In this post, I'll explain what are parallel lines, perpendicular lines, and slopes. I'll tell you how to determine the slope of a line and the relation between parallel and perpendicular lines.


3-3 Parallel Lines

41.The Reflexive, Symmetric, and Transitive Properties are true for parallel lines. You can see why in the example below:

Reflexive: AB || CD
Symmetric: If AB || CD, then CD || AB
Transitive: If AB || CD, and CD || EF, then AB || EF.

3-4 Perpendicular Lines

27.

28.The rungs on a ladder are perpendicular to the sides of the ladder. Parallel lines are lines that never intersect if they are extended forever. But the fact that the rungs are all perpendicular to the sides of the ladder helps prove that they are parallel to each other. This can be supported by Theorem 3-4-3.

3-5 Slopes

23. AB > 0-1

24. Lines that represent two cars driving at the same speed would be equal.

2 ways to determine a slope of a line

You can use the following formulas to find the slope of a line:

1.Using the two points' coordinates, you divide the difference of the y-coordinates with the difference of the x-coordinates to find your slope
2. Or use the equation: y=mx+b



The slopes of the lines on the left are the same because they are parallel to each other.
While the slope of the lines on the right, are negative reciprocals of each other because they are perpendicular. Meaning if one line's slope is -2, then the other is 1/2.

Post #3

In this post I will explain how the properties of equality and the properties of congruence are pretty much the same thing and I will also compare deductive proof and conjecture using inductive reasonig. 

For two segments to be congruent, they have to have equal lengths. Both properties of equality and congruence can be applied here. 

Reflexive Property

Equality: segment AB = segment AB

Congruence:  segment AB ≅ segment AB

Symmetric Property

Equality:segment AB = segment BA

Congruence: segment AB ≅ segment BA

Transitive Property

Equality: segment AB= segment CD, segment CD = segment EF, segment AB=EF

Congruence:segment AB≅ segment CD, segment CD  ≅segment EF, segment AB≅EF

Deductive proof is facts used to show that something is true and a conjecture is just an opinion based on incomplete information. These statements were formed using inductive reasoning. 

In conclusion, properties of equality and congruence mean the same thing when it comes to segments. The difference between deductive proof and conjectures is that deductive proof is proved to be true while conjectures aren't proven to be true.

Geometric Reasoning

In this topic, I learned how to easily identify biconditional statements and their conditionals and converse. I also learned what makes a definition a good one.

Biconditional Statements

 

(Biconditional) A statement is a biconditional statement if and only if it's two conditionals in the form "p if and only if q."

(Conditional) If a statement is biconditional, then it is two conditionals in the form "p if and only if q."

(Converse) If a statement is two conditionals in the form "p if and only if q," then it is biconditional.

Angle Bisector

Definition: a line that divides an angle into two equal parts

A definition is good if it reversible and can be written as a true biconditional:
A line is an angle bisector if and only if it divides an angle into two equal parts.

Traffic Ticket

You will get a traffic ticket if and only if you are speeding.

1)If you get a traffic ticket, then you are speeding.(false)

2) If you are speeding, then you will get a traffic ticket.(true)

The biconditional is false because there are numerous other reasons for receiving a traffic ticket such as driving under the influence, cell phone usage, or violating a car registration/ driver's license requirement. 

In conclusion, you can identify a biconditional statement using two conditionals. Also, if you want to know if a definition is a good definition it has to be reversible and can be written as a true biconditional.

Redesigning a Cereal Box

Redesigning a Cereal Box

I'm Cassidy, and welcome to my first ever blog post. Today's blog will be all about how I redesigned a cereal box which gave me less surface area and an increase in volume. This project gave me a deeper insight to the relationship between the surface area and volume of a cereal box. I also learned how changing the dimensions of a cereal box can benefit cereal box manufacturers.

What I noticed about the cereal box is that the product inside does not take up the entire space. Manufacturing of the cardboard for the cereal box costs the company money, especially with the excess surface area. The more cardboard there is, the more money is being spent.

Brands such as Kellogs and General Mills are known for their environmentally friendly cereal boxes. The boxes are made from 100 percent recycled cardboard. Some brands, specifically Kellogs, uses Number 2 plastic in their packaging which is commonly recycled. Therefore, the packaging is environmentally friendly.

Decreasing the excess packaging would save cereal companies loads of money. Reducing the surface area means you are using less cardboard and less money to manufacture the cereal box. This can be achieved by decreasing the height and length,and increasing the width of the box while trying not to lower the volume. This process will help reduce the packaging but you will still be able to package the same amount of product.

The design of a cereal box affects its shelf storage by how much space it takes. The surface area and volume contribute to how much space it takes. The size and design on the box contributes to its visual appeal. A small and plain looking box wouldn't catch the attention of a customer as much as a bigger and fancier looking one.

The surface area of a box is pretty much how much cardboard it takes to form the box. The surface area is calculated by finding the area of all 6 sides and adding them together. The volume of the box is how much space is within the box and this is calculated by multiplying the length, width, and height. They are both measurements of something and need the length, width, and height to be calculated. 

In conclusion, cereal boxes usually have excess packaging. To reduce this excess packaging, they have to minimize the surface area which can increase the volume of the box. A smaller surface area would save companies money. Boxes would be even more environmentally friendly by not wasting unnecessary cardboard.