Download 11.1 Basic Notions

11.1 Basic Notions
Euclid’s Axioms
1
There is exactly one line that contains any two points.
Euclid’s Axioms
1
There is exactly one line that contains any two points.
2
If two points line in a plane then the line containing these points
lies in the plane.
Euclid’s Axioms
1
There is exactly one line that contains any two points.
2
If two points line in a plane then the line containing these points
lies in the plane.
3
If two distinct planes intersect, they do so at a line.
Euclid’s Axioms
1
There is exactly one line that contains any two points.
2
If two points line in a plane then the line containing these points
lies in the plane.
3
If two distinct planes intersect, they do so at a line.
4
There is exactly one plane that contains any three colinear points.
Euclid’s Theorems
1
A line and a point determine a unique plane.
Euclid’s Theorems
1
A line and a point determine a unique plane.
2
Two distinct parallel lines determine a unique plane.
Euclid’s Theorems
1
A line and a point determine a unique plane.
2
Two distinct parallel lines determine a unique plane.
3
Two distinct intersecting lines determine a unique plane.
Basic Terms
Definition
A point is a location in space and has no dimension. We use a capital
letter to denote a point.
Basic Terms
Definition
A point is a location in space and has no dimension. We use a capital
letter to denote a point.
Definition
A line is an infinite collection of colinear points that has no depth but
has length. We label a line with either a lowercase script letter l or two
capital letters, denoting two points on the line.
Basic Terms
Definition
A point is a location in space and has no dimension. We use a capital
letter to denote a point.
Definition
A line is an infinite collection of colinear points that has no depth but
has length. We label a line with either a lowercase script letter l or two
capital letters, denoting two points on the line.
Order does←→
not matter
as the line is infinite in both directions. We
←→
could use AB or BA
More Terms
Definition
Colinear means that the points lie on the same line.
o
•A
•B
•C
Which are colinear?
/
Still More Terms
Definition
A ray is a subset of a line consisting of an initial point and extending
infinitely in one direction.
A•
•B
−→
−→
Is this ray properly named as AB or BA?
/
Still More Terms
Definition
A ray is a subset of a line consisting of an initial point and extending
infinitely in one direction.
A•
/
•B
−→
−→
Is this ray properly named as AB or BA?
Definition
A line segment is a subset of a line consisting of two endpoints.
o
•A
•B
Which is the correct way to name this line segment, AB or BA?
/
You Guessed It ... More Terms
Definition
A plane is an infinite collection of coplanar lines that has no thickness
but has two dimensions, length and width.
We either label a plane with a lowercase Greek letter α or at least 3
points.
You Guessed It ... More Terms
Definition
A plane is an infinite collection of coplanar lines that has no thickness
but has two dimensions, length and width.
We either label a plane with a lowercase Greek letter α or at least 3
points.
Definition
Intersecting lines are two lines in the same plane that share exactly
one common point.
Definition
Concurrent lines are more than two lines in the same plane that share
exactly one point.
Intersecting Lines v. Concurrent Lines
What is the difference between these two sets of lines?
_
o
?
/
•

o
_
•
/
And Still More Terms
Definition
Perpendicular lines are two lines in the same plane that intersect at a
right angle.
And Still More Terms
Definition
Perpendicular lines are two lines in the same plane that intersect at a
right angle.
Regardless of which type of intersection, two lines that intersect do so
at a
.
And Still More Terms
Definition
Perpendicular lines are two lines in the same plane that intersect at a
right angle.
Regardless of which type of intersection, two lines that intersect do so
at a
.
Definition
Parallel lines are two lines in the same plane that do not intersect.
And Still More Terms
Definition
Perpendicular lines are two lines in the same plane that intersect at a
right angle.
Regardless of which type of intersection, two lines that intersect do so
at a
.
Definition
Parallel lines are two lines in the same plane that do not intersect.
Definition
Skew lines are lines that are noncoplanar, meaning they cannot
intersect because they are in different planes.
Equal Measure v. Congruence
We have to be careful with notation ...
Two geometric structures that are same are congruent and have equal
measure.
Equal Measure v. Congruence
We have to be careful with notation ...
Two geometric structures that are same are congruent and have equal
measure.
For example, if AB and CD are exactly the same, other than the letters
we use, we could say the following:
AB ∼
= CD
AB = CD
mAB ∼
= mCD
Clarifications on Intersections
1
Lines intersect at a point.
Clarifications on Intersections
1
Lines intersect at a point.
2
Planes intersect at a line.
