6.1-6.3: Planes. 11-30-10, 25 days till Christmas excluding today

Once I finished my biology notes, I moved on to this task.

Here is what we learned...

1) not all planes fly

a) a geometric plane is a 2D surface that extends infinitely in all directions. It is defined by any 3 or more points where at least 1 is non-colinear.

b) When planes intersect, the intersection is a line. If a line intersects a plane not containing it, the inter section is exactly one point.

2) Any two lines prove a plain. Parallel, intersecting... BUT WAIT! NOT SKEW! You could probably come up with over 9000 planes to two lines as long as they are not skew.

Let this remind you that skew does not work.

a) When a line intersects a plane, the point of intersection is a "foot." For a line to be perpendicular to a plane, it must be perpendicular to every line passing through the foot.
3) A plane is parallel to a line if they never intersect. A plane is parallel to another plane if they do not intersect, as well.

(They aren't touching)
a) in this picture, with closer review, you'll notice that the parallel planes are being cut by a transversal plane. WHAT DOES THIS MEAN? Not a double rainbow, but all of the lines of intersection are parallel. Makes enough sense, right?

I hope you find this new blog much more helpful.
-Shane McPartlin
Remember: everyday is only as good as you make it.

Chapter 3 Review

 I had to do this blog on a chapter 3 review.
Chapter 3 was all about congruent figures. 

We learned how to prove triangles congruent by three postulates. Those were the SSS, SAS, and ASA postulate. 

SSS-
Given :  AB ≅ DE
BC ≅ EF
AC≅DF

Prove: ΔABC ≅ ΔDEF  

 Sense we know that if 3 corresponding sides are congruent we know that the triangles are congruent by SSS.











SAS- Given :  AB ≅ DE
BC ≅ EF
B ≅ E
Prove: ΔABC ≅ ΔDEF

 We can prove two triangles congruent if we have one angle in between two congruent sides. So by SAS we can prove these two triangles congruent. 










ASA-Given :  AB ≅ DE
B ≅ E
A ≅ D
Prove: ΔABC ≅ ΔDEF

 We can prove two triangles congruent if we have one congruent side in between two congruent angles.  So by ASA we can prove triangles congruent.











My definitions are in simple terms.  The exact definitions for each postulate is as follows. 
SSS postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
SAS postulate: If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. 
ASA : If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. 

Next we learned about circles.
Theorem 19: all radii of a circle are congruent. This is helpful in proving triangles congruent. 
We also learned about CPCTC. Which means that the corresponding parts of congruent triangles are congruent.  Which means that if two triangles are congruent that every corresponding side and angle are congruent. 
Here is a proof that involves both CPCTC and radii.  (O MEANS CENTER OF CIRCLE)

Given: OP
Prove: AB ≅ CD

 In this we know that there are 4 radii, and that they are all congruent.  We also know that                      CPD ≅APB. So by SAS we can prove the triangles congruent. And sense we learned CPCTC we know that sense the two triangles are congruent, that every corresponding side and angle is congruent too.  So we can prove that AB≅CD.






Next we learned about medians and altitudes.  
Altitude: of a triangle is a line segment drawn from any vertex of the triangle to the opposite side.  The altitude is perpendicular to one of the sides.  Every triangle has three altitudes.  Also if used in a proof: altitude -implies- perpendicular-implies- right angle.
Example:










Medians: of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. Every triangle has three medians.  Also if used in a proof: median -implies- segment divided into two congruent segments.
Example:

In isosceles triangles the altitude is the median, and vice versa.  

Next we learned about auxiliary lines. An auxiliary line is a line segment drawn that do not appear in the original figure.  Such that they connect two points already in the diagram, that help to prove what needs to be proven. 

Auxiliary lines can help prove proofs.  Draw them when needed.










Proof examples:

Given: CD and BE are altitudes of ΔABC
AD≅AE
Prove: DB ≅ EC

In this proof we use the postulate of ASA to prove triangles congruent.  Then we use CPCTC to prove segments congruent.  But then we go BEYOND CPCTC to prove more.  This proof involved altitudes, ASA, CPCTC.








Given: CFD ≅ EFD
FD is an altitude
Prove: FD is a median

In this proof we have to prove that an altitude is a median (which means it's isosceles).  We prove the triangles congruent, then use CPCTC to prove it is a median. 









Next we learned about overlapping triangles.  They are still proven the same way, but sometimes it's hard to recognize where the triangles are.  You may want to redraw them, or make the lines more apparent. 

 This is an example of an overlapping triangle. 
ΔCBF and ΔFDE are overlapping.  I'm not going to prove this one because you solve it the same basic way. By using SSS, SAS, or ASA. 







In the next section we just learned about types of triangles.
Scalene triangle- a triangle i which no two sides are congruent.










Isosceles triangle-a triangle in which at least two sides are congruent.










Equilateral triangle-a triangle in which all sides are congruent. (all equilateral triangles are equiangular)













Equiangular triangle-a triangle in which all angles are congruent. (all equiangular triangles are equilateral)







Acute triangle-a triangle in which all angles are acute.(angles<90)








Right triangle-a triangle in which one of the angles is a right angle.












