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9.1: Introduction to Geometry

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Introduction

The Skateboard Park

Marc and Isaac are working on a design for a new skateboard park. The city council of their town has agreed that the skateboard park is in need of renovation. Marc and Isaac have offered to help draw some initial plans to present at the next meeting. They are a little nervous about their design and about their presentation. Isaac’s mom offers to let them use some of her design paper and the two boys begin sketching their plan at the kitchen table.

“It definitely needs more rails,” Isaac says.

“What is a rail?” asks Isaac’s mom who glances at the design over her son’s shoulder.

“You know Mom, the sides don’t connect or cross,” Isaac says.

“Well, if that is what you want, your drawing is not accurate.”

Isaac looks down at the drawing. His mom is right. The rails don’t look correct.

To draw these rails, Isaac and Marc will need to understand the basics of Geometry. Pay attention to this lesson and you will understand how to help them with their design at the end.

What You Will Learn

In this lesson you will learn to complete the following:

  • Identify points, rays, lines and segments using words and symbols.
  • Identify intersecting and parallel lines.
  • Identify angles by vertex and ray.
  • Draw angles using a protractor.

Teaching Time

I. Identify Points, Rays, Lines and Segments using Words and Symbols

We have been working with numbers and operations in each of our previous chapters. In this chapter, we will begin working with the basics of geometry.

Geometry is a part of mathematics concerned with questions of size, shape and position of figures and with their location in space. This lesson is going to focus on some of the building blocks of geometry.

There are a lot of vocabulary words in this lesson. We use pictures, definitions and symbols to help us to understand things in geometry. Keep your notebook handy to take notes during this lesson!

Geometric Figures

As we work with the geometric figures below, we will discuss three things about each. We will discuss the description or definition, what the figure looks like and finally how to “name” it.

The first geometric figure to learn about is a point. A point is a definite place in space that doesn’t have a size or shape.

Here is point A plotted on the graph. Notice that the point does name a location, but it does not have a size or shape.

We can name the point Point A. Naming a geometric figure is a way to identify it in a mathematical sentence.

Next, we can learn about a ray. Often we think of a “ray of sunshine.” A ray has an endpoint but extends in one direction indefinitely. Here is a picture of a ray.

Notice that this ray has two points. It has one point, point A, that is the endpoint and one point, point B, which is on the line.

To name the ray, we use the letters of the two points and a symbol. The symbol looks like a small ray that is above the letters, \overrightarrow{AB}.

Our third geometric figure is a line. We often think of a line as looking like this:

However, in geometry, this would be a line segment. A line segment has two endpoints. Because this line segment does not have arrows on the ends, it means that the ends stop. A line segment is a set of connected points, meaning that while we see a straight line segment here, it is really a whole bunch of connected points. Two of the points on the line have been named. They are points A and B.

We can name this line segment by using a small line segment above the two endpoints. The symbol is the small line segment \overline{AB}. When you see this symbol, you know that you are working with a line segment.

If the example above is a line segment, what does a line look like?

A line has an arrow on each end. A line is also a set of connected points, but the line does not end, as indicated by the arrows. A line goes on and on and on indefinitely. Two of the points have been named on the line. These are the two points that we will use to name the line. The symbol for a line is a small line with arrows on the end. The symbol goes above the named points on the line to name the line, \overleftrightarrow{CD}.

See what you can remember, and answer the following questions about geometric figures.

1. Which figure has two arrows on the ends?

2. is what kind of figure?

3. is what kind of figure?

Take a few minutes to check your work with a friend.

II. Identify Intersecting and Parallel Lines

In the last section you learned about lines and line segments. When lines intersect, sometimes we need to describe how they do so. Two of the descriptions are intersecting lines and parallel lines.

Intersecting lines are lines that cross at some point. You can think of an intersection in a pair of streets to help you remember intersecting lines.

Here you can see that the streets of this highway intersect just as two intersecting lines intersect or cross. Here is an example of intersecting lines that you would see in geometry.

The lines intersect or cross at one point. We call this point the point of intersection. Sometimes lines will intersect with other lines at more than one point.

Parallel lines do not cross or intersect EVER. They are equidistant.

In the sport of gymnastics, gymnasts use parallel bars to perform. Notice that the parallel bars are two bars that do not connect. They are an equal distance apart and will never cross or intersect.

