Working With Unusual Figures in Geometry Questions
Today’s article comes courtesy of Veritas Prep GMAT instructor, Valerie Browning.
In Geometry, we often come across unusual figures. This can throw off our mind a bit, but it is important to remember: just use what you already know. Don’t let the unusual shapes take up too much time on the GMAT. Let’s take the following example, very similar to a problem a student emailed me this week.
In this figure, what is the sum of angles a, b, c, d, and e?
We obviously don’t memorize anything about five pointed stars, since it would be so infrequently used, but let’s take a look at what we do know here.
We know that there are 5 triangles and a pentagon in this shape.
We know that each triangle has 180 degrees in it.
We know that a pentagon has 540 degrees in it. (You might know a formula for this, or you might be able to draw a line through a pentagon and make it into a four-sided figure and a triangle, and then know that 180 degrees in the triangle plus 360 degrees in the quadrilateral mean that we have 540 degrees total.)
We know that there isn’t a “There is not enough information given to answer the question” answer choice, so we MUST have enough here.
We also know that a straight line has 180 degrees.
Let’s look at the pieces here.
Since we don’t know anything about the relative sizes of the triangles involved or the angles within them, we can assume that this problem will be the same for ANY five-pointed star. Let’s make this one a “regular” five pointed star, then, to make it easier on ourselves. By “regular” I mean one that has identical triangles for each of the points, and one that has a pentagon with five equal angles in the center of the figure.
Now, if the pentagon has five equal angles, then each one MUST be equal to 108, since 540 divided by 5 is 108.
Now, take a look at the angles of the pentagon as they relate to the triangles. We can take one of the non-labeled points of each triangle plus one of the pentagon angles and make 180 (a straight line) in each triangle.
If these triangles are all isosceles, as we have already decided, and the pentagon is regular, then in each triangle, we have a pair of 72 degree angles at the base of each triangle.
I’ve labeled one of them here.
This means that e, in this case, must be equal to:
If e is 36 and all the triangles are the same, then we have degrees in the sum of all the points.
Let’s talk through what this means for other problems.
Always break the shapes into pieces you recognize.
Look for patterns. If the figure isn’t drawn to scale, but there is a possible answer (i.e. no “we don’t have enough information” answer choice) then we can make it a regular figure (like an equilateral triangle) to make solving simpler. This makes for fewer variables, and fewer calculations, which is always a good thing.
Use the rules you know – when in doubt, go with straight lines and triangles, since the GMAT writers seem so fond of these!
For more GMAT tips like this one, make sure to visit the Veritas Prep blog!