A line is a one-dimensional abstraction—infinitely long with no width. Two points determine a straight line; given any two points, there is exactly one straight line that passes through them.
A line segment is a section of a straight line of finite length with two endpoints. A line segment is named for its endpoints, as in segment AB below. The midpoint is the point that divides a line segment into two equal parts.
Example:
In the figure above, A and B are the endpoints of the line segment AB, and M is the midpoint (AM = MB). What is the length of AB?
Since AM is 6, MB is also 6, so AB is 6 + 6, or 12.
Two lines are parallel if they lie in the same plane and never intersect each other regardless of how far they are extended. If line ℓ1 is parallel to line ℓ2, you write ℓ1 || ℓ2.
Two lines are perpendicular if they intersect at a 90° angle. The shortest distance from a point to a line is the line segment drawn from the point to the line such that it is perpendicular to the line. If line ℓ1 is perpendicular to line ℓ2, you write ℓ1 ⊥ ℓ2. If ℓ1 ⊥ ℓ2 and ℓ2 ⊥ ℓ3, then ℓ1 || ℓ3.
An angle is formed by two lines or line segments intersecting at a point. The point of intersection is called the vertex of the angle. Angles are measured in degrees (°).
Angle x, ∠ABC, and ∠B all denote the same angle shown in the diagram above.
An acute angle is an angle whose degree measure is between 0° and 90°. A right angle is an angle whose degree measure is exactly 90°. An obtuse angle is an angle whose degree measure is between 90° and 180°. A straight angle is an angle whose degree measure is exactly 180° (half of a circle, which contains 360°).
The sum of the measures of the angles on one side of a straight line is 180°.
The sum of the measures of the angles around a point is 360°.
a + b + c + d + e = 360
Two angles are supplementary if together they make up a straight angle (i.e., if the sum of their measures is 180°). Two angles are complementary if together they make up a right angle (i.e., if the sum of their measures is 90°).
A line or line segment bisects an angle if it splits the angle into two smaller, equal angles. Line segment BD below bisects ∠ABC, and ∠ABD has the same measure as ∠DBC. The two smaller angles are each half the size of ∠ABC.
Vertical angles are a pair of opposite angles formed by two intersecting line segments. At the point of intersection, two pairs of vertical angles are formed. Angles a and c below are vertical angles, as are b and d.
The two angles in a pair of vertical angles have the same degree measure. In the previous diagram, a = c and b = d. In addition, since ℓ1 and line ℓ2 are straight lines, a + b = c + d = a + d = b + c = 180.
In other words, each angle is supplementary to each of its two adjacent angles.
If two parallel lines intersect with a third line (called a transversal), the third line will intersect each of the parallel lines at the same angle. In the figure below, a = e because the transversal intersects lines ℓ1 and ℓ2 at the same angle. Since a and e are equal, and c = a and e = g (vertical angles), you know that a = c = e = g. Similarly, b = d = f = h.
In other words, when two parallel lines intersect with a third line, all acute angles formed are equal, all obtuse angles formed are equal, and any acute angle is supplementary to any obtuse angle.
Now let’s use the Kaplan Method on a Problem Solving question dealing with lines and angles:
When analyzing a geometry question that includes a figure, you’ll want to consider the question stem and the figure together. Here, the question stem tells you that ℓ1 and ℓ2 are parallel. This is the time to use Critical Thinking to help you come to an efficient solution. Since ℓ1 and ℓ2 are parallel, you know that the corresponding angles created by the transversals (intersecting lines) will be of equal measure. Specifically, that means:
a = c
b = d
Looking at the figure again, you see that angles a and b, plus that third angle between them, compose a straight line, which is 180°. Since angle e is the “vertical angle” of that third angle, their measures are the same. So you can deduce:
a + b + e = 180
Your task is to determine which of the three Roman numeral statements are equal to 180. You just deduced a + b + e = 180, so if a statement is equal to a + b + e, then it will also be equal to 180.
Which Roman numeral statement should you check first? You could start with Statement III, which looks similar to what you’re after. It also appears in the most answer choices (tying Statement II), so that’s another good reason to start with Statement III.
Does c + d + e = a + b + e? You figured out earlier that a = c and b = d. Substituting a for c and b for d, you confirm that c + d + e does in fact equal a + b + e, which means that it also equals 180. The right answer must contain Statement III, so you can eliminate (A) and (B). (You can also use Critical Thinking to note that c, d, and e are vertical angles for the triangle in the middle of the picture; the angles in a triangle always sum to 180°.)
Looking at the answers that remain, you see that Statement II appears twice, whereas Statement I appears only once. Evaluate Statement II next.
Does c + e = a + b + e? You deduced earlier that c + d + e = a + b + e. The only way, then, that c + e would also equal a + b + e is if d = 0. You have no idea what its value is, but it’s definitely not 0. So Statement II is not part of the right answer. That lets you eliminate (D) and (E), leaving only the correct answer, (C).
You could also have solved by picking numbers. A Roman numeral statement will be part of the right answer only if it must equal 180. So if you can pick numbers to get a statement to equal something other than 180, you can eliminate it. Let’s say that you pick c = 30 and d = 40. Since a and c are corresponding angles, a = 30. Similarly, b = 40. That means that the unnamed angle between a and b must equal 180 − (30 + 40), or 110. Since e has the same measure, e = 110. Quickly plugging those numbers into the Roman numeral statements gives the following:
After you eliminate Statements I and II, only choice (C) is possible.
Finally, reread the question stem, making sure that you didn’t miss anything about the problem. You’re looking for the angles in the figure that add up to 180. You’ve determined that a + b + e = 180 and that a + b + e = c + d + e. Therefore, c + d + e = 180, making (C) the only correct answer.
Note: Not drawn to scale.
