Angles and Lines
Angles and Lines
An angle is a geometric figure consisting of two lines, rays, or line segments that share a common endpoint called a vertex:
In the angle above, the vertex is point A. The angle can be called either angle CAB or angle BAC. The only rule for naming an angle is that the vertex must be the middle “initial” of the angle. The SAT may also refer to angles using symbols: .
Degrees
Angles are measured in degrees, which have nothing to do with Nelly or temperature. Geometric degrees are sometimes denoted by this little guy: º. There are 360º in a complete rotation around a point (that’s why a circle has 360º).
Two Lines Meet in a Bar...
When two lines meet, they produce angles. And when two lines meet, they form four angles! That must be exhausting.
These aren’t just any old four angles, either. Together, the angles encompass one full revolution around the point of intersection of the two lines. So, the four angles produced by two intersection lines total 360º: angle a + b + c + d = 360º.
If you know the value of three of the four angles formed by intersecting lines, you can always find the value of the fourth.
Types of Angles
The different types of angles are named and categorized according to their number of degrees.
Zero Angles
A zero angle has, you guessed it, . To visualize a zero angle, first picture two lines that form some angle greater than . Then imagine one of the lines rotating toward the other until they both fall on the same line. The angle they create has shrunk from its original measure to, forming a zero angle:
Right Angles
For some reason, an angle with a measure of 90º is called a right angle. For some other reason, right angles are symbolized with a square drawn in the corner of the angle. Whenever you see that reliable little square, you know you’re dealing with a right angle.
Right angles are extremely important on the SAT. They appear in math questions all the time. Knowing their special properties will help you solve right angle questions. We give you a detailed look at those properties a little later in this chapter. For now, just remember: Always be on the lookout for right angles on the SAT.
Straight Angles
An angle with a measure of 180º is called a straight angle. It looks just like a line. Don’t confuse straight angles with zero angles, which look like a single ray.
Acute and Obtuse Angles
An angle can also be classified according to whether its measure is greater or less than 90º. If an angle measures less than 90º, it’s called an acute angle. If it measures more than 90º, it’s called an obtuse angle. Right angles are neither acute nor obtuse. They’re just right. In the picture below, is acute, while is obtuse.
Complementary and Supplementary Angles
Special names are given to pairs of angles whose sums equal either 90º or 180º. Angles whose sum is 90º are called complementary angles, while angles whose sum is 180º are called supplementary angles.
In the picture above, and are complementary, since together they make up a right angle. Angles and are supplementary, since they make up a straight line.
On the SAT, you’ll have to use the rules of complementary and supplementary angles to figure out the degree measure of an angle.
In the diagram below, AC is a line. What is x in degrees?
The picture tells you that is 113º, but how many degrees is Well, since you know that AC is a line, must be a straight angle (meaning it equals 180º). So and are supplementary angles that add up to 180º. To find out the value of you can simply take 180º and subtract 113º. = 67º.
Vertical Angles
When two lines (or line segments) intersect, the angles that lie opposite each other, called vertical angles, are always equal.
Angles and are vertical angles and are therefore equal. Angles and are also vertical (and equal) angles. We promise that the SAT will ask you at least one question involving vertical angles. Promise.
Parallel and Perpendicular Lines
Pairs of lines that never intersect are parallel. Parallel lines appear to line up right next to each other because they never meet in space. However, on the SAT, you can’t assume two lines are parallel just because they look parallel. The SAT will tell you if two lines are parallel.
Lines (or segments) are perpendicular if their intersection forms a right angle. And if one of the angles formed by the intersection of two lines or segments is a right angle, then all four angles created will also be right angles. By the way, this also shows that the degree measurement of four angles formed by two intersecting lines will add up to 360º, since 90º + 90º + 90º + 90º = 360º.
As with parallel lines, don’t assume that lines on the SAT are perpendicular unless the SAT tells you they are. The SAT will tell you either in words (“lines a and b are parallel”) or by using the little reliable box to show that the angles are 90º.
Parallel Lines Cut by a Transversal
A transversal is a line that cuts through two parallel lines. The SAT loves to cut parallel lines with transversals. Who knows why? Not us. But we know how to get those questions right, and you will too.
A transversal creates eight angles when it intersects with two parallel lines. The eight angles created by these two intersections have special relationships to each other.
You now have a choice to make: (1) spend all day figuring out these relationships, or (2) use our list.
Good choice:
  • Angles 1, 4, 5, and 8 are equal to each other because they’re vertical angles.
  • Angles 2, 3, 6, and 7 are equal to each other because they’re vertical angles.
  • The sum of any two adjacent angles, such as 1 and 2 or 7 and 8, equals 180º, because these are supplementary angles.
By using these three rules, you can figure out the degrees of angles that may seem unrelated. For example, since angles 1 and 2 sum to 180º, and since angles 2 and 7 are equal, the sum of angles 1 and 7 also equals 180º. The SAT will almost definitely include a question that asks you to solve for an angle whose measurement at first glance seems impossible to determine.
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