Obtuse triangle-a triangle in which one of the angles is an obtuse angle.(90<obtuse<180)

We also learned about Angle-Side theorems. 

-If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is the opposite the longer side.
-If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the longer side is opposite the larger angle. 
Also shorter side opposite shorter angle, and shorter angle opposite shorter side.
SOOO basically this-

There was also another theorem about angles that we learned before this section that was-if two sides of a triangle are congruent, the angles opposite the sides are congruent. (congruent angles -> opposite sides congruent too)












Proof example:
Given: AB ≅AC  
Prove: 1≅2

HL postulate- if there exists a correspondence between the verticies of two right triangles such that the hypotenuse and a leg of one triangle are congruent to he corresponding parts of the other triangle, the two right triangles are congruent. 

so even thou it does not fall into SSS, SAS, or ASA, these triangles are congruent by HL.  There is only one line that fits the distance created by the hypotenuse and one of the legs if it is a right triangle. 







Proof:
Given: GJ is the altitude of HK
HG≅KG
Prove: ΔHGJ≅ΔKGJ

And we are finally done....YAY!!!!!!!!! (sorry the pictures are soo small)


-Jennifer Kendall

Ch 2 Review

I lost the game <----- Highlight first

As being one of the final four, I am proud to introduce to you,

CHAPTER 2 BASIC CONCEPTS AND PROOFS

The first section is 2.1 "Perpendicularity"

Perpendicular- lines, rays, or segments that intersect at all right angles

http://www.mathopenref.com/perpendicular.html

... And thats all about all there is for the first quick section.

The second section is 2.2 "Complementary and Supplementary Angles"

Complementary Angles- are two angles who sum is 90 degrees

Supplementary Angles- are two angles whose sum is 180 degrees

http://farm3.static.flickr.com/2682/4264858681_6342a820b6_z.jpg?zz=1

http://www.mathsisfun.com/geometry/complementary-angles.html

http://www.mathsisfun.com/geometry/supplementary-angles.html

The third section is 2.3 "Drawing Conclusions"

Procedures for D.C.

1. memorize theorems, definitions, and postulates.

2. Look for key words and symbols in the Given

3. Think of all theorems, definitions, and postulates that involve those keys

4. decide which t, d, p allows you to draw the conclusion

5. Draw a Conclusion, and justify

The fourth section is 2.4 "Congruents Supplements and Complements"

Theorem 4: If angles are supplementary to the same angle, then they are congruent.

" " 5: If angles are supplementary to congruent angles, then they are congruent.

" " 6: If angles are complementary to the same angle, then they are congruent.

" " 7: If angles are complementry to congruents angles, then they are congruent.

The fifth section is 2.5 "Addition and Subtraction Properties"

Theorem 8: If a segment is added to two congruent segments the sums are congruent. (Addition Property)

Theorem 9: If an angle is addes to two congruent angles, the sums are congruent. (Add. post.

Theorem 10: If congruent segments are added to congruent segments, the sums are congruent. (Add. post.)

Theorem 11: If congruent angles are added to congrunet angles, the sums are congruent.

Theorem 12: If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)

Theorem 13: If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subt. Prop.)

The sixth section is 2.6 "Multiplication and Division Properties"

Theorem 14: If segments (or angles) are congruent, their like multiples are congruent. (Muliplication Property)

Theorem 15: If segments (or angles) are congruent, their like divisions are congruent. (Division Property)

The seventh section is 2.7 "Transitive and Subtraction Properties"

Theorem 16: If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (Transitive Property)

Theorem 17: If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. (Trans. Prop.)

For the final section in Ch. 2, 2.8 "Vericles Angles"

Opposite rays- two collinear rays that have a common endpoint and extend in different directions

Verticle Angles- two angles formed by two rays forming the sides of one and the rays forming the sides of the other are opposite rays.

Theorem 18: Verticle Angles are congruent.

http://www.mathsisfun.com/geometry/vertical-angles.html

HopinG thAt it has been ten MinutEs.

And if you don't know what the Game is
http://en.wikipedia.org/wiki/The_Game_(mind_game)

Yes, it has its own website http://www.losethegame.com/

-Peter Kessel

Chapter 1 Review

Well in chapter 1 we started off by learning that when saying that when objects are the same they are congruent. __~ __ ~
Such as AB = CD or <1=<2

When we are comparing two numbers, values or lengths we say that they are equal.
Such as AB=CD or m<1=m<2

We also learned about a Postulate vs. a Theorum
A postulate(or axiom) is a statement that is accepted as true without proof.
A theorum is a statement that can be proven to be true

Conditional Statements:
Conditional- p→q
Where p is the hypothesis and q is the conclusion.
An example of this is the conditional statement that if you live in Birmingham this
implies that you live in Michigan
Converse- q→p
Such as if you live in Michigan, you live in Birmingham, which is sometimes true.
Inverse- ~p→~q
Such as if you don't live in Birmingham, you don't live in Michigan, which is also
sometimes true
Contrapositive- ~q→~p
Such as if you don't live in Michigan, this implies you don't live in Birmingham, which
is true. If the conditional statement is true, this means that the contrapositive is true.

