**Lines & Angles**

When two straight lines meet at a point, they form an angle. The point is called the vertex of the angle, and the lines are called the sides of the angle.

The angle to the right can be identified in three ways:

1. ∠x

2. ∠B

3. ∠ABC or ∠CBA

When two straight lines meet at a point, they form four angles. The angles opposite each other are called vertical angles, and they are congruent (equal). In the figure to the right, a = b, and c = d.

a = b and c = d

Angles are measured in degrees, °. By definition, a circle has 360°. So an angle can be measured by its fractional part of a circle. For example, an angle that is $\frac{1}{360}$ of the arc of a circle is 1°. And an angle that is $\frac{1}{4}$ of the arc of a circle is $\frac{1}{4}\times 360=90\xb0$.

There are four major types of angle measures:

An **acute angle** has measure less than 90°:

A **right angle** has measure 90°:

An **obtuse angle** has measure greater than 90°:

A **straight angle** has measure 180°:

**Example:**

In the figure to the right, if the quotient of a and b is 7/2, then *b* =

Since *a* and *b* form a straight angle, *a + b* = 180. Now, translating “the quotient of *a* and *b* is 7/2” into an equation gives $\frac{a}{b}=\frac{7}{2}$. Solving for a yields $a=\frac{7}{2}b$ . Plugging this into the equation *a + b* = 180 yields

$\frac{7}{2}b+b=180$

7*b* + 2*b* = 360

9*b* = 360

*b* = 40

The answer is 40.

**Example:**

**Column A**

*y*

**Column B**

90

According to the figure above, it is true that:

(A) Column A is larger

(B) Column B is larger

*x*and 2

*y*– 40 represent vertical angles, 4

*x*= 2

*y*– 40. Since 3x and y form a straight angle, 3

*x*+

*y*= 180. This yields the following system:

4

*x*= 2

*y*– 40

3

*x + y*= 180

Since you have two equations with the same variables, you can solve for y by isolating a variable (either x or y) in both equations and setting the equations equal to each other. After you have solved for one of the variables, plug it into one of the original equations to solve for the other variable.

Isolating y in both equations from above yields the following equations:

*y*= 2

*x*+ 20

*y*= 180 - 3

*x*

Now, by setting these two equations equal to each other you can solve for x.

2

*x*+ 20 = 180 - 3

*x*

Solving for x yields x = 32

Plug the value of x into one of the original equations to solve for y. Solving this system for *y* yields *y* = 84. Hence, Column B is larger and the answer is (B).

Two angles are supplementary if their angle sum is 180°:

Two angles are complementary if their angle sum is 90°:

*l*

_{1}⊥

*l*

_{2}

Perpendicular lines meet at right angles.

**Caution:**Since figures are not necessarily drawn to scale on the GRE, do not assume that two lines that appear to be perpendicular are in fact perpendicular. You must see a small box at the angle, or the perpendicular symbol (⊥), or be told that the lines meet at right angles.

**Two lines in the same plane are parallel if they never intersect. Parallel lines have the same slope.**

When parallel lines are cut by a transversal, three important angle relationships exist:

Alternate interior angles are equal.

Corresponding angles are equal.

Interior angles on the same side of the transversal are supplementary.

The shortest distance from a point to a line is along a new line that passes through the point and is perpendicular to the original line.