Angles - GMAT Math Study Guide

Definitions

• Angle - A figure formed by two lines sharing a common endpoint.
• Ray - A straight line extending from a point.
• Acute Angle - An angle less than 90° and greater than 0°
• Obtuse Angle - An angle between 90° and 180°
• Right Angle - An angle that is exactly 90°
• Complementary Angles - Two angles whose sum is 90°
• Supplementary Angles - Two angles whose sum is 180°
• Adjacent Angles - Two angles that share a side (i.e., are adjacent).
• Angle Bisector - A ray that divides an angle into two equal parts.
• Vertical Angles - The two non-adjacent angles that share a vertex when two lines intersect.
• Alternate Interior Angles - When two parallel lines are cut by a transversal, the two angles on opposite sides of the transversal and between the parallel lines (an example below).
• Alternate Exterior Angles - When two parallel lines are cut by a transversal, the two angles on opposite sides of the transversal and outside the parallel lines (an example below).

Types of Angles

Acute Angle

An acute angle is one whose measurement is less than 90°. Angle A, shown below, is an acute angle.

Obtuse Angle

An obtuse angle is one whose measurement is greater than 90° and less than 180°. Angle B, shown below, is an obtuse angle.

Right Angle

A right angle is one whose measurement is exactly 90°. Angle C, shown below, is a right angle.

Note: The lines that surround the letter C are the formal way of denoting a right angle. If you see this symbol, you know that you are dealing with a right angle. You cannot assume an angle is right simply because it appears to be right.

Intersecting Lines

When two lines intersect, they form four angles. Angles that are not adjacent (i.e., do not share a common side) are called vertical angles and these angles are congruent (or equal in measurement).

Due to the properties of intersecting lines, vertical angles are congruent. Consequently, angle D = angle F and angle G = angle E.

Parallel Lines Cut by a Transversal

When two parallel lines are cut by a transversal (i.e., a third line intersects the two parallel lines), a number of relationships exist between the resulting angles.

• Alternate Interior Angles Are Equal: K = L; O = J
• Alternate Exterior Angles Are Equal: H = M; N = I
• Corresponding Angles Are Equal: K = N; J = M; H = O; I = L
• Non-Alternate Interior Angles Are Supplementary: L + J = 180; K + O = 180

It is important to note that if two lines cut by a transversal have any of the above properties, then the two lines must be parallel. For example, if alternate interior angles are equal, then the two lines cut by a transversal must be parallel.

Sum of Angles

The sum of the interior angles of a polygon is 180*(n-2) where n is the number of sides. For example, the sum of the interior angles of a triangle is 180 = (180)(3-2) while the sum of the interior angles of a square is 360 = (180)(4-2).

Types of GMAT Problems

1. Problems Involving Various Aspects of Angles.

   Many times, a problem will require the use of several properties of angles. If stuck on a problem, always write out as much of the information as can be deduced from the given information.

   

   A)	20°
   B)	55°
   C)	35°
   D)	125°
   E)	45°

   Correct Answer: D

   1. Line l is parallel to line m since the unlabeled transversal forms the same degree angle with both lines l and m; (the angles are 90 degrees in this instance, but the angle measurement does not have to be 90 in every instance of a parallel line cut by a transversal; in the general case, the angles cut by the transversal simply must be the same measurement, which guarantees that they will never intersect and, therefore, the lines cut by the transversal will be parallel).
   2. Angle Y shares the same relationship with lines n and m as the 55° angle shares with line n and l. Since lines l and m are parallel, y = 55°
   3. Since m is a line, adjacent angles along m must sum to 180. Thus, x + y = 180.
   4. Substituting in 55 for y, x + 55 = 180
      x = 125°
   5. Note: You could also solve this problem by using the fact that all triangles must sum to 180 and there is a triangle in the center of the picture. The diagram is set up so that seeing this way of solving the problem is more difficult, so if you cannot see it or do not understand the logic below, do not be concerned.
      The top angle will be 90-55 = 35
      The bottom left angle will be 90
      The bottom right angle will be 180-x
      
      35 + 90 + (180-x) = 180
      125 + 180 - x = 180
      125 - x = 0
      125 = x
2. Determining if Lines are Parallel

   At times, diagrams in GMAT problems make lines appear parallel. However, you cannot assume that two lines are parallel or an angle is a right angle simply because it looks that way. In order to determine whether lines are parallel, assess the relationship between the angles.

   Which of the following depicts line l parallel to line m?
   

   A)	A
   B)	B
   C)	C
   D)	A and C
   E)	B and C

   Correct Answer: C

   1. If two lines are parallel, their consecutive interior angles sum to 180. In this case, angles 2 and 3 are consecutive interior angles.
   2. Consequently, in order to solve the problem, we must find the measurements of angles 2 and 3 and see if they add to 180.
   3. Since the sum of angles along a line must equal 180, we can write the following equations:
      (Angle 1) + (Angle 2) = 180
      (Angle 2) = 180 - (Angle 1)
      
      (Angle 3) + (Angle 4) = 180
      (Angle 3) = 180 - (Angle 4)
   4. In figure A, angle 2 = 60 and angle 3 = 110; 60 + 110 = 170
      In figure B, angle 2 = 60 and angle 3 = 100; 60 + 100 = 160
      In figure C, angle 2 = 70 and angle 3 = 110; 70 + 110 = 180
   5. Figure C contains parallel lines while figures A and B do not.

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