Knowt4.9(8)
Geometry
**Euclidean Geometry**
\-Collinear: Points that lie on the same line
\-Coplanar: Contained within the same plane
\-Deductive reasoning: The process of utilizing facts, properties, definitions, and theorems to form a logical argument
\-Postulate: A statement accepted without proof; also known as an axiom
\-Theorem: A statement that has been proven based on previous theorems, postulates, or axioms
\
Euclid(Lived in Alexandria around 300 BCE)
\
*Undefinable Notions*
There are some concepts in Euclidean geometry that are considered undefinable, but are used to define other foundational objects.
Undefinable because there’s no clear definition.
\
*Point*
Location on coordinate plane designated by an ordered pair(x.y)
\-Label: P
\-0 dimensions
\
*Line*
Infinite series of point
\-No beginning or end
\-Label: italicized *m(or lowercase letter)* / two points on a line and mark them with PQ(one-dimensional)
\
*Plane*
Infinitely many lines, and has a length and width(two-dimensional)
\-No beginning or end
\-Label: Script/uppercase italicized letter. *P*
\
*Analyzing the Meaning of Distance*
\-Circular logic is inevitable here as you could change a number line so defining distance on one is impossible
\
*Postulates for Undefinable Notions*
Point existence postulate for lines: A line contains at least two points
\-Points on the same line are collinear
\
Unique line postulate: Through any two points there exists one and only one line
\
*Postulates for Undefinable Notions*
Point existence postulate for places: A plane contains at least three noncollinear points.
Objects that are on the same plane are coplanar.
\
Unique plane postulate: Through any three noncollinear points there exists one and only one plane
\
Flat plane postulate: If two point are in a plane, then the line that contains those two points lies entirely in that plane
(Lines are contained within the plane)
\
*Intersections of Geometric Figures*
If two distinct lines intersect, then they intersect at one and only one point.
\
One intersection: Two lines forming an X shape, making one intersection
\
No intersection: Called parallel lines
(Thought as extending infinitely always the same distance apart)
\
*Intersections of Geometric Figures*
If two planes intersect, then their intersection is a line
\-Envision a room and see a wall intersecting the floor, forming a line.
\
If a plane and a line intersect, they intersect at one point
\-Envision a piece of paper and pushing a pencil through it
\
**Defining Terms**
\-Angle: A figure formed by two rays that share a common endpoint
\-Circle: The set of all points in a plane that are a given distance away from a given point called the center
\-Line segment: A part of a line that has two endpoints and a specific length(CD and DC both work)
\-Parallel lines: Lines that lie in the same plane and do not intersect
\-Perpendicular lines: Lines that intersect to form right, or 90-degree, angles
\-Ray: Part of a line that has one endpoint and extends indefinitely in one direction(A; point B in between the arrow and the endpoint. AB)
\
*Rays and Line Segments*
Identify the rays and line segments shown in the diagram.
Rays are part of a line so one could start at D and extend through E; ray DE
\
*Precisely Defining Mathematical Terms*
How can we define the mathematical terms parallel lines and perpendicular lines?
*Parallel Lines:*
\-Lines on the same plane that never intersect(arrows in the middle)
*Perpendicular Lines:*
\-Lines that intersect at 90 or at right angles(coplanar; can lie in different intersecting planes)
\
*Parallel and Perpendicular Lines*
Lines outside of the plane aren’t parallel to lines inside the plane
\
*Precisely Defining Mathematical Terms*
How can we define the mathematical terms angle and circle?
\
Angle
\-Two rays with a common endpoint
\
Distance around an arc can be measured when given specific endpoints, but not precisely defined.
\
Undefined terms:
\-Point: Locations. A, B, C
\-Line: AB(Line with another line on top with arrows on each end), or lowercase letters
\
Defined terms:
\-Line segment: Portion of a line with two endpoints AB(Line on top without arrows)
\-Ray: One endpoint. CD(Ray on top). Ray DC would be in the opposite direction
\-Angle: Formed by two rays that meet at a common endpoint. DCE; C being the common endpoint
\
**Measuring Length and Angles**
*Measuring Segments*
\-Ruler postulate: The distance between any two points can be measured by finding the absolute value of the difference of the coordinates representing the points.
