What are the special parallelograms?

A rhombus is a parallelogram with four congruent sides and angles

- The diagonals of a rhombus are perpendeicular
- The diagonals bisect the angles
- They bisect each other


A rectangle is a parallelogram with four congruent angles also called an equiangular parallelogram

-The diagonals of a rectangle are congruent ands bisect each other

A square is an equilateral rectangle or regular quadrilateral or an equiangular rhombus

-A square's diagonals are congruent, perpendicular & they bisect each other.


Resources:
http://www.analyzemath.com/Geometry/rhombus_1.gif
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg33UNJHn2qgmKLlEmfdl5gWiTqg0So6hdwpnVIRRolCI0UH9mwuPEoAus_ZS4N8xf0WvRyEGnlT32YVvap1K_IBXX252UFw9gHPG3LWsDWa9QtAn0bu6XSTzaweBK8oPRZ6_Qqi8rFma1y/s1600/rectangle+18.PNG
http://www.google.com/imgres?um=1&hl=en&sa=X&tbo=d&biw=1366&bih=618&tbm=isch&tbnid=ZdYwe4YDYLWyGM:&imgrefurl=http://www.geom.uiuc.edu/~dwiggins/conj28.html&docid=L8um8FYz5pA4mM&imgurl=http://www.geom.uiuc.edu/~dwiggins/proof29.GIF&w=364&h=196&ei=LWj5UOXEL6mB0QGooIDIBw&zoom=1&iact=hc&vpx=2&vpy=179&dur=1887&hovh=156&hovw=291&tx=225&ty=48&sig=114690899695513863037&page=1&tbnh=137&tbnw=256&start=0&ndsp=18&ved=1t:429,r:0,s:0,i:122

Are there any other special shapes?

How do we use the properties of parallelograms?

Parallelogram: A quadrilateral with 2 pairs of parallel sides.

 Properties

• The opposite sides are congruent.  
• Consecutive angles are supplementary. Example: 
• The opposite angles are congruent. 
• Diagonals bisect each other 
• Not always congruent
• Not always perpendicular 

http://www.wyzant.com/Help/Math/Geometry/Quadrilaterals/Properties_of_Parallelograms.aspx
http://ef004.k12.sd.us/ch8notes.htm
http://www.ck12.org/user%3AYWtlZWxlckBhY2VsZnJlc25vLm9yZw../book/ACEL-Geometry-2012-2013/r2/section/6.5/Proving-Quadrilaterals-are-Parallelograms/

How do we use properties of trapezoids?

12.7.12

First thing to know is what is a trapezoid.

• A trapezoid has exactly one pair of parallel lines

 

Properties of trapezoids

1. Trapezoid Consecutive Angles Conjecture                                                                                         -The consecutive angles between the bases of a trapezoid are 180 degrees (supplementary).

     2. Isosceles Trapezoid Conjecture 

         - The base angles of an isosceles trapezoid are congruent.

<A is congruent to <D.

Try solving this:  Find x.

  

1st step: 2x+5=3x+2

2nd step:subtract 2x from both sides

                                                    Equation should look like this: 5=x+2

3rd step: subtract 2 from both sides and you get your answer.

                                                    Answer is 3+x

* You don't do this to 4x-2 and 5x+3 because those are parallel not congruent.

1st picture from http://www.google.com/imgresum=1&hl=en&tbo=d&biw=1517&bih=693&tbm=isch&tbnid=sGz7fgFw41GoMM:&imgrefurl=http://www.ck12.org/concept/Trapezoids---Intermediate/&docid=eeMQ8dVhvkcDNM&imgurl=http://www.ck12.org/flx/show/image/user%25253Ack12editor/201209241348511282764513_dff7a8f97ef56047035788567c45fc33-201209241348512460267699.png&w=708&h=511&ei=MabCUMukGPC10AGlqYG4Aw&zoom=1&iact=hc&vpx=4&vpy=274&dur=292&hovh=191&hovw=264&tx=128&ty=97&sig=108818101793549926599&page=2&tbnh=145&tbnw=202&start=27&ndsp=38&ved=1t:429,r:42,s:0,i:215

2nd picture from http://www.geom.uiuc.edu/~dwiggins/conj22.html

  

WHAT ARE THE RHOMBUS'S CONJECTURES?

