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What is the value of w in terms of x and y?
Note: Figure not drawn to scale
repeated angles in nested triangles
Here's the same diagram with letters. First of all, look at the big triangle, triangle ACE. We know the three angles in this or any triangle must add up to 180°. Let's say that ∠E = k. Then we know
x + y + k = 180, or k = 180 – x – y
Now, look at the angles around point F. Those three angles form a straight line, that is to say, a 180° angle, so the sum of the three angles there must also be 180°. Since one is x and one is y, the other has to be k --- ∠DFE = k
Now, look at triangle DEF. The sum of the three angles in this triangle must also be 180°.
w + k + k = 180°
w = 180 – 2k
Now, we have an expression for w in terms of k. To express w in terms of x & y, we need to substitute the expression for k above, k = 180 – x – y
w = 180 – 2(180 – x – y)
w = 180 – 360 + 2x + 2y
w = 2x + 2y – 180
Answer = (A)
FAQ: Can we assume that figure ABCDEF is a triangle? Is this assumption valid? If the figure is not drawn to scale and no angle is specified exactly, how can we assume that the overall figure is a triangle and not a polygon that is very similar to a triangle?
A: That's a really great question!
We can always assume that lines on the test that look like straight lines are straight lines unless we are told otherwise; we can assume that there are no hidden "bends." So by these assumptions, we know that the big triangle, is, in fact, a triangle.
FAQ: How do we know that angle DFE is equal to k?
A: There are two rules we need to use here:
- The angle of a line is 180˚
- The sum of the three angles in a triangle is 180˚
Now, notice in the question that the triangle ACE has three angles: x, y, and the angle of CEA. We'll call that third angle k.
So x + y + k = 180, because they are the three angles of a triangle.
Meanwhile, near the point F, we see three angles that add up to a line. The first two are given as x and y (the same as the angles in the corners of triangle ACE!). The third angle, DFE, is not given. But we know that x + y + DFE = 180, because they add up to a line.
So, then, we know that x + y + k = 180 and that x + y + DFE = 180. That must mean that k = DFE.
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.