Geometry

Words of “Wiz-dom”-If there is anything important that can be said about all geometry problems it’s that they are going to be two-step problems, sometimes more. Think back to the lesson about how to do word problems. Remember that you have some given information and a question you have to answer. To answer the question, there is some information that you need that is not given directly. You need to calculate it by using the given information. It is called “calculated” or “derived” information. Geometry questions are the ones that will require you to use the Word Problem Paradigm most frequently.

Using Diagrams

Words of “Wiz-dom”-Diagrams are the key to most geometry problems. Sometimes they are provided. Sometimes you have to draw them based on information in the narrative part of the question. On the ACT, the diagrams are not necessarily drawn to scale. However, I think the ACT test writer tells you the illustrations aren’t necessarily to scale in case he goofs. They are so close to scale that you can treat them as such. This means that you can use the diagrams to estimate answers and to check your work.

When drawing your own diagrams, you should draw them quickly but as close to scale as you can. When you add something to an existing diagram, attempt to draw it to scale. (Be sure you don’t waste time trying to be an architect, but being close to scale helps.)

Further, when you draw diagrams based on narrative information, draw all possible diagrams. Sometimes the information you are given allows you to draw two or more versions of the diagram. When you are creating your own, you need to make sure you have considered all possibilities.

More words of “Wiz-dom” on using diagrams

Words of “Wiz-dom”-The test writer draws odd shapes and wants to see if you can break them down into regular shapes: triangles, quads, and circles. For example:

The strategy for these questions is to look at the unusual shape and break it down into normal geometric shapes. In the first example, you are now working with a square and a triangle. In the second example, you are now working with a circle and a square. The second example is a case of “overlapping” figures. The question may be, “What is the area of the unshaded region?”

Answer to area of unshaded region

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

obtuse angle > 90o

acute angle < 90o

2 complementary angles equal 90o

2 supplementary angles equal 180o

Let’s see what you remember about parallel lines.

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

Sample Questions

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

A. 45
B. 90
C. 120
D. 135
E. 150

Answer to question #1

2.  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 #2


Note: Remember that The Wiz’s figures are not drawn to scale.

3.  What is the value of x?

A. 35
B. 90
C. 115
D. 120
E. 160

Answer to question #3

4.  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 #4

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

A. 10
B. 45
C. 65
D. 75
E. 115

Answer to question #5

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

A. 10
B. 45
C. 65
D. 90
E. 180

Answer to question #6

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