Points, Lines, Planes and Angles

State another name for the angle.

$\angle LES$

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle 2$
  

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle LAS$
  

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle ATL$
  

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle S$
  

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle LTS$
  

State whether the angle appears to be acute, right, obtuse, or straight.

• $\angle EDT$
  

Complete. $m\angle 3+m\angle 4=m\angle ?$

Complete. $m\angle ALS-m\angle 2=m\angle ?$

Complete. $m\angle 1=m\angle 2$, then ? bisects ?.

Complete. $m\angle LDA+m\angle ADE=\underset{¯}{?}$

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$90°$ angle

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$45°$ angle

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$45°$ angle

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$\mathbf{45}\mathbf{°}$angle

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$\mathbf{150}\mathbf{°}$angle

Without measuring, sketch each angle. Then use a protractor to check your accuracy.

$\mathbf{10}\mathbf{°}$angle

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

In the plane represented by your paper, how, many lines can you draw through $\mathit{O}$ that will form a $\mathbf{90}\mathbf{°}$ angle with $\overrightarrow{\mathit{O}\mathit{B}}$?

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

Using a ruler, draw a large triangle. Then use a protractor to find the approximate measure of each angle and compute the sum of the three measures. Repeat this exercise for a triangle with a different shape. Did you get the same result?

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

Find $\mathit{m}\mathbf{\angle }\mathbf{2}$, $\mathit{m}\mathbf{\angle }\mathbf{3}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}$ when the measure of $\mathbf{\angle }\mathbf{1}$ is:

a. 90    

b. 93

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

Express $\mathit{m}\mathbf{\angle }\mathbf{2}$, width="43" height="20" style="max-width: none; vertical-align: -4px;" $\mathit{m}\mathbf{\angle }\mathbf{3}$ and $\mathit{m}\mathbf{\angle }\mathbf{4}$ in terms of $\mathit{t}$ when $\mathbf{\angle }\mathbf{1}\mathbf{=}\mathit{t}$.

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

A careless person wrote, using the figure shown,  

$\mathit{m}\mathbf{\angle }\mathit{A}\mathit{O}\mathit{B}\mathbf{+}\mathit{m}\mathbf{\angle }\mathit{B}\mathit{O}\mathit{C}\mathbf{=}\mathit{m}\mathbf{\angle }\mathit{A}\mathit{O}\mathit{C}$.

What part of the Angle addition Postulate did that person overlook?

$\overleftrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathbf{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathbf{7}\mathit{x}\mathbf{+}\mathbf{3}$, $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{6}\mathit{x}\mathbf{+}\mathbf{7}$

$\overleftrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathit{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of$\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{6}\mathit{x}$, $\mathit{m}\mathbf{\angle }\mathit{K}\mathit{A}\mathit{T}\mathbf{=}\mathbf{90}\mathbf{-}\mathit{x}$

$\overleftrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathit{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathbf{5}\mathit{x}\mathbf{-}\mathbf{12}$,  $\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{6}$

 $\overleftrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathit{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathit{x}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{4}\mathit{x}$

$\overleftrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathbf{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{8}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{116}$

$\overrightarrow{\mathit{A}\mathit{L}}$ bisects $\mathit{\angle }\mathit{K}\mathit{A}\mathit{T}$. Find the value of $\mathit{x}$.

$\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathit{x}\mathbf{+}\mathbf{12}$, $\mathit{m}\mathbf{\angle }\mathbf{3}\mathbf{=}\mathbf{6}\mathit{x}\mathbf{-}\mathbf{20}$

$\overrightarrow{\mathit{O}\mathit{C}}$ bisects $\underrightarrow{\angle AOB}$, $\overrightarrow{OD}$ bisects $\angle AOC$, $\overrightarrow{OE}$ bisects $\angle AOD$, $\overrightarrow{OF}$ bisects $\angle AOE$ and  $\overrightarrow{OG}$ bisects $\angle FOC$.

1. If $m\angle BOF=120$, then $m\angle DOE=\underset{¯}{?}.$
2. If $m\angle COG=35$, then $m\angle EOG=\underset{¯}{?}$.

Theorem 1-1 states that two lines intersect at exactly one point. The diagram suggests what would happen if you tried to show two “lines” drawn through two points. State the postulate that makes this situation impossible.

State Postulate 6 using the phrase one and only one. 

Reword the following statement as two statements, one describing the existence and the other describing uniqueness:

A segment has exactly one midpoint.

Postulate 6 is sometimes stated as “Two points determine a line.”

Restate Theorem 1-2 using the word determine.

Postulate 6 is sometimes stated as “Two points determine a line.”

Do two intersecting lines determine a plane?

Postulate 6 is sometimes stated as “Two points determine a line.”

Do three points determine a line?

Postulate 6 is sometimes stated as “Two points determine a line.”

Do three points determine a plane?

State a postulate, or part of a postulate, that justifies your answer to each exercise.

Name two points that determine line l.

State a postulate, or part of a postulate, that justifies your answer to each exercise.

Name three points that determine plane M.

State a postulate, or part of a postulate, that justifies your answer to each exercise.

Name the intersection of the planes M and N.

State a postulate, or part of a postulate, that justifies your answer to each exercise.

Does $\overleftrightarrow{AD}$ lie in plane M?

State a postulate, or part of a postulate, that justifies your answer to each exercise.

Does plane N contain any points not on $\overleftrightarrow{AB}$?

Surveyors and photographers use a tripod for support.

Why does a three-legged support work better than one with four legs?

Surveyors and photographers use a tripod for support.

Explain why a four-legged table may rock even if the floor is level.

A carpenter checks to see if a board is warped by laying a straightedge across the board in several directions. State the postulate that is related to this procedure.

State Theorem 1-2 using the phrase one and only one.

Reword Theorem 1-3 as two statements, one describing the existence and the other describing uniqueness.

Planes $\mathbf{M}$ and $\mathbf{N}$ are known to intersect.

1. What kind of figure is the intersection of $\mathbf{M}$ and $\mathbf{N}$?
2. State the postulate that supports your answer to part (a).

Points $\mathbf{A}$ and $\mathbf{B}$ are known to lie in a plane.

1. What can you say about $\overleftrightarrow{\mathbf{A}\mathbf{B}}$?
2. State the postulate that supports your answer to part (a).

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.

Write the postulate that assures you that $\overleftrightarrow{AC}$ exists.

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 $\overleftrightarrow{AC}$.