# 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. ∠

2. ∠

3. ∠

*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,

*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 1/360 of the arc of a circle is 1˚. And an angle that is 1/4 of the arc of a circle is .

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, 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 a/b = 7/2. Solving for a yields a = 7b/2. Plugging this into the equation a + b = 180 yields

7b + 2b = 360

9b = 360

b = 40

Example

Column A | Column B |

y | 90 |

Since 4
4

3
Solving this system for

Two angles are supplementary if their angle sum is 180˚:
Two angles are complementary if their angle sum is 90˚:

*x*and 2*y*– 40 represent vertical angles, 4*x*= 2*y*– 40. Since 3*x*and*y*form a straight angle, 3*x*+*y*= 180. This yields the following system:*x*= 2*y*– 403

*x*+*y*= 180*y*yields*y*= 84. Hence, Column B is larger and the answer is (B).Two angles are supplementary if their angle sum is 180˚:

Perpendicular lines meet at right angles.

**Caution:** Since figures are not necessarily drawn to scale on the test, 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:

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.