Question

If one of the angles of a quadrilateral measures 75 degrees and the degree of the other angles are in ration of 2 : 4: 9, what is the degree measure of the largest angle of the quadrilateral?

Answer 1

a. 171
b. 168
c. 135
d. 105
e. 63

Answer 2

The sum of the other angles is 360 - 75 = 285.

So the ratios split 285.

As 2+4+9 = 15, and 285/15 = 19.

So the ratios in proportion to 285 are 2*19 : 4*19 : 9*19 = 38: 76: 171.

Hence A is the correct answer....

Answer 3

@Sakigurl

If this is an SAT problem, you would not necessarily want to work it in the way you would in Geometry class.

In Geometry class, you might think as follows:

Well, there are 360 degrees as the sum of the interior angles of a quadrilateral. Then, 75 degrees has already been taken up by that 75 degree angle which leaves 285 degrees for the sum of the other three angles.

Those angles are in an extended ratio which means the three angles have some common factor. The common factor might be something like ten, [20:40: 90 ] and "fit" the 2:4:9 ratio BUT those angles do not sum to 285.

So, then, an equation is written.

Let one angle of the quadrilateral be 2x, the second, 4x, and the third 9x. Then,
2x + 4x + 9x = 285 and after the crinkum-crankum work, x = 19. That gives the angle measures of 38, 76, and 171 as Vishal has written above.

Do we really need to crank out all those measures to solve the SAT problem?

The question asks about the largest angle. Before doing the Algebra, you know it is going to be the larger of 75 and whatever 9x turns out to be. So, once you get the 19 and see that 9x = 171, that is the largest angle.

Vishal has done an excellent job of streamlining the problem's solution. Would that be considered as a complete solution in Geometry class. That depends on the solution instructions and teacher expectations.

The SAT designers are not going to look at any work. In practicing for the SAT, you can decide just how much work you need to show for you to get the correct answer.