Day 1 – Lines, Circles, Angles

TEXT: Elementary College Geometry by Africk

Geometry Labs by Henri Picciotto. Videos from Khan Academy.

Geometry, literally the measure of the earth, begins with some commonplace but surprisingly abstract objects. A point takes up no space. Think location:

• for a plane (using the Cartesian coordinate system): given an x and y, start at the origin, move right (or left if negative) x units, then turn 90° left and go up (or down if negative) y units.

• for the earth: latitude is north or south of the equator, longitude is east or west of Greenwich, England, both measured in degrees, with 90° the furthest north and south you can go (the poles) and 180° the furthest east and west you can go (approximately the international date line  in the western Pacific Ocean). New York city is 40.5° north and 73.5° west. Here are locations of some US cities and world cities. (Instead of decimal degrees, the measures are given as degrees and minutes. For example, 40.5° is 40 degrees and 30 minutes or 40°30′.) Here is a point and click tool for finding the longitude and latitude of any location on earth.

A line is straight, long and thin. It is a collection of points and can be defined or nailed down by giving 2 points.

Here are some additional concepts/terminology having to do with lines that we will need.

• The intersection is the point where 2 lines meet or in general, where 2 objects overlap.
• Two lines are perpendicular if they intersect (meet) at right angles.
• A line segment is a finite portion of a line. Its length can be measured with a ruler.
• A ray is a portion of a line which extends indefinitely in exactly one direction.
• An endpoint is the beginning or end of a line segment or ray.

Given a line segment, the midpoint is the location halfway between the endpoints. The perpendicular bisector is a line perpendicular to a line segment at its midpoint. For those of you who did line and compass constructions in a math class in high school, check out this animation of the construction of a perpendicular bisector.

A circle is the set of points a fixed distance (radius) from a fixed point (center). Circles are  the straight “lines” used to travel around a sphere (ball). The lines of latitude and longitude discussed earlier are circles.  The diameter of a circle is a line passing through its center from one point on the circle to another point on the circle. In terms of distance, it is twice the radius. Note that the center is always the midpoint of a diameter.

The perimeter of a planar figure is the length around its edge. The circumference is the perimeter of a circle. Take a circular object such as a cylinder (can)  or sphere and put a string around it. Open the string and compare it with the diameter (which is easy to measure for cylinder, not so easy with a sphere). It turns out to be slightly more than 3 times as big. The actual number is represented by the Greek letter π or pi (pronounced pie) . In summary, C=π D.

Angle, perhaps the most subtle of the basic concepts, is the join of 2 rays at their endpoints. When the 2 rays

• coincide, the angle is 0°;
• are perpendicular (like a grid system, avenue and street), the angle is 90° and is a right angle;
• form a straight line, the angle is 180° and is a straight angle.

An angle that is

• less than 90° is acute;
• between 90° and 180° is obtuse.
  

In the second half of the above video, 2 important concepts are covered: supplements, 2 angles that sum to 180° (fit together to form a straight angle) and complements, 2 angles that sum to 90° (fit together to form a right angle). Such angles are said to be supplementary, respectively, complementary.

The angle bisector is a ray which divides an angle evenly in 2. Its construction with compass and straightedge is one of the easier ones.

The protractor is a device  used to measure an angle. We will on occasion use protractors in class, so if you are not familiar with how to use them, try the “math is fun” tutorial, which includes an exercise to try your skill. While you measure each of the interior angles of the 4 sided figure (a quadrilateral), check whether the angle is acute or obtuse. Finally, the 4 angles should sum to 360°. (We will learn why this is true later in the course.)

To explore angles, we will do lab exercises in class using different colored pattern blocks. If you miss class, we highly suggest that you arrange to work with one of the instructors to complete the exercises.

Warmup LAB 1.1 procedure and questions:

Place pattern blocks around a point so that a vertex (corner) of each block touches the point and no space is left between the blocks. The angles around the point will add to 360°. Try blocks that are

• all the same color;
• of 2 colors;
• more than 2 colors.

LAB 1.2 will be given as a worksheet. Here are the topics explored:

1. Measures of angles in common symbols and objects.
2. Measures of the angles for the shapes used in LAB 1.1
3. Creating a primitive protractor using the tan shaped block and using it to measure the angles of the shapes used in LAB 1.1
4. Using a protractor to measure the angles of arbitrary triangles. One way to classify triangles is by the largest of its angles: acute, right or obtuse. Can you classify the triangles?
5. Use the protractor to draw 3 angles.

Additional questions for the laboratory:

1. Which of the questions in problem 1 did you find the most difficult? What made the others easier?
2. Of the 5 exercises, which did you find the most difficult? For which one do you think you learned the most?