Chapter 1- 2000

Chapter 1  -  Introduction to Geometry

 

Day 1 - Half day- Introduction to the course, books distributed, expectations for student achievement.

Day 2 -  Load sketchpad onto computers, read section 1.1 for homework. Place the following vocabulary into notes.   

Point 

Line 

Plane 

Segment 

Endpoint

Number line

Ray 

Angle

Vertex

Side

Union

Intersection

Triangle

 

 

1.1    The objective of this section is to familiarize the student with the basic terms of geometry.

Undefined Terms:  POINT    LINE    PLANE

Point - description- it is a position with no size, thickness or shape

Points are represented by dots and labeled with capital letters.

Line- description - it is a series of points extending on a straight path infinitely in opposite directions.

Lines are labeled either with a lower case letter.  line m or by two points on the line with a double arrow over the top         

Plane - a flat surface extending infinitely

Planes are labeled with a capital letter in the corner or by 3 points on the plane.                  

Number Lines are used to pair numbers with each point on a line.  The coordinate of a point is the number assigned to the point.

Defined Terms

line segment - a portion of a line containing two points and all points between them

endpoints - the two points that mark the boundaries of the segment

      or 

A segment is labeled by its endpoints with a bar over the two

ray- portion of a line beginning at a point and extending infinitely in one direction.

endpoint is where the ray begins

labeled by endpoint(1st)  and a point on the ray with an arrow pointing to the right above the two

Angle - a figure formed by two rays with the same endpoint.

Vertex - endpoint of the rays on the angle

sides - rays that make up the angle

 

Angles are labeled either by 3 letters, the vertex  or by a number in the interior of the angle. The middle letter must be the vertex of the angle when using 3 letters.  You may not use just the vertex if more than one angle is present at that vertex see below

intersection - points that figures have in common

union - all points that are contained in either figure

 

Collinear - can be contained on the same line

Non-collinear - cannot be contained on the same line

Triangle - the union of the segments joining 3 non-collinear points

A triangle is labeled by the 3 vertices

all refer to the same triangle above.

1-2 Objective - Be able to measure segments, angles and understand the concept of congruent.

Segments are generally measured in either standard units of  inches or using the metric system in centimeters.  If anyone has difficulty in using a ruler please see me.

Using sketchpad you will select the segment to be measured and chose from the measure drop down menu.  You will have to go into preferences under the display drop down menu and chose the standard of measurement and accuracy that you wish to have (nearest unit, tenth etc)

An angle's measure is determined by the rotation between the 2 rays.  A protractor is used to measure the rotation.  Angles have measures between 0 and 180 degrees.

If you do not know how to use a protractor go the following website and follow the directions 
http://www.ex.ac.uk/cimt/mepres/book7/bk7i5/bk7_5i2.htm

Classification of angles

 

Acute angle  - angle whose measure is greater than 0° and less than 90°

 

 

 

 

Right angle – angle whose measure is 90°

  

 

 

 

 

Obtuse angle – angle whose measure is greater than 90° but less than 180°

 

 

Straight angle – angle whose measure is 180°

  

 

 

 

A question that is often asked is to find the measure of the angle formed by the hands of a clock at a certain time.  Since there are 12 numbers on the clock, each section between two numbers will have a 30 degree measure.  (360/12 = 30)  The problem is that when the minute hand moves, the hour hand moves slowly toward the next number.  If the time is 2:30, the hour hand will be located halfway between the two and three.

90 + 15 = 105 degrees

What ever fraction of the hour (in this case 30/60 or 1/2) is multiplied times the 30 degrees and this will give the number of degrees the hour hand has moved through toward the next number. 

Changing fractional degrees into minutes and seconds

If it is necessary to be more accurate than unit degrees, the fractional parts of a degree are divided into minutes and seconds.  A one degree rotation is split into 60 minutes and each minute is split into 60 seconds.  These rotations are minuscule when on paper, but when dealing with astronomy or longitude a fraction of a degree becomes a great distance the further out on the rays you travel.  Therefore a system of breaking down the degrees in portions is necessary.  To change a fraction of a degree into minutes, multiply the fraction times 60.
4 1/2 degrees = 4degrees 30 minutes

Go to the following site to see a good explanation of changing into degrees minutes and seconds.  There is even a converter on the site.  Your scientific calculators will also convert.

This site has a good explanation of changing fractions of degrees into minutes and seconds.
http://id.mind.net/~zona/mmts/trigonometryRealms/degMinSec/degMinSec.htm

 

 

1.3   Collinearity, Betweeness and Assumptions.

Collinear - able to be contained on one line.

Note-  Any two points are always collinear whether a line is shown in the diagram or not.

Noncollinear - not able to be contained on the same line.

E,G, and F are collinear.  

A, C, and B are noncollinear.

Any two points are always collinear even if a line is not shown. The key is they are able to be contained on a line.

Betweeness- If a point is between two other points it must lie on the same line.

If D is between A and C then AD + DC = AC      

      

Triangle Inequality Theorem- (proven later)- The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

 

If two sides of a triangle are given to find possible lengths of the third side, follow this rule.

The 3rd side must be greater than the difference between the two lengths given, and less than the sum.

example:  If two sides of a triangle are 4 and 9, the third side must be greater than 5(9-4) and less than 13 (9+4).

The most important part of this lesson is to determine what you can and cannot assume from a diagram given.

You are allowed to assume the following.

   a)  Position of points (collinear or noncollinear and betweeness)

   b) Straight angles and straight lines

You may not assume.

   a) Congruent angles and segments unless they are marked

   b) Right angles

   c) relative size of angles or segments

example:

 

Assumptions that can be made <BFC is a straight angle, A, B, and C are noncollinear,

 

Cannot assume AD = DB or that <AFB is a right angle

1.4 Beginning Proofs

Proof - in geometry is a logical organization of statements backed by geometric principles in order to reach a conclusion.