\-Acute angle: An angle measuring between 0 and 90 degrees
\-Adjacent angles: Two angles within the same plane that share a common side and vertex, but do not share any common interior points
\-Bisect: To divide into two congruent parts
\-Congruent angles: Two angles that have the same measure
\-Congruent segments: Two lines segments that have the same length
\-Midpoint: A point on a line segment that is equidistant from the two endpoints
\-Obtuse angle: An angle measuring greater than 90 degrees, but less than 180 degrees
\-Protractor: Tool used to measure an angle in degrees
\-Right angle: An angle in exactly 90 degrees
\-Straight angle: An angle whose measure is exactly 180 degrees
\-Reflex angle: An angle whose measures are strictly greater than 180 degrees but less than 360 degrees
\
PQ(bar on top)
\-P is at 15 at the number line
\-Q is at 35 at the number line
\
To get the absolute value, subtract 35 from 15. (Can subtract either way because absolute value always gives a non-negative number)
\-Can’t measure the length of a line
\-Can only measure the length of a line segment
\
PQ = 20 (add *units* if necessary)
\
*Segment Addition Postulate*
If L is between K and M, then KL + LM = KM. If KL + LM = KM, then L is between K and M
\
*Segment Addition Postulate*
Points R, S, and T are collinear. S is between R and T.
Example:
RT = 5x + 1
ST = 2x - 3
RS= 3x + 4
\
*Bisecting a Segment*
TR is bisected by segment AB, creating two congruent segments: PT and PR
(Congruent because they have the exact same length)
(Equal sign with a squiggle over top of it and that means congruence)
AB isn’t necessarily bisected by the line segment TR, because there’s nothing to indicate that the line segments PB and AP are congruent.
\
*Bisecting a Segment*
Midpoint theorem: If P is the midpoint of TR, then PT (congruent symbol) PR
PT and PR:
\-Are congruent halves of one whole
\-Have equal lengths
(Each smaller segment of a bisected segment is half the length of the longer segment)
PT = PR = ½TR
\
*Measuring Angles*
A protractor is a tool used to measure an angle in degrees
Protractor postulate: Given any angle, we can express its measure as a unique positive number from 0 to 180 degrees
(The orientation of the angle is irrelevant when it comes to measuring the angle)
(After measuring, symbolize it: *m*
\-Collinear: Points that lie on the same line
\-Coplanar: Contained within the same plane
\-Deductive reasoning: The process of utilizing facts, properties, definitions, and theorems to form a logical argument
\-Postulate: A statement accepted without proof; also known as an axiom
\-Theorem: A statement that has been proven based on previous theorems, postulates, or axioms
\
Euclid(Lived in Alexandria around 300 BCE)
\
*Undefinable Notions*
There are some concepts in Euclidean geometry that are considered undefinable, but are used to define other foundational objects.
Undefinable because there’s no clear definition.
\
*Point*
Location on coordinate plane designated by an ordered pair(x.y)
\-Label: P
\-0 dimensions
\
*Line*
Infinite series of point
\-No beginning or end
\-Label: italicized *m(or lowercase letter)* / two points on a line and mark them with PQ(one-dimensional)
\
*Plane*
Infinitely many lines, and has a length and width(two-dimensional)
\-No beginning or end
\-Label: Script/uppercase italicized letter. *P*
\
*Analyzing the Meaning of Distance*
\-Circular logic is inevitable here as you could change a number line so defining distance on one is impossible
\
*Postulates for Undefinable Notions*
Point existence postulate for lines: A line contains at least two points
\-Points on the same line are collinear
\
Unique line postulate: Through any two points there exists one and only one line
\
*Postulates for Undefinable Notions*
Point existence postulate for places: A plane contains at least three noncollinear points.
Objects that are on the same plane are coplanar.
\
Unique plane postulate: Through any three noncollinear points there exists one and only one plane
\
Flat plane postulate: If two point are in a plane, then the line that contains those two points lies entirely in that plane
(Lines are contained within the plane)
\
*Intersections of Geometric Figures*
If two distinct lines intersect, then they intersect at one and only one point.
\
One intersection: Two lines forming an X shape, making one intersection
\
No intersection: Called parallel lines
(Thought as extending infinitely always the same distance apart)
\
*Intersections of Geometric Figures*
If two planes intersect, then their intersection is a line
\-Envision a room and see a wall intersecting the floor, forming a line.
