SAT Tip: The Basics of SAT Geometry
This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.
In Euclidean geometry, a line is defined as having only one dimension: length. It is usually described by an equation. An angle is formed when two lines intersect. Here are the necessary fundamentals to improve your scores on SAT geometry questions.
The general equation for a line is: ax + by + c = 0 for all values (x,y) that are on that line. In this form, the slope is –a/b and the y-intercept is –c/b.
The most common equation for a line is called slope-intercept form: y = mx + b, for all values (x, y) on the line. Here, m is the slope and b is the y-intercept.
A modified version of slope-intercept form is called point-slope form: y – y1 = m (x – x1) + b. This equation is helpful if you are given two points on the line, (x, y) and (x1, y1).
When two lines or line segments intersect at at least one point, an angle forms. The point of intersection is called the vertex. Angles are measured in either degrees or radians (but usually in degrees).
An acute angle is an angle whose measurement in degrees is between 0 and 90. A right angle is an angle whose measurement in degrees 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 degrees.
All the angles on one side of a straight line sum to 180 degrees. In the image above, a + b + c = 180 degrees.
All the angles around one point must sum to 360 degrees. In the same image, a + b + c + d + e = 360 degrees
Perpendicular lines are formed when the angle between two lines is 90 degrees. The shortest distance from a point to a line is a line with a length such that the two lines form a 90 degree angle.
Two angles are supplementary if they share one line; i.e., if the sum of their angles is 180 degrees. Two angles are complementary if together they make a right angle; i.e., if the sum of their angles is 90 degrees.
To bisect an angle means to cut it in half. The two smaller angles will then have the exact same measurement.
If two parallel lines intersect with a third line, the third line is called a transverse line. All vertical (opposite) angles are equal and all corresponding (located in the same relative position on the other parallel line) angles are also equal. Additionally, opposite interior angles are also equal. Finally, each adjacent angle is supplemental to the angle next to it since together, they form a line or 180 degrees.
In the figure to the left, x and y are parallel lines, and z is the transverse line. Therefore:
a = d = e = h
c = b = g = f
a = d, e = h, c = b, and g = f because they are vertical angles.
d = e and c = f because they are opposite interior angles.
a = e, d = h, c = g and b = f because they are corresponding angles.
Now you’re ready to tackle any geometry questions asking about parallel angles and lines.
Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.
For more SAT advice from Veritas Prep, watch “SAT Tip: How The Calculator Can Hurt Your Score On the SAT Math Section”