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*<EFG.)

(Which means the measure of angle EFG)

*Adjacent Angles*

<JHK and <GHK are adjacent angles. They share vertex H, have ray HK in common, and do not overlap.

*Angle Addition Postulate*

The measure of an angle created by two or more adjacent angles is equal to the sum of the measures of the individual angles.

-<JHG will measure 22` + 130`

-*m*<JHG = 152`

(*m*<JHK + *m*<KHG = *m*<JHG)

*Applying the Angle Addition Postulate*

Which expression reads *m*<AED?

If m<AED = 132`, what is the degree measure of the smallest angle in the diagram?

(3x + 4) + (5x) + (2x + 8) = 132

*m*<AEC =

*m*<CEB =

*m*<BED =

*Angle Bisectors*

An angle bisector divides an angle into two congruent angles.

**Introduction to Proof**

-Conjecture: A statement though to be true but not yet proved true or false

-Deductive reasoning: The process of utilizing facts, properties, definitions, and theorems to form a logical argument

-Reflexive property: The property that states that for any real number x, x = x; or that a figure and its parts (e.g., sides, angles, triangles, etc.) are congruent to themselves

-Substitution property: The property stating that if two values are equal, then they are interchangeable in an equation; or if two figures are congruent, then they are interchangeable in a statement

-Symmetric property: The property that states that the left and right sides of an equation or congruence statement are interchangeable

-Transitive property: The property states that for all real numbers

*x*, *y*, and *z*, if *x* = *y* and *y* = *z*, then *x* = *z*; or if two figures(or sides, angles, etc.) are each congruent to a third figure, then the two figures are congruent

*Properties of Equality and Congruence*

Reflexive property(X = X):

-<ABC ~ <ABC

-*m*<ABC = *m*<ABC

Symmetric property:

-If XY ~ MN, then MN ~ XY

(Either term on both sides of an equality or congruence statement can just be interchanges)

Deductive reasoning: The process of utilizing facts, properties, definitions, and theorems to form a logical argument.

-Justify our steps

-Prove that two segments are congruent

Ex. Julia hates getting wet, and has a rule to always have an umbrella when it rains. If you see it raining, you can assume Julia has her umbrella.

Inductive reasoning:

Making generalizations based on observations and patterns

-A lot of basketball players are 6ft tall. So when you hear about a basketball player, you’ll assume they’re 6ft tall.

(Not guaranteed to be correct)

*Proofs*

Proofs involve:

-Given information, in words or a diagram

-A statement to be proven

-An argument using deductive reasoning and justification of steps in a logical order

-A conclusion

*Two-Column Proofs*

Given: AB = CD and EF = CD

Prove: AB = EF

*Paragraph Proofs*

Given: AB = CD and EF - CD

Prove: AB = EF

We are given that AB = CD and EF = CD. By the definition of congruence,

AB = CD and EF = CD(AB equals the length of CD, EF equals the length of CD)

Can substitute EF from the expression.

(Couldn’t do that with the congruency statements because substitution property is only for equality.)

Using the substitution property, we conclude the length of AB = EF

*Diagrams*

Can assume:

-Adjacent angles

-Vertical angles

-Linear pairs

-Collinear points

-Opposite rays

Cannot assume:

-Segment measure

-Angle measure

-Parallel lines

-Perpendicular lines

*Linear Pairs and Vertical Angles*

-Adjacent angles: Two coplanar angles with a common side, a common vertex, and no common interior points

-Congruent angles: Two angles that have the same measure

-Linear pair: Two adjacent angles whose noncommon sides are opposite rays

-Vertical angles: Opposite angles formed by two intersecting lines

*Vertical Angles*

Vertical angles are opposite(nonadjacent)angles formed by two intersecting lines.

*Vertical Angles Theorem*

If two angles are vertical angles, then they are are congruent angles

*Finding Unknown Angle Measures*

Can we find all the other angle measures using properties of linear pairs and vertical angles?

(Refer to notebook)

**Complementary and Supplementary Angles**

-Complementary angles: Two angles whose measures have a sum of 90 degrees

-Linear pair: Two adjacent angles whose noncommon sides are opposite rays

-Opposite rays: Rays that are collinear and have the same endpoint but run infinitely in opposite directions

-Supplementary angles: Two angles whose measures have a sum of 180 degrees

Linear pair postulate: If two angles form a linear pair, then they are supplementary.

Supplementary angles are two angles whose measures have a sum of 180.

(They don’t have to make a linear pair, they just need their measures to sum up to 180 degrees).

Complementary angles are two angles whose measures have a sum of 90

(They don’t have to be adjacent)

*The Congruent Complements Theorem*

If two angles are complements of the same angle or of congruent angles, then the two angles are congruent.

-If <1 and <2 are complements, and <3 and <2 are complements, then <1 = <3

-If <4 and <5 are complements, <6 and <7 are complements, and <5 = <6, then <4 = <7

*The Congruent Supplements Theorem*

Congruent supplements theorem: If two angles are supplements of the same angle or of congruent angles, then the two angles are congruent.

-If <1 and <2 are supplements, and <3 and <2 are supplements, then <1 = <3

-If <4 and <5 are supplements, <6 and <7 are supplements, and <5 = <6, then <4 = <7

**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*<EFG.)

(Which means the measure of angle EFG)

*Adjacent Angles*

<JHK and <GHK are adjacent angles. They share vertex H, have ray HK in common, and do not overlap.

*Angle Addition Postulate*

The measure of an angle created by two or more adjacent angles is equal to the sum of the measures of the individual angles.

-<JHG will measure 22` + 130`

-*m*<JHG = 152`

(*m*<JHK + *m*<KHG = *m*<JHG)

*Applying the Angle Addition Postulate*

Which expression reads *m*<AED?

If m<AED = 132`, what is the degree measure of the smallest angle in the diagram?

(3x + 4) + (5x) + (2x + 8) = 132

*m*<AEC =

*m*<CEB =

*m*<BED =

*Angle Bisectors*

An angle bisector divides an angle into two congruent angles.

**Introduction to Proof**

-Conjecture: A statement though to be true but not yet proved true or false

-Reflexive property: The property that states that for any real number x, x = x; or that a figure and its parts (e.g., sides, angles, triangles, etc.) are congruent to themselves

-Substitution property: The property stating that if two values are equal, then they are interchangeable in an equation; or if two figures are congruent, then they are interchangeable in a statement

-Symmetric property: The property that states that the left and right sides of an equation or congruence statement are interchangeable

-Transitive property: The property states that for all real numbers

*x*, *y*, and *z*, if *x* = *y* and *y* = *z*, then *x* = *z*; or if two figures(or sides, angles, etc.) are each congruent to a third figure, then the two figures are congruent

*Properties of Equality and Congruence*

Reflexive property(X = X):

-<ABC ~ <ABC

-*m*<ABC = *m*<ABC

Symmetric property:

-If XY ~ MN, then MN ~ XY

(Either term on both sides of an equality or congruence statement can just be interchanges)

Deductive reasoning: The process of utilizing facts, properties, definitions, and theorems to form a logical argument.

-Justify our steps

-Prove that two segments are congruent

Ex. Julia hates getting wet, and has a rule to always have an umbrella when it rains. If you see it raining, you can assume Julia has her umbrella.

Inductive reasoning:

Making generalizations based on observations and patterns

-A lot of basketball players are 6ft tall. So when you hear about a basketball player, you’ll assume they’re 6ft tall.

(Not guaranteed to be correct)

*Proofs*

Proofs involve:

-Given information, in words or a diagram

-A statement to be proven

-An argument using deductive reasoning and justification of steps in a logical order

-A conclusion

*Two-Column Proofs*

Given: AB = CD and EF = CD

Prove: AB = EF

*Paragraph Proofs*

Given: AB = CD and EF - CD

Prove: AB = EF

We are given that AB = CD and EF = CD. By the definition of congruence,

AB = CD and EF = CD(AB equals the length of CD, EF equals the length of CD)

Can substitute EF from the expression.

(Couldn’t do that with the congruency statements because substitution property is only for equality.)

Using the substitution property, we conclude the length of AB = EF

*Diagrams*

Can assume:

-Adjacent angles

-Vertical angles

-Linear pairs

-Collinear points

-Opposite rays

Cannot assume:

-Segment measure

-Angle measure

-Parallel lines

-Perpendicular lines

*Linear Pairs and Vertical Angles*

-Adjacent angles: Two coplanar angles with a common side, a common vertex, and no common interior points

-Congruent angles: Two angles that have the same measure

-Linear pair: Two adjacent angles whose noncommon sides are opposite rays

-Vertical angles: Opposite angles formed by two intersecting lines

*Vertical Angles*

Vertical angles are opposite(nonadjacent)angles formed by two intersecting lines.

*Vertical Angles Theorem*

If two angles are vertical angles, then they are are congruent angles

*Finding Unknown Angle Measures*

Can we find all the other angle measures using properties of linear pairs and vertical angles?

(Refer to notebook)

**Complementary and Supplementary Angles**

-Complementary angles: Two angles whose measures have a sum of 90 degrees

-Linear pair: Two adjacent angles whose noncommon sides are opposite rays

-Opposite rays: Rays that are collinear and have the same endpoint but run infinitely in opposite directions

-Supplementary angles: Two angles whose measures have a sum of 180 degrees

Linear pair postulate: If two angles form a linear pair, then they are supplementary.

Supplementary angles are two angles whose measures have a sum of 180.

(They don’t have to make a linear pair, they just need their measures to sum up to 180 degrees).

Complementary angles are two angles whose measures have a sum of 90

(They don’t have to be adjacent)

*The Congruent Complements Theorem*

If two angles are complements of the same angle or of congruent angles, then the two angles are congruent.

-If <1 and <2 are complements, and <3 and <2 are complements, then <1 = <3

-If <4 and <5 are complements, <6 and <7 are complements, and <5 = <6, then <4 = <7

*The Congruent Supplements Theorem*

Congruent supplements theorem: If two angles are supplements of the same angle or of congruent angles, then the two angles are congruent.

-If <1 and <2 are supplements, and <3 and <2 are supplements, then <1 = <3

-If <4 and <5 are supplements, <6 and <7 are supplements, and <5 = <6, then <4 = <7