Points, Lines, Planes and Angles

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name a plane that contains $\overleftrightarrow{\mathbf{A}\mathbf{C}}$.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name a plane that contains $\overleftrightarrow{AC}$ but that is not shown in the diagram.

Complete.

If $\mathbf{m}\mathbf{\angle }\mathit{H}\mathit{O}\mathit{K}\mathbf{=}\underset{\mathbf{¯}}{\mathbf{?}}$, and $\mathbf{\angle }\mathbf{HOK}$ is called a(n)  $\mathbf{\angle }\underset{\mathbf{¯}}{\mathbf{?}}$ angle.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name the intersection of plane DCFE and plane ABCD.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name the intersection of plane $\mathbf{DCFE}$ and plane $\mathbf{ABCD}$.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name four lines shown in the diagram that don’t intersect plane EFGH.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name four lines shown in the diagram that don’t intersect plane $\mathbf{EFGH}$.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name two lines that are not shown in the diagram and that don’t intersect plane EFGH.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name two lines that are not shown in the diagram and that don’t intersect plane $\mathbf{EFGH}$.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name three planes that don’t intersect $\overleftrightarrow{EF}$ and don’t contain $\overleftrightarrow{EF}$.

In Exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it by using four points, no three of which are collinear.

Name three planes that don’t intersect $\overleftrightarrow{\mathbf{EF}}$ and don’t contain $\overleftrightarrow{\mathbf{EF}}$.

If you measure $\angle EFG$ with a protractor you get more than $90°$. But you know that $\angle EFG$ represents a right angle in a box. Using this as an example, complete the table.

If you measure $\mathbf{\angle }\mathbf{EFG}$ with a protractor you get more than $\mathbf{90}\mathbf{°}$. But you know that $\mathbf{\angle }\mathbf{EFG}$ represents a right angle in a box. Using this as an example, complete the table.

State whether it is possible for the figure described to exist. Write yes or no.

Two points both lie in each of the two lines.

State whether it is possible for the figure described to exist. Write yes or no.

Two points both lie in every two lines.

State whether it is possible for the figure described to exist. Write yes or no.

Three points all lie in each of the two planes.

State whether it is possible for the figure described to exist. Write yes or no.

Three points all lie in each of the two planes.

State whether it is possible for the figure described to exist. Write yes or no.

Three non-collinear points all lie in each of two planes.

State whether it is possible for the figure described to exist. Write yes or no.

Three non-collinear points all lie in each of two planes.

State whether it is possible for the figure described to exist. Write yes or no.

Three non-collinear points all lie in each of the two planes.

State whether it is possible for the figure described to exist. Write yes or no.

Two points lie in a plane X, two other points lie in a different plane Y, and the four points are coplanar but not collinear.

Surveyors and photographers use a tripod for support.

Think of the intersection of the ceiling and the front wall of your classroom as line $\mathbf{I}$. Let the point in the center of the floor be point $\mathbf{C}$.

1. Is there a plane that contains line $\mathbf{I}$ and point $\mathbf{C}$?
2. State the theorem that applies.

State whether it is possible for the figure described to exist. Write yes or no.

Two points lie in a plane $\mathbf{X}$, two other points lie in a different plane $\mathbf{Y}$, and the four points are coplanar but not collinear.

Points $R$, $S$ and $T$ are non collinear points.

1. State the postulate that guarantees the existence of a plane $X$ that contains $R$, $S$ and $T$.
2. Draw a diagram showing plane $X$ containing the non collinear points $R$, $S$ and $T$.
3. Suppose that $P$ is any point of $\overleftrightarrow{RS}$ other than $R$ and $S$. Does point $P$ lie in plane $X$? Explain.
4. State the postulate that guarantees that $\overleftrightarrow{TP}$ exists.
5. State the postulate that guarantees that $\overleftrightarrow{TP}$ is in plane $X$.

Points $A$, $B$,  $C$ and $D$ are four non-coplanar points.

a. State the postulate that guarantees the existence of planes $ABC$, $ABD$, $ACD$, and $BCD$.

b. Explain how the ruler postulate guarantees the existence of a point $P$ between $A$ and $D$.

c. State the postulate the guarantees the existence of plane $BCP$.

d. Explain why there are an infinite number of planes through $\overline{BC}$.

State how many segments can be drawn between the points in each figure. No three points are collinear.

b. 4 points $\underset{¯}{?}$ segments.

d. 6 points $\underset{¯}{?}$ segments.

e. Without making a drawing, predict how many segments can be drawn between seven points, no three of which are collinear.

f. How many segments can be drawn between $n$ points, no three of which are collinear?

