When two lines intersect each other, they create four angles:
In this case, the angles opposite one another are called vertical angles. As you can see, vertical angles are by definition always equal to each other. Therefore, in this case, 1 = 2, and 3 = 4. Another way to say this is that both "little angles" will equal each other and both "big angles" will equal each other. Also, note that a "little angle" plus a "big angle" will always equal 180°.
Parallel Lines
Two lines that never intersect and never get closer or farther away from one another are called parallel lines. || is the symbol for parallel lines. In the figure to the right, A || B, or line A is parallel to line B.
When two parallel lines are cut by a third line, they form a system of Vertical Angles.
In this figure, we can see that 1 =4 and 5 =8. But, because they are formed by two parallel lines, they are all equal to each other, so: 2 = 3 = 6 = 7.
There are many terms to describe these angles, such as “alternate interior” or “alternate exterior,” but these terms are not used on the test. For the GRE, it is simply enough to know that all the little angles will always be equal to each other and all the big angles will always be equal to each other.
Additionally, you should realize that any little angle added to any big angle will always equal 180°.
Perpendicular Lines
When two lines intersect each other at a 90° angle, they are
said to be perpendicular to one another.
is the symbol for perpendicular, and mn
means that m and n are perpendicular to each other.
Midpoint and Bisect
Two more vocabulary words you need before you can finish this section. Midpoint is the center point
of any line. In the example below, point B is the midpoint
of line AC.
Bisect means “to cut in half”. Anything may be bisected: a line, an angle, a circle, a square. In the figure above, point B “bisects” line AC.
Example 1
A complement of an angle the second of a pair that adds to 90°. A supplement is the second of a pair that adds to 180°. If the complement of an angle is one quarter of its supplement, what is the angle?
Solution
Let x be the angle.
Its complement y is y = 90 - x
Its supplement z is z = 180 - x
90 - x = (180 - x)/4
If y = z/4, we can set this.
360 - 4x = 180 - x
Multiply both sides by 4.
180 = 3x
Subtract 180 from both sides and add 4x to both sides.
is the symbol for perpendicula
