Points, Lines, Planes and Angles

Draw a line $\overleftrightarrow{\mathit{A}\mathit{B}}$. Choose a point O between A and B. Use a protractor to investigate the following questions. 

In the plane represented by your paper, how, many lines can you draw through O that will form a $\mathbf{30}\mathbf{°}$ angle with $\overleftrightarrow{\mathit{O}\mathit{B}}$?

a. Complete.

b. Study the pattern in the four cases shown, and predict the number of angles formed by six noncollinear rays that have the same endpoint.

c. Which of the expression below gives the number of angles formed by noncollinear rays that have the same endpoint?

$\mathit{n}\mathbf{-}\mathbf{1}$ $\mathbf{2}\mathit{n}\mathbf{-}\mathbf{3}$ ${\mathit{n}}^{\mathbf{2}}\mathbf{-}\mathbf{3}$ $\frac{\mathit{n}\mathit{(}\mathit{n}\mathit{-}\mathit{1}\mathit{)}}{\mathit{2}}$

Find the value of the variable.

$\left(4f+5\right)+\left(5f+40\right)=180$

Suppose that the diagram showed,  all the points that are 5 cm from P. What geometric figure would the points form?

In a more complete diagram, would there be a redpoint 15 cm from both F and T? How many such points?

It appears as if points P, X, and T might lie on a straight line. Use a ruler or the edge of a sheet of paper to see if they do.

It looks as if P might be equidistant from F and X. Is it?

Copy and complete the table. Refer to the diagram on pages 1 and 2.

Copy the points $F$, $T$ and $P$ from the diagram on page 2. If you lay your paper over the page, you can see through the paper will enough to get the points?

1. Draw a line to indicate all points equidistant from $F$ and $T$.
2. Draw a circle to indicate points 6 cm from $P$. If you don’t have a compass, draw as well as you can freehand.
3. How many points are equidistant from $F$ and $T$, and are also 6 cm from $P$?

Repeat exercise 2, but use 2 cm instead of 6 cm.

There is a distance you could use in parts (b) and (c) of Exercise 2 that would lead to the answer one point in part (c). Estimate that distance.

Suppose Pat spoke a line l joining F and T while you thought of a line n joining F and T. Is it better to say that l and n are two different lines, or to say that we have one line with two different names?

Point X is equidistant from F and T. Furthermore, point Y is equidistant from F and T. Does that mean that X and Y are equally distant from F?

Suppose you were asked to find a point 5 cm from P, 5 cm from F, and 5 cm from T. Is there such a point?

Do you believe there is any point that is equidistant from P, F, and T?

Which is greater, the distance from R to S or the distance from T to U?

Which is greater, the distance from R to S or the distance from T to U?

Which is greater, the distance from A to B or the distance from A to C?

How does the area of the outer square compare with the area of the inner square?

Compare the areas of the red and blue regions. (Area of circle = .)

In diagrams a, b, c, and d are lengths. Which is greater, the product ab or the product cd?

A path between opposite vertices of the square is made up of hundreds of horizontal and vertical segments. (The diagram shows a simplified version.) What is the best approximation to the length of the path-24, 34, 44, or more than 44?

Classify the statement as true or false.

intersects plane M at point O.

Classify the statement as true or false.

$\overleftrightarrow{AB }$ Is in plane R.

Classify the statement as true or false.

Plane M intersects  at more than one point.

Classify the statement as true or false.

S contains $\overleftrightarrow{AB}$.

Classify the statement as true or false.

T, O, and R are collinear.

Classify each statement as true or false.

R and S contain D.

Classify the statement as true or false.

X, O, and Y are collinear.

Classify each statement as true or false.

D is on line h.

Classify the statement as true or false.

R, O, S, and W are coplanar.

Classify each statement as true or false.

h is in S.

Classify the statement as true or false.

R, S, T, and X are coplanar.

Classify each statement as true or false.

h is in R.

Classify the statement as true or false.

R, X, O, and Y are coplanar.

Classify each statement as true or false.

Plane R intersects plane S in $\overleftrightarrow{AB}$.

Does a plane have edges?

Classify each statement as true or false.

Point C is in R and S.

Can a given point be in two lines? In ten lines?

Classify each statement as true or false.

A, B, and C are collinear.

Can a given line be in two planes? In ten planes?

Classify each statement as true or false.

A, B, C, and D are coplanar.

The plane that contains the top of the box can be called plane ABCD.

Name a fourth point that is in the same plane as the given points.

A, B, C

Name a fourth point that is in the same plane as the given points.

E, F, H

Name a fourth point that is in the same plane as the given points.

D, C, H

Name a fourth point that is in the same plane as the given points.

A, D, E

Name a fourth point that is in the same plane as the given points.

B, E, F

Name a fourth point that is in the same plane as the given points.

B, G, C

The plane that contains the top of the box can be called plane ABCD.

Are there any points  besides C and G?