How do we use triangle congruency?

11.27.12

Corresponding Parts of Congruent Triangles are Congruent

Don't get what it means. No problem

•  It means that if the triangles are congruent, so are the parts of the triangle.

Still don't get it. Here is an example:

Given: Line Segment (LS) AB is congruent to LS DE; LS AB is perpendicular to LS BD; and, LS DE is perpendicular to LS BD. 

Prove: LS AC is congruent to LS EC 

Solution: 

*NOTE: AAS MEANS ANGLE ANGLE SIDE EXAMPLE: 

CPCTC MEANS CONGRUENT PART OF CONGRUENT TRIANGLES ARE CONGRUENT.

 Problem from :http://www.algebra.com/algebra/homework/Geometry-proofs/Geometry_proofs.faq.question.221424.html

Are there any other triangle congruency shortcuts?

Aim: What are the congruence shortcuts?

11.14.12

Side-Side-Side Triangle Postulate(SSS)- If 3 sides of a triangle are congruent to 3 sides of another triangle, those two triangles are congruent.

Side-Angle-Side Triangle Postulate(SAS)-If 2 sides of a triangle and the INCLUDED ANGLE are congruent to the 2 sides of another triangle and its INCLUDED ANGLE, then the two triangles are congruent.
* Congruent angle must be between the congruent sides.

Angle-Side-Angle Congruence Postulate(ASA)-Triangles are congruent if any two angles and their INCLUDED SIDE are equal in both triangles.

Angle-Angle-Side Congruence Postulate(AAS)- Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles.

What "congruence shortcuts" dont work on triangles?

Aim: How do we use the triangle inequality theorem?

11.5.12

Triangle Inequality Theorem: The sum of any 2 sides of a triangle must be greater than the measure of the third side.

Example: 

Side AB is 25

Side BC is 30

Side AC is 50

The sum of AB and BC is greater than AC.

Is there a similar theorem for angle in a triangle?

How do we construct an angle bisector?

10.26.12

1. Create an angle ABC(any size, any degree) 

2. Put the sharp end of your compasses at point B. Draw an arc that intersects both sides of the angle. This makes one point on each side of the angle. 


3. Draw an arc from each of these points so that the arcs intersect.

4. Draw the ray from the vertex of the angle to the intersection of the two arcs drawn on the last step.










So we just did all of this:










http://www.shmoop.com/basic-geometry/angles.html
http://planetmath.org/CompassAndStraightedgeConstructionOfAngleBisector.html
http://www.ck12.org/concept/Congruent-Angles-and-Angle-Bisectors---Intermediate/r2/
http://mathforum.org/sanders/geometry/GP05Constructions.html

HOW TO CONSTRUCT A 30 DEGREE ANGLE USING JUST A COMPASS AND A STRAIGHTEDGE? 

How do we construct the basic geometric figures?

The geometric figure that were constructing is an equilateral triangle.
We are making  an equilateral triangle with a compass.

1. Start with the line segment. 
2. Put the needle of your compass at A and draw an arc.
3. Put the needle of your compass at B and draw an arc.
4. Draw the segments from the two endpoints to the point where the two arcs intersects.     

  

You should end with this :

• http://strader.cehd.tamu.edu/Mathematics/Geometry/PolygonLesson/Details/line_segment.html
• http://www.astarmathsandphysics.com/gcse_maths_notes/gcse_maths_notes_constructing_an_equilateral_triangle_60_degree_angle.html

HOW DO WE CONSTRUCT AN ISOSCELES TRIANGLE?

How do we use special segments in triangles?

The special segments we use are the median, angle bisector, and altitude. Median: connects the vertex of an angle to the midpoint of the opposite side. Angle Bisector: bisects angle in half, creating two congruent angles. Altitude: the height of a triangle. meets the "ground" at a right angle. 

Median: 








Angle Bisector:

                 


Altitude: 




http://www.atlantic.k12.ia.us/~period5/Units/Triangles/specilsegmntsoftriangles.htm

http://en.wikipedia.org/wiki/Angle_bisector_theorem