Geometry

Activity 1

a. Draw as many lines as possible which are uniquely determined by the points $A$, $B$, $C$, and $D$ given below (do not move the points).

b. How many lines did you draw?

c. Could you have instead used counting/combinatorial techniques to answer the question in part (b), without drawing the lines? If so, how?

d. Can you move the points so that they uniquely determine exactly one line? Two lines? Other? If so, explain what you did.

The geometry standards include the following:
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

How should a student think about a point, a line, distance along a line and distance around a circular arc? Intuitively and with the following assumptions about their relationships.

i. A line is a set of points, which contains at least two points.

ii. Two distinct lines intersect in at most one point.

iii. Every pair of distinct points lies on at least one line.

Definition Collinear points are points that lie on the same line.

Activity 2

In this activity, you will experiment with ideas from the Ruler Postulate.

Ruler Postulate The points of a line can be placed in correspondence with the real numbers so that:
i. to every point of the line there corresponds exactly one real number called its coordinate;
ii. to every real number there corresponds exactly one point called the graph of the number on the line;
iii. to each pair of points there corresponds a unique real number called the distance between the points; and
iv. for any two distinct points and of a line, the line can be coordinatized in such a way that $A$ corresponds to $0$ and $B$ corresponds to a positive number.

Do the following:
i. Drag $\overleftrightarrow{AB}$ to the given axis.
ii. What are the coordinates of points $A$, $B$, $C$, and $D$?
iii.What are the distances between the pairs of points? Can you drag the points to change the distances between them?
iv. Can you move $\overleftrightarrow{AB}$ to satisfy part (iv) of the Ruler Postulate?

Theorem A line is an infinite set of points.

The theorem follows from parts (i) and (ii) of the Ruler Postulate since the set of real numbers is infinite.

Definition On a coordinatized line the distance between two ponts having coordinates $x$ and $y$ is $|x-y|$.

Question In the figure above, what is the distance between points $A$ and $B$?

Definition Let $A$, $B$, and $C$ be three collinear points on a coordinatized line. Then point $B$ is between points $A$ and $C$ if and only if the coordinate of $B$ is between the coordinates of $A$ and $C$.

Question In the figure above, is point $B$ between $A$ and $C$? Is there a point between $A$ and $D$? If so, name it. If not, why?

Definition The line segment (or segment) with endpoints $A$ and $B$ is the set whose elements are distinct points $A$ and $B$ and all points between $A$ and $B$. It is denoted by $\overline{AB}$.

Definition The length of a segment is the distance between its endpoints. The length of $\overline{AB}$ is denoted by $AB$ and is a nonnegative real number.

Betweenness Theorem If a point $B$ is between points $A$ and $C$, then $AB+BC=AC$.

Question What are the values of $AB$, $BC$, and $AC$ for the image above? Do these values satisfy the Betweenness Theorem?

Definition Point $M$ is the midpoint of $\overline{AB}$ if and only if $M$ is in $\overline{AB}$ and $AM=MB$.

Question What is the midpoint of $\overline{AB}$ above? Drag point $C$ there.

Activity 3

Definition The ray with endpoint $A$ and which contains point $B$ is the union of $\overline{AB}$ and the set of points for which $B$ is between each element of the set and $A$. It is denoted by $\overrightarrow{AB}$.

a. Draw as many rays as possible which are uniquely determined by the points in the figure below.

b. How many rays did you draw? Did you draw the same number of rays as lines? Why or why not?

Theorem On a ray there is exactly one point at a given distance from the endpoint.

Activity 4

In the activity that follows, you will experiment with ideas from the Protractor Postulate. First we need a definition.

Definition An angle is the union of two rays which have the same endpoint.

Protractor Postulate
To each angle, say $\angle{ABC}$, there corresponds a unique, nonnegative real number which is less than or equal to 180, called the measure of the angle and denoted by $m\angle{ABC}$ so that each of the following statements holds.

i. If both sides of an angle are the same ray, then the measure of the angle is $0$.
ii. If the sides of an angle are opposite rays, then the measure of the angle is $180$.

In the figure below, move the slider to illustrate parts (i) and (ii) of the Protractor Postulate.

Activity 5

Protractor Postulate (continued)
iii. If $\overrightarrow{BD}$ is in the interior of $\angle{ABC}$, then $m\angle{ABD}+m\angle{DBC}=m\angle{ABC}$.
iv. In each half-plane of a line $\overleftrightarrow{AB}$ there is exactly one ray whose union with $\overrightarrow{AB}$ is an angle having a given measure, $\theta$, between $0$ and $180$.

Activity 6

Definition Suppose $A$, $B$, $C$, $D$,... are coplanar points. A path is the union of line segments connecting all of these points in any order.

Definition A polygon is a path consisting of three or more line segments (called sides) such that each line segment intersects exactly two others, one at each of its endpoints.