so x + y must be greater than y. The correct answer is (B).
By the way, you’ve just proven that the exterior angle (w) is equal to the sum of the two remote interior angles (x + y). This is true in every case:
This is a good rule to know!