First, a couple of relatively easy practice problems.
1) In the diagram above, what is the measure of angle y?
Statement #1: x = 30°
Statement #2: line AB is parallel to line CD
2) In the diagram above, line m and line n are parallel. Given that angle a = 40° and angle c = 55°, what is the measure of angle b?
- (A) 85°
- (B) 90°
- (C) 95°
- (D) 100°
- (E) angle b cannot be determined from the information given
3) In the diagram above, angle measures in degrees are marked as shown, and segment BC is parallel to line AD. What is the measure of angle E?
- (A) 30°
- (B) 35°
- (C) 40°
- (D) 45°
- (E) 50°
Solutions will come at the end of this article.
Basic Geometry on the GMAT
Some of the most fundamental geometry facts have to do with the special properties of parallel lines. These facts include what angles are equal, and which angles have other mathematical relationships, if the lines are parallel. The special properties of parallel lines are also directly connected to one of the most famous theorems in Geometry, the 180°-Triangle Theorem:
The sum of all three angles in any triangle equals 180°.
This is a fact true, not just for certain triangles, but for every possible triangle. A more dramatic way to say this would be: God Himself would not be able to create a triangle in the plane the sum of whose angles is not 180°.
(Cool fact that is 110% irrelevant to the GMAT: in some alternate, non-Euclidean geometries, there are no parallel lines possible, and in these geometries, the 180°-Triangle Theorem does not hold. Consider the surface of the Earth, which is approximately spherical. Consider a triangle formed by three points: (1) the North Pole, (2) the intersection of the Prime Meridian and the Equator, and (3) the intersection of the 90° West meridian and the Equator. That’s a triangle with three right angles!!)
Close but no cigar!
Important idea #1 in this context is: while there are a number of special geometry facts that are true for parallel lines, absolutely none of them are true for lines that are almost parallel. Almost parallel is absolutely worthless in geometry. This has an important implication for diagrams. Unless otherwise specified, all diagrams on the GMAT Problem Solving are drawn as accurately as possible. BUT, if two lines look parallel, you can’t assume they are parallel. Two lines that look parallel could be half a degree off from being truly parallel — that difference would not be visually apparent, but none of the special parallel-line facts would be true if the two lines are not exactly parallel. Your eyes can deceive you on this. You have to see, printed in black & white: the lines are parallel. Otherwise, you can’t assume anything.
If two lines are parallel
OK, if we are guaranteed that the lines are parallel, and another line intersects these parallel lines, what do we know? This diagram summarizes everything you will need to know.
Notice that we could divide these eight angles into “big” angles (angles 1 & 4 & 5 & 8) and “small” angles (angles 2 & 3 & 6 & 7). Here’s what’s true:
1. All the big angles are equal
2. All the small angles are equal
3. Any big angle plus any small angle equals 180°
There are all kinds of fancy geometry names these angles had back in high school geometry — for example, angles 3 and 6 are “alternate interior angles” (does that bring back pre-prom memories?) —- but for the purpose of the GMAT, you don’t need to know any terms more technical than “big angles” and “small angles.” Keep it simple. J
It may that this refresher cleared up a few things for you. If you found the three questions at the beginning of this article challenging, then take another look at them before reading the solutions below. Here’s a slightly more challenging question along the same lines, for practice.
Practice question solutions
1) First of all, notice from the diagram: if the lines are parallel, then the two angles would be equal, x = y; but, if the lines are not parallel, we can conclude absolutely nothing about x & y. Furthermore, a visual assessment is not enough — yes, the lines look parallel, but that’s not a guarantee that they are parallel, and without this guarantee, we can do nothing.
Statement #1: Here, we know angle x, but we don’t know whether the lines are parallel, so we can conclude nothing. Alone & by itself, this statement is insufficient.
Statement #2: Now, we know the lines are parallel, but we don’t know the values of any variables. Alone & by itself, this statement is insufficient.
Combined Statements: Now, we know the lines are parallel, and we know x = 30°, so this means y = 30°. We now have definitive information that allows us to answer the prompt question. Together, the statements are sufficient.
Answer = C
2) Imagine we constructed a new line, parallel to lines m & n, through the vertex of the “crook” between the lines. This splits angle b into two smaller angles, b1 & b2.
Notice, by the parallel lines properties, b1 = a and b2 = c, so b = b1 + b2 = a + c. This means b = 40° + 55° = 95°. Answer = C
3) This is a very tricky one. First of all, in triangle ABC, the sum of the three angles must be 180°. We are given two angles, so we know the third angle, the angle at vertex C, must be 40°. Now, because segment BC is parallel to line AD, we know this angle at C, 40°, must be equal to angle EAD. Therefore, angle EAD = 40°. Now, we know two of the three angles in triangle EAD, and we know their sum must be 180° also, so the angle at E must be 45°. Answer = D