Blog Entry #4

The Converse of the Corresponding Angles Postulate is used for constructing parallel lines. The Parallel Postulate swears that for any line, you can construct a parallel line through a point that is not on the line.

Following postulates and theorems, can help prove lines parallel, and prove angles congruent.
Other properties may apply to lines and could be used to prove lines parallel.

 Page 168 #42, the question asks if the information that is given allows you to conclude that the two lines A and B are parallel.

     Yes, they are parallel and the given information allows you to conclude that.

     Line A is intersected by a line (vertical angles.) The angle measures 125 degrees and one of these pairs are its opposite. Vertical angles have equal measures, so that means that the measure of the opposite angle 125 = 125 degrees. The vertical angle of angle 125 and angle 55 are considered same-side interior angles since they lie on the same side of the transversal between  lines A and B. Same-side interior angles are supplementary, they equal 180 when added (125 + 55 = 180) By using the Converse of the Same-Side Interior Angles Theorem, we can conclude that line A is parallel to line B.



     Page177 #27,( I drew this out but my camera isn’t working) In my drawing, segment AB is a perpendicular bisector of segment XY BUT segment XY is not a perpendicular bisector of segment AB.

    Page 177 #28 The rungs of the ladder are parallel because the two rungs are perpendicular to the same side. Theorem 3-4-3 states that, "If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other." By using this, it’s safe to say that the two coplanar lines are the rungs and the line is the side of the ladder. This means that the rungs are parallel.
    

    To find the slope of a line, you can use any two points on the line.

     If line W has a slope of a/b, then the slope of a line perpendicular to line W is -b/a. Two nonvertical lines are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines are perpendicular.

     Page 186 #23, the question asks for an inequality for the slope of a line perpendicular to line AB, while line AB has a slope greater than 0 and less than 1.

    By looking at the problem, the slope of a line greater than 0 would be m > 0. The slope of a line less than one would be m < 1. Since the problem gives two values in one sentence, we form a compound inequality. So the slope of line AB would be 1 > m > 0

     

     When perpendicular lines are multiplied, the product formed is -1. So this means that if the slope of line AB is 1 > m > 0, the slope of a line perpendicular to line AB should be its negative response. 1(x) = -1.The new slope should include -1 > m. 0(x)= -1 results with x = -1/0, a.k.a. undefined. After taking out undefined, the slope of a line perpendicular to line AB would be -1 > m. The slope of a line perpendicular to line AB is m < -1

     Page 186 #24) asks, if two cars are driving at the same speed, what is true about the lines that represent the distance traveled by each car?

   Both cars should pass the same distance with the time, if both cars are travelling at the same speed. The slope represents the speed. Because the speeds are the same, the slopes should be the same. When following the Parallel Line Theorem, we can conclude that the two lines that represent the distance are parallel, since the two lines have the same slope, and lines with the same slope are parallel lines.

     What are two ways to determine the slope of a line?

     One way to find the slope of a line is by using the slope formula. The slope formula is m = y2 - y1 / x2 - x1. To find y2, y1, x2, and x1, you can use any two points on the line. After choosing two points, put them into the formula and solve. The result is the slope of the line.

     Another way to find the slope of a line is by using a graph. When looking at the graph choose any two points. From there we can determine the slope using the ratio of rise to run. The rise is the difference in the y-values of any two points on the line, while the run is the difference in x-values. Find the rise by counting how many spaces the point was moved vertically.Up is positive, down is negative. After finding the rise, find the run. Count the spaces horizontally until you get to your second point. Left is negative, right is positive. The ratio of rise to run should be expressed as a fraction or whole number. This is the slope.

      Draw graphs of pairs of parallel lines and perpendicular lines and explain how their slopes are related.

Parallel Lines:

   As you can see in the photo, parallel lines don't intersect. 

They extend forever and never cross. For them to never cross, they would need to have the same slope, or else they would end up crossing. Because of this, parallel lines have the same slope. 

 

Perpendicular Lines:

     The slopes of perpendicular lines are negative reciprocals. When multiplied their product is -1. 

Algebraic Proof: Blog Entry #3

Throughout this blog entry, I will be identifying some of the properties of equality and the properties of congruence and reviewing them to use and write them as algebraic proofs.

Proof uses logic, definitions, properties, and proven statements to show that a conclusion is true. When writing proofs, it’s important to give justifications to show that each step is valid. These justifications can be made by using any piece of information that can be obtain from a problem.

Solving an equation uses proof, because several properties are used to solve the equation. Using  the Properties of Equality can help prove a statement to be true.

Example:

 Segment congruence means two line segments are congruent if they have the same length.

