Lines and Angles


Words of “Wiz-dom”-In some of the easiest problems, you will use given distances between points on a line to calculate distances between other points. This is generally a straightforward process unless you have to create your own diagram.

Try this.

Note: None of the Wizard’s figures are ever drawn to scale!

Line segment MO is 40. O is the midpoint of

MP and NO is half as long as OP.

How long is MP?

Answer #1

What is the value of ?

Answer #2

What is the value of MN + OP?

Answer #3

Words of “Wiz-dom”-Remember:

360o around a point
180o on one side of a line at any point
perpendicular = 90o angle = a right angle
obtuse angle > 90o
acute angle < 90o
2 complementary angles equal 90o
2 supplementary angles equal 180o

A “bisector” divides an angle (or anything else) into two equal pieces.

  • Parallel lines are always equal distant from one another and they will never cross each other. For example, lines l and m are parallel.

  • A transversal is a straight line that crosses parallel line as in the diagram above.

  • Vertical angles are ones that are opposite one another and equal in size. For example, y and c in the diagram are vertical angles.

  • Corresponding angles have the same relative position when a transversal crosses parallel lines and they are equal. For example, z and x are corresponding angles as are a and c.

    If lines l and m are parallel and z equals 40 degrees,

how many degrees is x?

Answer #4

how many degrees is a?

Answer #5

how many degrees is a plus x?

Answer #6

which is larger, a + x or b + c?

Answer #7

Sample Questions

8.  If lines a, b and c are parallel, how many degrees is x + y – z?

(A) 45
(B) 90
(C) 120
(D) 135
(E) 120

Answer to Question #8

13.  If  l and m are parallel lines, which of the following must be true?

I.    a + d = w + z
II.   x – y = a – d
III.   a + c > w + z

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) III only

Answer to Question #13

Note: Figure not drawn to scale.

4.  What is the value of x?

Answer to Question #4

14.  In the figure above, if y/3 and z/7 are integers, what is the smallest possible value of y/3 + z/7?

(A)  2
(B)   3
(C)   7
(D) 10
(E)  21

Answer to Question #14

5.  In the figure above, three lines intersect at a point. What is the value of 75 – a?

(A) 115
(B) 75
(C) 65
(D) 45
(E) 10

Answer to Question #5

10. Four lines intersect at a point creating a set of angles that are equal. What is the size of one of the angles?

Answer to Question #10

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