\
If a plane and a line intersect, they intersect at one point
\-Envision a piece of paper and pushing a pencil through it
\
**Defining Terms**
\-Angle: A figure formed by two rays that share a common endpoint
\-Circle: The set of all points in a plane that are a given distance away from a given point called the center
\-Line segment: A part of a line that has two endpoints and a specific length(CD and DC both work)
\-Parallel lines: Lines that lie in the same plane and do not intersect
\-Perpendicular lines: Lines that intersect to form right, or 90-degree, angles
\-Ray: Part of a line that has one endpoint and extends indefinitely in one direction(A; point B in between the arrow and the endpoint. AB)
\
*Rays and Line Segments*
Identify the rays and line segments shown in the diagram.
Rays are part of a line so one could start at D and extend through E; ray DE
\
*Precisely Defining Mathematical Terms*
How can we define the mathematical terms parallel lines and perpendicular lines?
*Parallel Lines:*
\-Lines on the same plane that never intersect(arrows in the middle)
*Perpendicular Lines:*
\-Lines that intersect at 90 or at right angles(coplanar; can lie in different intersecting planes)
\
*Parallel and Perpendicular Lines*
Lines outside of the plane aren’t parallel to lines inside the plane
\
*Precisely Defining Mathematical Terms*
How can we define the mathematical terms angle and circle?
\
Angle
\-Two rays with a common endpoint
\
Distance around an arc can be measured when given specific endpoints, but not precisely defined.
\
Undefined terms:
\-Point: Locations. A, B, C
\-Line: AB(Line with another line on top with arrows on each end), or lowercase letters
\
Defined terms:
\-Line segment: Portion of a line with two endpoints AB(Line on top without arrows)
\-Ray: One endpoint. CD(Ray on top). Ray DC would be in the opposite direction
\-Angle: Formed by two rays that meet at a common endpoint. DCE; C being the common endpoint
\
**Measuring Length and Angles**
*Measuring Segments*
\-Ruler postulate: The distance between any two points can be measured by finding the absolute value of the difference of the coordinates representing the points.
\-Acute angle: An angle measuring between 0 and 90 degrees
\-Adjacent angles: Two angles within the same plane that share a common side and vertex, but do not share any common interior points
\-Bisect: To divide into two congruent parts
\-Congruent angles: Two angles that have the same measure
\-Congruent segments: Two lines segments that have the same length
\-Midpoint: A point on a line segment that is equidistant from the two endpoints
\-Obtuse angle: An angle measuring greater than 90 degrees, but less than 180 degrees
\-Protractor: Tool used to measure an angle in degrees
\-Right angle: An angle in exactly 90 degrees
\-Straight angle: An angle whose measure is exactly 180 degrees
\-Reflex angle: An angle whose measures are strictly greater than 180 degrees but less than 360 degrees
\
PQ(bar on top)
\-P is at 15 at the number line
\-Q is at 35 at the number line
\
To get the absolute value, subtract 35 from 15. (Can subtract either way because absolute value always gives a non-negative number)
\-Can’t measure the length of a line
\-Can only measure the length of a line segment
\
PQ = 20 (add *units* if necessary)
\
*Segment Addition Postulate*
If L is between K and M, then KL + LM = KM. If KL + LM = KM, then L is between K and M
\
*Segment Addition Postulate*
Points R, S, and T are collinear. S is between R and T.
Example:
RT = 5x + 1
ST = 2x - 3
RS= 3x + 4
\
*Bisecting a Segment*
TR is bisected by segment AB, creating two congruent segments: PT and PR
(Congruent because they have the exact same length)
(Equal sign with a squiggle over top of it and that means congruence)
AB isn’t necessarily bisected by the line segment TR, because there’s nothing to indicate that the line segments PB and AP are congruent.
\
*Bisecting a Segment*
Midpoint theorem: If P is the midpoint of TR, then PT (congruent symbol) PR
PT and PR:
\-Are congruent halves of one whole
\-Have equal lengths
(Each smaller segment of a bisected segment is half the length of the longer segment)
PT = PR = ½TR
\
*Measuring Angles*
A protractor is a tool used to measure an angle in degrees
Protractor postulate: Given any angle, we can express its measure as a unique positive number from 0 to 180 degrees
(The orientation of the angle is irrelevant when it comes to measuring the angle)
(After measuring, symbolize it: *m*
4.9(8)