Some of the other things we learned were that points on the same line are called collinear

We learned the triangle inequality theorum, which statest that the sum of any two sides of a triangle must be larger than the third.

Thank you all for letting me put this off until the very end, I hope you enjoy this blog post.

I won the second game.
-Andrew Barton

Chapter 4 Review -- Lines in the Plane

4.1 -- Detours and Midpoints

A detour proof is a proof that involves proving more than one pair of triangles congruent to solve the problem.

Procedure for Detour Proofs:

1. Determine which triangles you must prove to be congruent to reach the required conclusion.

2. Attempt to prove the triangles congruent. If you can't, take a detour. (See steps 3-5 below)

3. Establish which parts of the diagram you must prove to be congruent to prove the congruence of the triangles.

4. Find a pair of triangles that you can prove congruent and contain a pair of parts need for the main proof. (See step 3)

5. Prove triangles in step 4 congruent.

6. Use CPCTC and complete proof.

Midpoints sometimes need to be located in geometry problems. They can be determined by the following formula :

4.2 -- Case of the Missing Diagram

Sometimes geometry problems will not have a diagram but you will be given a statement and are expected to set up your own problem. It is important that you can effectively set up a problem correctly. There are several techniques you can use.

"If...then..." -- The given information is usually found in the "if" clause and what we need to prove is usually in the "then" clause.

Example: If the base of an isosceles triangle is extended in both directions,

then the exterior angles formed are congruent.

Given:△ABC Iso.

Prove:∠ABX≅∠ACY

If the word "then" is left out, or the conclusion comes before the hypothesis, the hypothesis is always follows the word "if" and always contains the givens.

Example: If a triangle is isosceles, the triangle formed by its base and the angle bisectors of its base angles is also isosceles.

Given:△ABC iso. Base BC

CX bis ∠ACB

BX bis∠ABC

Prove:△BXC iso.

If "if" and "then" are left out, this is a hint that the statement begins with the given information and ends with the conclusion.

Example: The median to the base of an isosceles triangle divides the triangle into two congruent triangles.

Given:△XYZ iso. Base YZ

XW median to YZ

Prove:△XYW≅△XZW

4.3 -- A Right-Angle Theorem

Theorem -- If two angles are both supplementary and congruent, they are right angles.

Example:

In the proof above, the use of the theorem is highlighted.

4.4 -- The Equidistance Theorems

The distance between an object is the length of the shortest path joining them.

Postulate:

A line segment is the shortest path between two points.

If two points are the same distance from a third point, the third point is said to be equidistant from the first two points.

The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.

-- In an isosceles triangle, the median is the altitude, and thus implies that it is the perpendicular bisector.

Theorems:

If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment.

On isosceles triangle LMN, if LP is the perpendicular bisector, we can determine that MP is congruent to PN because perpendicular bisector implies congruent segments. We can also imply that angle LPM and angle LPN are right angles because perpendicular bisector implies right angles.

4.5 -- Introduction to Parallel Lines

A plane is a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface.

A plane only has length and width, both infinite. A plane has no thickness.

If points, lines, etc. lie on the same plane, they are said to be coplanar. If points, lines, etc. don't lie on the same plane, they are noncoplanar.

Transversals

In the figure above, AB is the transversal of lines EF and CD. A transversal is a line that intersects two coplanar lines in two distinct points.

The area shaded green is considered the interior of the figure. The area shaded blue is considered the exterior region.

In the figure, angles 1 and 5 are corresponding angles. Look for an F shape to find corresponding angles. Angles 3 and 6 are alternate interior angles. Look for an N or Z shape. Angles 2 and 7 are alternate exterior angles.

Parallel lines are two coplanar lines that do not touch.

4.6 -- Slope

The slope (m) of a nonvertical line, segment, or ray containing (x1, y1) and (y1, y2) is defined by the following formula:

It doesn't matter which point is chosen as X1 or Y1.

When the slope formula is used on a vertical line, the denominator is zero. Since division by zero is undefined, the line has an undefined slope. When the formula is used on a horizontal line, the slope is zero.

Summary

Rising line --> positive slope

Horizontal line --> zero slope

Falling line --> negative slope

Vertical line --> undefined slope

Slopes of Parallel and Perpendicular Lines

Theorem: If two nonvertical lines are parallel, they have equal slopes.

The converse of that theorem is also true. (If the slopes of two nonvertical lines are equal, then the lines are parallel.)

Theorem: If two lines are perpendicular and neither is vertical, each line's slope is the opposite reciprocal of the other's.

The converse is also true. (If a line's slope is the opposite reciprocal of another line's slope, the two lines are perpendicular.)

Sorry for this ridiculously long blog post! Hope it helped.

-Olivia Sheridan