By knowing this, we could come to the conclusion that each property of equality have their own corresponding property of congruence. Properties of equality do apply to segments as well, and also three properties of congruence does too.

Reflexive Property of Equality: Segment AB = Segment AB

Reflexive Property of Congruence: Segment AB ≅ Segment AB

Symmetric Property of Equality: If Segment AB = Segment CD, then Segment CD = Segment AB

Symmetric Property of Congruence: If Segment AB ≅ Segment CD, then Segment CD ≅ Segment AB

Transitive Property of Equality: If Segment AB = Segment CD and Segment CD = Segment EF, then Segment AB = Segment EF

Transitive Property of Congruence: If Segment AB ≅ Segment CD and Segment CD ≅ Segment EF, then Segment AB ≅ Segment EF

  From the definition of segment congruence, the properties of equality and the properties of congruence are the same. It is true since they prove to be the same to each other, and the definitions, statements, and examples of each property are the same. We were able to  prove that these properties are true for segments .

 A deductive proof uses logic and reasoning to come to a valid conclusion. A conjecture based on inductive reasoning, is a statement believed to be true based on patterns formed from multiple observations. Using deductive reasoning and inductive reasoning is different, the processes use different methods to achieve solutions.

Blog Entry #2

35.) Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 

Biconditional: A statement is a biconditional if and only if it is written in the form "p if and only if q."
Conditional: If a statement is written in the form "p if and only if q," then it is a biconditional statement. (Truth Value: True)
Converse: If a statement is written in the form "p if and only if q," then it is a biconditional. (Truth Value: True

The biconditional is true since both the conditional and the converse are true.

36.) Use the definition of an angle bisector to explain what is meant by the statement "A good definition is reversible."

Definition: An angle bisector is a ray that divides an angle into two congruent angles.
A good definition is reversible since its statement has true value as a conditional statement and as a converse.
Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles.

Conditional: If a ray is an angle bisector, then it divides an angle into two congruent angles; True

Converse: If a ray divides an angle into two congruent angles, then it is an angle bisector; True

The definition of an angle is a good definition because it is reversible. Both conditional and converse are true, so that means the whole definition is true. 

41.) Write the two conditional statements that make up the biconditional "You will get a traffic ticket if and only if you are speeding." Is the biconditional true or false? Explain your answer.

Conditional: If you get a traffic ticket, then you are speeding.
Truth Value: False; you can get a ticket for other issues, like passing a red light.

Converse: If you are speeding, then you get a traffic ticket. 
Truth Value: False; You only get a traffic ticket when caught speeding. Those who don't get caught speeding does not get a traffic ticket.

The biconditional is false because both conditional and converse were false.
For the biconditional to be true, the conditional and converse have to both be true. But since they are both not true, the biconditional is false.

Redesigning Cereal Box

Hi! My name is Erika Mendiola but you can call me Erk for short. I am currently 14 years old, turning 15 this November 27th :) I love cooking, baking, & reading books. I am a Sophomore at Mount Carmel School, Saipan. This is my blog for Mrs.Buenaflor's 4th period Geometry class.

A few weeks ago, my class was given a project called, Cereal Investigation: Maximizing Volume, Minimizing surface area. We had to calculate all the dimensions of the cereal box and create a smaller box that could still fit in the cereal.

The cereal box that I used was Cheerios. What I noticed about the package was that half of it was unnecessary. The cereal box had half of Cheerios (which was on the bottom of the box of course,) and half air which filled the top.

If you think about the cereal box and how it was made, you'll know that the box was made from trees. If all Cheerios cereal, and other cereals, wasn't filled with half of air, think about how many trees that would've been saved from being cut down just to make the cereal box when half of it was really unnecessary. Making the cereal box smaller would save money, trees, and it would be very economically friendly.

What I, and everyone, could do to change the packages to reduce the excess packaging is to find a way to redesign a smaller cereal box, that could fit the same amount of cereal into it. I believe that the factories that make the cereal boxes should do this too.

The new cereal box I made could hold the same amount of Cheerios. It saves up space since it is smaller, and if cereal boxes were smaller, people would be amazed at how small the box is and how much cereal it holds!

Making the new cereal box took me many tries till I finally got it right. Without the surface area, you would not be able to find out if the new cereal box design takes up less space than the original cereal box design. And without the volume, you can't tell if the new box could hold the same amount of cereal or not.

Redesigning a new cereal box was very frustrating for me. But it did make me realize that the cereal boxes that factories made today are not economically friendly. If factories redesigned cereal boxes, it would save money, time, and trees. It would also be much better for the environment.



This is my redesigned cereal box: