Studybitten

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Using Tools Of Geometry

Duplicate Segments and Angles

Sketch: free hand, no tools

Draw: use a ruler and a protractor (more accurate)

Construct: use a compass and a straight edge

Constructing Perpendicular Bisectors

Perpendicular Bisector: cuts a segment in half (creates a midpoint), and is perpendicular (creates 90 degree angles)

Median of a Triangle: Connects a vertex to the midpoint of the opposite side

Midsegment of a Triangle: connects midpoints of the sides of the triangle. It is parallel and half the length of the third side

Constructing Points of Concurrency

Concurrent: a set of lines that have a point in common

Circumcenter: point of concurrency for the perpendicular bisectors of a triangle

The circumcenter is equidistant from the vertices of the triangle

The circumcenter is on the inside of an acute triangle, the outside of an obtuse triangle, and on the midpoint of the hypotenuse of a right triangle

The circumcenter is the center of the circumscribed circle

Incenter: point of concurrency for the angle bisectors of a triangle

The incenter is equidistant from the sides of the triangle

The incenter is the center of the inscribed circle

It is always inside the triangle

The incenter is only on Euler's Line in an isosceles triangle

Orthocenter: point of concurrency for the altitudes of the triangle

The orthocenter is on the inside of an acute triangle, on the outside of an obtuse triangle, and on the right angle of a right triangle

The circumcenter, incenter and orthocenter are the same on an equilateral

Euler's Line: connects the circumcenter, orthocenter and centroid

The distance from the circumcenter to the centroid is one-half the distance between the centroid and the orthocenter. So the centroid divides Euler's line into a ratio of 2:1

Learn to construct: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwi3vuevydrTAhXKilQKHc3_DokQFggnMAA&url=https%3A%2F%2Fwww.khanacademy.org%2Fmath%2Fgeometry-home%2Fgeometric-constructions&usg=AFQjCNHzFlLSqF8JTjxVQbiWpOyts2z_Bg&sig2=6JOf4rpdcM7B74LgUGt9tg

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Angle Relationships

Linear Pair Conjecture:

If two angles are a linear pair, they add up to 180 degrees.

Vertical Angles Conjecture:

If two angles are vertical angles, then they have equal measures (are congruent).

Transversal:

A line that cuts through two other lines.

Corresponding Angles:

Angles in the same location but different intersection.

1 and 5, 2 and 6, 3 and 7, 4 and 8

Alternate Interior Angles (AIA):

Are angles between two lines, and on opposite sides of a transversal.

3 and 6, 4 and 5

Alternate Exterior Angles (AEA):

Are angles outside two lines and opposite sides of a transversal.

1 and 8, 2 and 7

Same Side Interior Angles (SSIA):

Are angles inside two lines, and on the same side of a transversal.

3 and 5, 4 and 6

Same Side Exterior Angles (SSEA):

Are angles outside two lines, and on the same side of the transversal.

1 and 7, 2 and 8

Parallel Lines Conjecture:

If parallel lines are cut by a transversal, then Corresponding Angles, AIA and AEA are congruent.

If parallel lines are cut by a transversal, then SSIA and SSEA are supplementary.

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Reasoning in Geometry

Inductive Reasoning

Inductive Reasoning: reasoning based on patterns

Conjecture: a rule based on inductive reasoning

Patterns:

Add 2

Add 5

Add 2 to each number you add by

          +2 +4 +6 +8 +10

Multiply by 2, Subtract 1

          x2  -1 x2 -1 x2 -1  x2

Powers:

-1, 1, 5, 11, 19, 29...

          +2 +4 +6 +8 +10

             +2 +2 +2 +2

1, 4, 9, 16, 25…

         +3 +5 +7 +9

            +2 +2 +2

11, -4, -14, -16, -7, 16, 56, 116, 199, 308

           -15  -10  -2  +9 +23 +40 +60 +83 +109

                +5 +8 +11 +14 +17 +20 +23 +26

                    +3 +3  +3   +3   +3   +3   +3

Finding the General (nth) Term

Take a sequence of numbers:

20, 27, 34, 41, 48

Look at the relationship between the term and value:

Term:     1     2     3     4     5     6     20            n    

Value:   20   27   34   41   48   55   153         7n+13

             +7   +7    +7  +7   +7   +7

Ask yourself what happens from term to term:

n represents the term number

7(n) + k

7(1) +k = 20

7 + k = 20

-7 = -7

k = 13

Check:

7(2) + 13 = 27

14 + 13 = 27

27 = 27

7(20) + 13 = 153

140 + 13 = 153

153 = 153

Mathematical ModelingParty Handshake Problems

10 people attend a party. Each person shakes everyone else's hand, how many handshakes take place?

1 person can't shake his own hand meaning each person shakes 9 hands, and each handshake accounts for two people. This reasoning gives you this equation:

10 (10-1)

_________

2

10 (9)

_________

2

90

______

2

45 handshakes

The equation for party handshakes is:

n (n-1)

________

2

Venn Diagram Problems

36 people are in a class. 24 of them are taking chemistry, 22 are taking calculus, 16 are taking both. How many students are enrolled in neither chemistry nor calculus.

chemistry                                            both                                            calculus

8                                               16                                               6

neither

6

Start from the middle and work your way out, there are 16 students taking both and 24 are taking chemistry meaning that 8 are only taking chemistry. Use the same type of logic to solve the rest.

Matrix Problems

Match the driver to the car given the following clues:

No first name initial is the same as the initial of the car

No person has the same number of letters as the car

Brian's brother was mad when Brian didn't buy a Toyota

Fred drove Kevin's BMW to the concert

                            Toyota        Kia        BMW        Ford

Tom                         X              X             X              O

Kevin                       X              X             O              X

Brian                       X               O             X             X

Fred                       O               X             X              X

Use the clues to narrow down you choices until the answer becomes clear

Deductive Reasoning

Deductive Reasoning: the process of showing how certain statements follow from directly agreed upon assumptions and proven facts

Solving problems algebraically is deductive reasoning:

2 (3x – 1) + 4x = 8x + 24                           Given

6x – 2 + 4x = 8x + 24                    Distributive Property

10x – 2 = 8x + 24                    Simplify/Combine Like Terms

2x – 2 = 24                             Subtraction Property of Equality

2x = 26                                     Addition Property of Equality

x = 13                                       Division Property of Equality

Some mathematical proofs are also deductive reasoning

Angles A, B and C form Triangle ABC

m∠A + m∠C = 90 degrees                         Given

m∠B + m∠C = 90 degrees                         Given

m∠A + m∠C = m∠B + m∠C               Substitution

m∠A = m∠B                           Subtraction Property of Equality

∠A ≅ ∠B                                     Definition of Congruence

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Introduction to Geometry

Introducing Geometry

Point: is the most basic building block of geometry. It has no size, only location. You represent a point with a dot and label it with a capital letter. EX:

P

Line: a straight, continuous arrangement of infinitely many points. It is infinite in length and has no thickness. Name a line by giving the letter names of any two points on the line and by placing the line symbol above the letters.

Plane: is like a flat surface that extends infinitely along its length and width. It has no thickness, represent a plane with a four sided figure and name it using a script capital letter or three points on it

Collinear: On the same line

Coplanar: On the same plane

Line Segment: consists of  two points called the endpoints of the segment and all the points between them that are collinear with the two points. To name line segment AB, use a segment symbol and the two endpoints.  Order does NOT matter. 

Midpoint: the point on a segment that is the same distance from both endpoints.

Bisector: Something that divides a segment into two congruent segments.

Congruent: Same size and shape

Ray: a part of a line that begins at a point and extends infinitely in one direction. A ray is named with two letters. The first letter is the endpoint, the second letter is any other point on the line and thearrow above always points to the right.

Angle: two rays with a common endpoint (the vertex). The rays are the sides of the angle.

∠ABC

Angle Bisector: a ray that divides an angle into two equal parts.

Parallel Lines: lines that are coplanar and never intersect

Skew Lines: Lines that are on different planes

Perpendicular Lines: Lines that intersect at a 90 degree angle

Right Angle: an angle that measures 90 degrees

Acute Angle: an angle that measures less than 90 degrees

Obtuse Angle: an angle that measures more than 90 degrees, but less than 180 degrees.

Complementary Angles: are a pair of angles that have a sum of 90 degrees

Supplementary Angle: are a pair of angles that have a sum of 180 degrees

Vertical Angle: are angles formed by two intersecting lines; they share common vertex but not a common side

Linear Pair of Angles: angles that share a vertex and a common side and their non-common sides form a line

http://studyasesthetic.blogspot.com/2017/05/introducing-geometry.html

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Studying

"Our higher education system is one of the things that makes America exceptional. There's no place else that has the assets we do when it comes to higher education. People from all over the world aspire to come here and study here. And that is a good thing." - Barack Obama

Many people have bad studying habits including a few of my own friends. People try to have good habits but they don't know how to form them, or they have these good habits but still don't understand the content that they are learning. My goal is to help with both of those problems, feel free to ask questions but I do have a limited education but I will try my best to get the answers you need.

And no, I will not do your homework for you.