There are two forms of proofs we will be working with.  Paragraph and two column

The form covered in this section is 2 column.

5 Parts of a proof

1. Given information

2. Prove statement

3. Diagram

4. Statement Column

5. Reason Column

The following is a proof of the theorem- If two angles are right angles then they must be congruent.  Note it has all of the 5 parts

.

Once have proven a theorem you may use it as a reason in another proof.

The 4 types of reasons that may be used in a proof are.

1.    Given Information

2.    Definitions

3.    Postulates (statements accepted as true without any proof - such as through 2 points exactly one line may be drawn)

4.    Theorems - statements that are proven

The second theorem in this section is- If two angles are straight angles then they are congruent.

This can be proven the same way as above, just substitute, straight for right angles and replace 90 with 180.

1.5-Division of Segments and Angles

Midpoint of a segment - the point that divides a segment into 2 congruent segments.

If AM = MB then M is the midpoint of AB

If M is the midpoint of AB then AM = MB

Definitions are reversible.

Every segment has exactly one midpoint.

Trisect - If a segment is divided into 3 congruent segments, then it is said to be trisected.

AB = BC = CD

Bisector of a segment-  Any line, ray, segment or plane that intersects a segment only at its midpoint is the bisector of the segment.

Angle Bisector - The bisector of an angle is the ray that divides the angle into two congruent angles.  

Every angle has exactly one bisector.

Trisect - If an angle is trisected, then two rays divided the angle into 3 congruent angles. The rays are said to be trisectors of the angle.

1.6  Paragraph Proofs

Paragraph proofs are not as formal as two column proofs.  If on a test, you get the general idea of how to prove something or you are short on time, revert to a paragraph proof and write in simplest form how you would solve the proof. 

If you write a paragraph proof first, it helps you to organize your thoughts to put in two column form

Below is an example of a proof solve first in paragraph form and then in 2 column. 

Paragraph form: 

If the larger segments are congruent and the segment BC is subtracted from both of them the remaining segments must be congruent.

2 column form: 

 

 

1.7 Deductive Structure

Recognize the difference between deductive reasoning and inductive reasoning.

Deductive Reasoning is a system of thought based on statements that have already been proven or accepted as fact.

ex.   If you add to odd numbers together, the solution is always even.  John is given a problem of adding 111 and 47 together.  He concludes that his answer will be even.

Inductive Reasoning is a system of thought based on observation. 

ex.  Peter uses his calculator to add 3+5 and gets 8. He repeats the process adding two odd numbers and continues to get an even number.  He concludes that everytime you add two odd numbers together your solution is even. 

The 4 parts of the geometric deductive system are

Undefined terms - point, line, and plane

Definitions- descriptions of terms

Postulates- statements accepted as true without any proof

Theorems- statements that are proven

The definitions, postulates and theorems may be written in if ____ then____ form

The if_________then ______________form of a statement is called a conditional.

Conditional - statements that are written in if___ then form

What comes after the if is the hypothesis.

What comes after the then is the conclusion.

The hypothesis contains the given information and the conclusion contains the prove.

If the hypothesis and the conclusion are exchanged, the sentence is called the converse.

Example:  Given the conditional below:

If two angles are both right angles then they are congruent.

The hypothesis is 2 angles are both right angles

The conclusion is they are congruent.

The converse is If two angles are congruent then they are both right angles. 

Note the converse is false.  Two angles of 30 degrees each are congruent but not right.

When you give an example to prove a statement incorrect it is called a counterexample

To reason from a conditional often a Venn diagram is made. 

Example.  If a student is late 5 times then the student will be given a detention. 

The conclusion is always in the big circle.  The hypothesis in the small circle.

If there is only one place to put the person, then a conclusion can be made.

John has 5 tardies.  Conclusion:  He has a detention.

Mary has a detention.  No conclusion: she might have gotten a detention for some other reason.

Linda does not have a detention.  Conclusion:  She does not have 5 tardies.

Peter does not have 5 tardies.  No conclusion:  Might have a detention for another reason.

1.8  Logic Statements

 

There are two more ways to manipulate a conditional.

 

Inverse:  Negation of the conditional. ( the opposite of the hypotheses and conclusion).

In logic symbols a negation is ~ p then ~q

 

Contrapostive:  the negation of the converse of the conditional. (~q then ~p)

 

Conditional:       If  an angle measures 40° , then the angle is acute.

Converse:           If an angle is acute, then it measures 40°.

Inverse:              If an angle does not measure 40°, then it is not acute.

Contrapositive.  If an angle is not acute, then it does not measure 40°

 

 Note in this case  the Converse and the Inverse are false statements.  To prove they are false a counter example is given.  This is an example that disproves the statement.

“An angle that is acute could be an angle that measures 50°.”

 

If the original conditional is true then the contrapositive must be true.  They are what is referred to as LOGICALLY EQUIVALENT.

 

Chain Reasoning.  When more than one conditional is given and a connection between the conclusion of one to the hypotheses of the next is made, it is referred to as chain reasoning.  Below is an example of chain reasoning.

 

If  the team scores enough points, then they will win the game.

If the team wins the game, then they will go to the districts.

If the team goes to the districts, then they will miss school.

 

Conclusion:  If the team scores enough points, then they will miss school.

 

Example using logic symbols    p then q,  q then r, r then t   Conclusion: if p then t.

 

If the 2nd  conditional had been ~r then ~q, it could be turned into q then r as the contrapositive and the conditional are logically equivalent.