Math Study Guide for the SAT Exam

Geometry: Lines and Angles

Items relating to geometry on the SAT will assume you know the basic geometry concepts from pre-high school math and that you learned how to expand upon these concepts during geometry courses in high school. Here are some things to review.

Lines

Lines are one-dimensional figures that extend through two points to infinity. Consequently, lines do not have a defined length. Though this is an important concept in geometry, calculating measurements with lines often requires you to rely on what you learned in algebra.

Line Segments

The portion of a line between, and including, two points is called a line segment. Line segments have a definite length. On the SAT, you will be asked to determine the length of line segments based on given information. For instance, consider the line segment \(AC\) below, which includes point \(B\). If line segment \(AC\) measures \(12\), line segment \(AB\) measures \(3x\), and line segment \(BC\) measures \(4x\), what is the length of \(AB\)?

Since \(B\) lies along \(AC\), the length of \(AB\) combined with the length of \(BC\) yields the length of \(AC\). As a result, an algebraic equation can be generated from the information provided:

\[AB + BC = AC\] \[3x + 4x = 12\] \[7x = 12\] \[x = \frac{12}{7}\]

With this information, we can substitute the known value of \(x\) into the expression to determine the length of \(AB\):

\[3x = 3 \cdot \frac{12}{7} = \frac{36}{7}\]

Line Segment Midpoints

The midpoint of a line segment is the point along the segment that divides the segment into two equal portions.

Consider segment \(XZ\) below, which has the midpoint \(Y\). If \(XY\) measures \(3x + 1\) and \(YZ\) measures \(x + 2\), what is the length of \(XZ\)?

Because \(Y\) is the midpoint, \(XY\) and \(YZ\) are of equal length. That means we can set their lengths equal to each other to solve for the unknown \(x\):

\[XY = YZ\] \[3x + 1 = x + 2\] \[2x = 1\] \[x = \frac{1}{2}\]

Now, we substitute this value into the sum of \(XY\) and \(YZ\) to determine the length of \(XZ\):

\[XY + YZ = XZ\] \[3 \cdot \frac{1}{2} + 1 + \frac{1}{2} + 2 = XZ\] \[\frac{3}{2} + \frac{1}{2} + 3 = XZ\] \[5 = XZ\]

Types of Lines

You will need to know these various types of lines and what defines them:

• Parallel lines are lines that never intersect because they have the same slope but different \(x\)- and \(y\)-intercepts when graphed on a coordinate plane. The only way that parallel lines can intersect is if they are the same line, in which case they intersect at all points.

• Perpendicular lines are lines that intersect at a \(90\)-degree angle. The slopes of perpendicular lines are negative reciprocals of each other. For example, if the slope of line \(a\) is \(2\), and line \(b\) is perpendicular to line \(a\), line \(b\) has a slope of \(-\frac{1}{2}\).

• A transversal line is a line that passes through two parallel lines. Because parallel lines have the exact same slope, the angles formed by a transversal and one of the parallel lines are equivalent in measure to the corresponding angles formed on the other parallel line. We discuss this topic in greater detail below.

Angles

When lines, line segments, or rays (a portion of a line with a fixed point on only one end) intersect, they generate angles. We define angles based on the degrees between the two lines. Straight angles measure \(180\) degrees and form a straight line. And a collection of angles around a point sum to \(360\) degrees (creating a circle).

Acute angles are less than \(90\) degrees, right angles are exactly \(90\) degrees, and obtuse angles are greater than \(90\) degrees.

Congruent and Adjacent Angles

When a transversal line passes through two parallel lines, it creates four pairs of corresponding angles that have specific relationships with each other. Each corresponding angle pair is congruent, meaning they have the same measure. In this diagram, the angles labeled \(a\), \(b\), \(g\), and \(h\) are congruent acute angles. The angles labeled \(c\), \(d\), \(e\), and \(f\) are congruent obtuse angles.

Furthermore, \(\angle a\) and \(\angle g\) are defined as a vertical angle pair, while \(\angle a\) and \(\angle h\) are alternate exterior angles, and \(\angle d\) and \(\angle e\) are alternate interior angles.

Complementary and Supplementary Angles

As stated, a straight line measures \(180\) degrees. Consider line \(PQ\) (shown below), which you may see designated by the symbol \(\overleftrightarrow{PQ}\). It has the ray \(RS\) (or \(\overrightarrow{RS}\)) emanating from point \(R\), which is somewhere between \(P\) and \(Q\). As you can see, two angles are created: one that is \(120\) degrees and one that is \(60\) degrees. These are known as supplementary angles, because when added together they equal \(180\) degrees.

In a similar fashion, angles that combine to \(90\) degrees are known as complementary angles.

Consider the following image with \(\angle X = 3x + 2\) and \(\angle Y = x + 6\). The two angles are complementary. Can you determine the value of \(x\)?

Because the angles are complementary, their sum is \(90\) degrees. So, you just need to set up an equation and solve for the unknown \(x\):

\[\angle X + \angle Y = 90^{\circ}\] \[3x + 2 + x + 6 = 90\] \[4x + 8 = 90\] \[4x = 82\] \[x = \frac{41}{2}\]