Parts (a) through (d) justify theorem 1-2: through a line and a point not in the line there is exactly one plane.

a. If $P$ is a point not in line $k$, what postulate permits us to state that there are two points $R$ and $S$ in line $k$?

b. Then there is at least one plane $X$ that contains points $P$, $R$ and $S$. Why?

c. What postulate guarantees that plane $X$ contains line $k$? Now we know that there is a plane $X$ that contains both point $P$ and line $k$.

d. There can’t be another plane that contains point $P$ and line $k$, because then two planes would contain non collinear points  $P$, $R$ and $S$. What postulate does this contradict?

Copy the grid system shown on the previous page onto a piece of graph paper. Then locate the following points.

a. A point $\mathit{T}$ five blocks due west of the centre of town.

b. A point $\mathit{U}$ five blocks east and two blocks south of the centre of town

c. A point $\mathit{V}$ two blocks west and one block north of your house, which is located at point $\mathit{P}$

Give the letter that names each point.

1. $\left(\mathbf{2}\mathbf{,}\mathbf{30}\mathbf{°}\right)$
   
2. $\left(\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{,}\mathbf{120}\mathbf{°}\right)$
   
3. $\left(\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{90}\mathbf{°}\right)$

Give the distance and angle for each point.

1. $\mathrm{C}$
   
2. $\mathrm{A}$
   
3. $\mathrm{T}$
   
   

When $\mathbf{RN}\mathbf{=}\mathbf{7}$, $\mathbf{NC}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{5}$ and $\mathbf{RC}\mathbf{=}\mathbf{18}$, what is the value of $\mathbf{x}$?

Give another way of naming each point.

a. $\left(1,-120°\right)$

b. $\left(2,300°\right)$

c. $\left(2.5,-180°\right)$

A point is given in the grid system. What would it be called in the distance-angle system? (Hint: see the discussion at the top of the page. Use a protractor and a centimetre ruler to help you answer the question.)

$\text{a}.\left(3,4\right),\text{ b}.\left(-2,5\right),\mathrm{c}.\left(4,0\right),\mathrm{d}.\left(8,-6\right)$

A point is given in the distance-angle system. What would it be called, approximately, in the grid system? (Hint: Use a protractor and a centimetre ruler to draw the triangle suggested by the angle and distance. Measure the sides of the triangle.)

$\mathrm{a}.\left(2,50°\right), \mathrm{b}.\left(1.5,-70°\right), \mathrm{c}.\left(3,90°\right), \mathrm{d}.\left(1,120°\right)$

Write three names for the line pictured.

Name the ray that is opposite to $\overrightarrow{\mathbf{NC}}$.

Is it correct to say that point $\mathbf{B}$ lies between points $\mathbf{N}$ and $\mathbf{C}$.

Complete.

$\mathbf{m}\angle \mathbf{1}\mathbf{+}\mathbf{m}\angle \mathbf{2}\mathbf{=}\mathbf{m}\angle \underset{¯}{\mathbf{?}}$

Complete.

If $\angle \mathbf{1}\cong \angle \mathbf{2}$, then $\underset{¯}{?}$ is the bisector of $\angle \underset{¯}{?}$.

Which of the four things stated can’t you conclude from the diagram?

1. $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are collinear.
2. $\mathbf{\angle }\mathbf{DBC}$ Is a right angle.
3. $\mathbf{B}$ is the midpoint of $\overleftrightarrow{\mathrm{AC}}$.
4. $\mathbf{E}$ is in the interior of $\mathbf{\angle }\mathbf{DBA}$.

Apply postulates and theorems to complete the statements.

Through any two points $\underset{\mathbf{¯}}{\mathbf{?}}$.

Apply postulates and theorems to complete the statements.

If points $\mathbf{A}$ and $\mathbf{B}$ are in-plane $\mathbf{Z}$, $\underset{\mathbf{¯}}{\mathbf{?}}$.

Apply postulates and theorems to complete the statements.

If two planes intersect, then                 

Apply postulates and theorems to complete the statements.

If there is a line j and a point P not in the line, then              

In the exercise answer on the basis of what appears to be true.

How many blue points are 1 cm from point $O$?

In the exercise answer on the basis of what appears to be true.

How many red points are 1 cm from point $O$?

In the exercise answer on the basis of what appears to be true.

How many red points are 2 cm from point $O$?

In the exercise answer on the basis of what appears to be true.

Each red point is said to be _____ from points $A$ and $B$.

Sketch and label the figures described.

Points $A$, $B$, $C$, and $D$ are coplanar, but $A$, $B$, and $C$ are the only three of those points that are collinear.

Sketch and label the figures described.

Line $l$ intersects plane $X$ in point $P$.