Hi, there. I'm happy to help.grandh01 wrote:In TRIANGLE FGH, if FG = x, GH = y, and FH = z, which of the three
angles has the greatest degree measure?
(1) x - y = y - z
(2) x > z
You said "In angle FGH", but assume you meant "in triangle FGH" --- otherwise, the question would make no sense. Be careful with all mathematical details, even the wording of math problems. Success in math absolutely depends on handling every detail with precision.
We want to know which angle in the triangle has the greatest measure.
Statement #1: x - y = y - z
Add y and add z, and we get
x + z = 2y
Of course, because it's a triangle, we know
x + y + z = 180
Because we know x + z = 2y, substitute the left side into this equation
x + y + z = x + z = y = 2y + y = 3y = 180
y = 60
Statement #1 allows us to calculate the numerical value of y. Now, going back to the statement, it says that x is just as far above 60 as z is below 60. Naively, you may think: this must mean x is the biggest angle. BUT, it could be the case that x = y = z = 60, an equilateral triangle, in which case the biggest angle is any of the three of them. Because that's a possibility, we don't have sufficient information to answer the question. This is not sufficient.
We have to be very careful to forget everything we observed in Statement #1, and not drag it accidentally into Statement #2 (a classic DS mistake!)
Statement #2: x > z
This is a comparatively paltry statement that tells us very little. Here are two triplets that work for a triangle:
x = 80, y = 50, z = 30
x = 2, 7 = 177, z = 1
In the first triplet, x is the largest angle. In the second, y is the largest. Since different answers to the prompt result from different choices consistent with this statement, this statement is not sufficient.
Combined Statements
Combined, we have a lot of info now --- we know that y - 60, and we know that x > z, so the equilateral triangle case is excluded. This means, x must be bigger than 60 and x must be smaller than 60. That automatically means: x is the biggest angle. We now can give a definitive answer to the prompt. Combined, the statements are sufficient.
Answer = [spoiler]C[/spoiler]
Does all this make sense?
Mike