Here is what parallel lines in geometry look like.

If we use a symbol for parallel lines, the symbol looks like this: \overleftrightarrow{AB} \parallel \overleftrightarrow{CD}. This means that line AB is parallel to line CD.

Identify which lines are parallel and which are intersecting in each picture.

1.

2.

Take a few minutes and check your work with a peer.

III. Identify Angles by Vertex and Ray

An angle is one of the key geometric figures that you will be working with during geometry. An angle is created when two rays connect at a common point.

You can see here that the two rays are connected at a common endpoint, called a vertex. This forms the angle.

This is angle ABC. The vertex B is always in the middle. The symbol for angle looks like a small angle. Here is how we can name the angle. \angle{ABC}

Angle ABC is named with this symbol.

Name these two angles on your own. Be sure that the vertex is in the middle.

1.

2.

Check your work.

IV. Draw Angles Using a Protractor

We just finished naming angles using points on the rays and the vertex. We can also measure angles. If you look at the two angles that you just named, you will see that they are different sizes. Angles are measured in degrees. The larger the angle the higher the number of degrees.

How can we measure angles?

Angles are measured using a special tool called a protractor.

Here is a picture of a protractor.

Notice that you can see all of the degrees on the protractor.

How can we use a protractor?

  1. First, you line up the vertex with the little hole in the middle of the protractor, then carefully align the bottom ray with the bottom line of the protractor.
  2. Then, you follow the top ray to the number of degrees that the angle measures.

Real Life Example Completed

The Skateboard Park

Have you figured out what is wrong with Isaac’s drawing? Think back to this lesson on geometry, reread the problem and underline any important information.

Marc and Isaac are working on a design for a new skateboard park. The city council of their town has agreed that the skateboard park is in need of renovation. Marc and Isaac have offered to help draw some initial plans to present at the next meeting. They are a little nervous about their design and about their presentation. Isaac’s mom offers to let them use some of her design paper and the two boys began sketching their plan at the kitchen table.

“It definitely needs more rails,” Isaac says.

“What is a rail?” asks Isaac’s mom, who glances at the design over her son’s shoulder.

“You know Mom, the sides don’t connect or cross,” Isaac says.

“Well, if that is what you want, your drawing is not accurate.”

Isaac looks down at the drawing. His mom is right. The rails don’t look correct.

If Isaac’s drawing is incorrect, then the rails in his drawing must not be parallel. Remember that parallel lines do not connect or cross in any way. When Isaac describes the rails to his mom it is clear that he wants them to be parallel. She says that his drawing is not accurate, so Isaac needs to redraw the rails and show that they do not connect.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Point
a location in space that does not have size or shape.
Ray
a line that has one endpoint and continues indefinitely in one direction.
Line
a set of connected points without endpoints.
Line Segment
a set of connected points with two endpoints.
Point of Intersection
the point where two intersecting lines meet.
Intersecting Lines
lines that cross or meet at some point
Parallel Lines
Lines that do not cross or meet EVER and are equidistant.
Angle
a geometric figure formed by two rays that connect at a single point or vertex.
Vertex
The point of intersection of the lines or rays that form an angle
Protractor
a tool used to measure an angle in terms of degrees.

Technology Integration

James Sousa, Points, Lines, and Planes

Other Videos:

  1. http://www.mathplayground.com/mv_using_protractor.html – Video on using a protractor and an introduction to acute, obtuse and right angles.

Time to Practice

Directions: Identify each of the following geometric figures.

1.

2.

3.

4.

Directions: Tell whether each picture shows parallel or intersecting lines.

5.

6.

7.

8.

Directions: Draw a picture to illustrate each of the named geometric figures.

9. \overrightarrow{AB}

10. \overleftrightarrow{CD}

11. \overleftrightarrow{DE}

12. \angle{ABC}

13. \angle{LMN}

14. \overline{XY}

15. \overrightarrow{PQ}

16. \overleftrightarrow{GH}

17. \overleftrightarrow{AB} \parallel \overleftrightarrow{DE}

18. \overleftrightarrow{LM} \parallel \overleftrightarrow{DE}

19. \overleftrightarrow{RS} \parallel \overleftrightarrow{TU}

20. \overline{DF} \parallel \overline{XY}

Directions: For #20 – 30 Have students draw angles and then measure each other’s using a protractor. Use peer checking for answer checking.

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