Clarifications on Intersections
1
Lines intersect at a point.
2
Planes intersect at a line.
3
Lines can be parallel (perpendicular) to a plane if the line is in a
plane that is parallel (perpendicular) to said plane.
Clarifications on Intersections
1
Lines intersect at a point.
2
Planes intersect at a line.
3
Lines can be parallel (perpendicular) to a plane if the line is in a
plane that is parallel (perpendicular) to said plane.
4
The angle where two planes meet at the line creating the
half-planes is called a dihedral angle. So if planes are
perpendicular, the dihedral angle measures 90◦ .
Angles
What is an angle?
7
A
•
B•
•
C
'
Angles
What is an angle?
7
A
•
B•
•
C
'
Definition
An angle is the intersection of two rays. The intersection of the rays is
at a point called the vertex and the straight parts of the rays are called
sides.
Angles
What is an angle?
7
A
•
B•
•
C
'
Definition
An angle is the intersection of two rays. The intersection of the rays is
at a point called the vertex and the straight parts of the rays are called
sides.
What can we name this angle?
Types of Angles
Acute angle:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle: angle measure between 90◦ and 180◦
Straight angle:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle: angle measure between 90◦ and 180◦
Straight angle: angle measure is 180◦
Reflex angle:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle: angle measure between 90◦ and 180◦
Straight angle: angle measure is 180◦
Reflex angle: angle measure is between 180◦ and 360◦
Complementary angles:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle: angle measure between 90◦ and 180◦
Straight angle: angle measure is 180◦
Reflex angle: angle measure is between 180◦ and 360◦
Complementary angles: two adjacent angles who’s measures add
to90◦
Supplementary angles:
Types of Angles
Acute angle: angle measure between 0◦ and 90◦
Right angle: angle measure is 90◦
Obtuse angle: angle measure between 90◦ and 180◦
Straight angle: angle measure is 180◦
Reflex angle: angle measure is between 180◦ and 360◦
Complementary angles: two adjacent angles who’s measures add
to90◦
Supplementary angles: two adjacent angles who’s measures add
to 180◦
Adding and Subtracting Angles
When talking about fractions of angles, we can use standard decimals
or we can use minutes and seconds, just like with time.
Compute
48◦ 350 27” + 37◦ 500 40”
Adding and Subtracting Angles
When talking about fractions of angles, we can use standard decimals
or we can use minutes and seconds, just like with time.
Compute
48◦ 350 27” + 37◦ 500 40”
Compute
48◦ 350 27” − 37◦ 500 40”
Angle Conversion
Convert
48◦ 350 27” to a number of degrees.
Angle Conversion
Convert
48◦ 350 27” to a number of degrees.
Convert
21.42◦ to degrees, minutes and seconds.
Finding Missing Angles
Example
If we are given that ∠ABC and ∠DBC are complementary and that
m∠ABC = 14 m∠DBC, what is the measure of the two angles?
Finding Missing Angles
Example
If we are given that ∠ABC and ∠DBC are complementary and that
m∠ABC = 14 m∠DBC, what is the measure of the two angles?
Example
Find the measure of ∠ABC and ∠DBE.
_
F
3x
o
•
•
C
A
90◦ D •
2x
•B
•
E
/
Circles and Arcs
Circles and Arcs
_
AC is called a minor arc because the associated central angle is less
than
180◦ .
_
ABC is called a major arc because the associated central angle is
greater than 180◦ .
Angles and Arcs
The measure of the arc associated with a central angle is equal to the
central angle.
Angles and Arcs
The measure of the arc associated with a central angle is equal to the
central angle.
The measure of the arc associated with an inscribed angle is equal to
twice the measure of the inscribed angle.
Angles and Arcs
Example
_
Suppose m∠CDF = 25◦ . Find the measure of the major arc CBF.
Angles and Arcs
Angles and Arcs
We are given that m∠CDF = 25◦ , and this is an inscribed angle.
There is a relationship between the central angle and inscribed angle
_
associated with the arc CF.
Angles and Arcs
We are given that m∠CDF = 25◦ , and this is an inscribed angle.
There is a relationship between the central angle and inscribed angle
_
associated with the arc CF.
_
So, mCF = 50◦ .
Angles and Arcs
We are given that m∠CDF = 25◦ , and this is an inscribed angle.
There is a relationship between the central angle and inscribed angle
_
associated with the arc CF.
_
So, mCF = 50◦ .
_
The measure of the major arc CBF is 360◦ − 50◦ = 310◦ .