Geometry Foundations

01.01

Geometry explores points, lines, planes, and the shapes formed by them, starting with three undefined terms:

  • Point: A location without size, represented by a capital letter (e.g., point A).

  • Line: An infinite collection of points extending in one dimension, named using two points or a lowercase letter (e.g., line m); points can be collinear or noncollinear.

  • Plane: A flat surface extending indefinitely in two dimensions, named by three noncollinear points or a capital script letter (e.g., plane P); points can be coplanar or noncoplanar.

Additional geometry concepts include:

  • Ray: Has one endpoint and extends indefinitely (e.g., ray AB).

  • Angle: Formed by two noncollinear rays sharing a common endpoint, measured in degrees (e.g., ∠B, ∠ABC).

  • Parallel Lines: Lines in the same plane that do not intersect, denoted as a ∥ b.

  • Perpendicular Lines: Intersect at right angles (90 degrees).

  • Circle: Defined by a center point and radius, named (e.g., circle P or ⊙P).

In geometry, postulates are accepted facts that require no proof (e.g., through any two points, there is exactly one line), while theorems are statements needing proof (e.g., the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2).

01.02

Congruency

So far you have gathered various terms to help you begin the next step in geometry, constructing.

line segment a with endpoints A and BArt showing congruency

Public Domain

Creating two objects, or figures, that are the same is pretty common in the real world. For example, an artist may want to make two or more figures the same in one painting. To do so, she may have to actually construct the figures to ensure they are congruent. So where would she begin?

What does it mean when two or more figures are congruent? Use any prior knowledge you may have or just guess whether each set of figures below is congruent or not.

Understanding Constructions

One of the unique attributes of Euclidean geometry is that most constructions can be made using just two simple tools: a compass and a straightedge. These tools are very old and were used by the Ancient Greeks in geometric constructions.

So why can't you use "modern" tools like a protractor and a ruler when you're making constructions? Well, you can thank a man named Euclid. He is the credited inventor of geometry! He wrote a book called "Elements," which covered lots of constructions using just these two simple tools.

Think about duplicating or copying a line segment. What if you traced all the way around and made a full circle? Watch the video to see this in action.

What are you really changing when you change the length of the segment?

As you start with constructions, you will need to know what an arc is. An arc is a part of a circle. You will use the intersection of arcs to determine points of interest needed to create constructed diagrams.

Using Technology

You may also need to become familiar with a dynamic drawing program as you progress through this course. A dynamic drawing program allows its user to construct geometric figures with specific geometric properties. Paint is a program that allows users to draw figures freehand on the computer, but it does not measure angles, construct perfect circles, or perform many other features of a dynamic geometric drawing program or certain advanced graphing calculators.

Bisecting Segments

Line segment AB intersected by a line

Think of bisecting like dissecting in biology. The prefix bi- means “two.” The root of the word, “sect,” means “cut.” When you are bisecting a figure, you are cutting it into two pieces. However, the two pieces must be the same length. A segment bisector is a line, point, segment, or ray that divides a segment into two equal pieces, each of which is half of the original segment. To do this, the bisector must find a specific point, the midpoint.

The image to the right shows bisected segment

line segment AB

. What do you notice about the image? Where do you think the resulting bisector intersects the original segment?

Follow along with the video below to create your own segment and segment bisector.

Bisecting a Line Segment with Technology

Now that you're starting to get the hang of using a drawing program, you can expand your knowledge a bit more. Your next task is to bisect a line segment. Remember, a bisector splits a line into two equal parts. Just like you used a compass and straightedge to bisect figures, you can use tools in a dynamic drawing program to construct bisectors.

Angle Constructions

  • Acute angles are angles that measure less than 90°.

  • Obtuse angles are angles that measure more than 90°, but less than 180°.

  • Right angles are angles that measure exactly 90°.

Did you notice the little square at the vertex of the angle? If that's there, then you know the angle measures 90°.

Straight angles are similar to straight lines. There is no bend, like when you straighten your arm, so it lies flat as a line and measures exactly 180°.

Finally, although there are right angles, there aren't any wrong ones!