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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, 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 =
  1. 30
  2. 35
  3. 40
  4. 46
  5. 50
Solution

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/2 + b = 180
 
7b + 2b = 360
 
9b = 360
 
b = 40
 
The answer is (C).

 

Example

In the figure, what is the measure of angle y ?

  1. 80
  2. 84
  3. 85
  4. 87
  5. 90
Solution
Since 4x and 2y – 40 represent vertical angles, 4x = 2y – 40.
Since 3
x and y form a straight angle, 3x + y = 180.
This yields the following system:
 
 
4x = 2y – 40
 
3x + y = 180
 
Solving this system for y yields y = 84.
 
Hence, the answer is (B).
 

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

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

 
Perpendicular 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.




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