Lines Within the Coordinate Plane

There are two equations that we have learned (so far) in geometry that help us with writing the equations of lines. From this, we know that if lines have the same slope, then they are parallel. If lines are perpendicular, then their slopes are negative reciprocals, etc.

Here's an example of perpendicular lines:




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Congruence of Triangles

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How do we say a triangle is congruent?


IF THE TRIANGLES:

- have corresponding angles
- have corresponding sides
- have side-side-side congruence
- have side-angle-side congruence
- have angle-side-angle congruence
- have angle-angle-side congruence
- have hypotenuse-leg congruence

Then they are congruent.

Defining Triangles

There are multiple types including:

- Acute Triangle: three acute angles
- Equiangular Triangle: three congruent acute angles
- Right Triangle: one right angle
- Obtuse Triangle - one obtuse angle



Holt Geometry Book

The definition of a triangle is a polygon that is three-sided.


Here are their classifications by their side lengths:

- Equilateral Triangle: three congruent sides
- Isosceles Triangle: at least two congruent sides
- Scalene Triangle: no congruent sides



Holt Geometry Book

Properties of Parallel Lines applied to Plane Geometry

Here is a list of some of the theorems/postulates that help us define parallel lines:
- converse of corresponding angles postulate
- converse of alternate interior angles theorem
- converse of alternate exterior angles theorem
- converse of same-side interior angles theorem
- parallel postulate

Parallel lines can apply to planes, because planes can be parallel to each other too.
Check out this example!


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Proporties of Parallel Lines and Euclidean Geometry

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Not to be confused with Blondie's album...



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The definition of parallel lines are lines that do not intersect and lie in the same plane.


When applying this concept of parallel lines to euclidean geometry, we could say that since there are multiple postulates and theorems (as you will see later) that help us with defining parallel lines, we can identify them in reality. For instance, if we know the properties of parallel lines, then we can show that train tracks (for example) are parallel to each other.


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Euclidean geometry goes hand in hand with the parallel postulate, which states that if there is a point that is parallel to a line, then there is exactly one line through it.

Developing and Expanding on Knowledge

Some of you may be thinking "How do we develop and expand concepts of what we've learned?" My friends, we do this by applying geometry to everyday life. Whether it is applying these concepts through our living styles, to just generally thinking about geometry, there are many ways that geometry is used in everyday life and can therefore be expanded upon. Don't believe me?

Holt Geometry Book

THAT ^^^ right there is proof (not A proof, but proof) that you, a mere common individual, uses geometry in real life. As a geometry student, I expand on my knowledge of flowcharts (in this case) by making myself a sandwich (preferably peanut butter, but since our school took that away, the sandwich option isn't as common).

Another way that I expanded my knowledge of geometry was by taking different types of conditional statements that I just happened to say in real life, and finding the converse, inverse, and contrapositive.

For example:

If I can go to the store, then I can by some cookies. (Conditional statement)

If I can by some cookies, then I can go to the store. (Converse)

If I cannot go to the store, then I cannot by some cookies. (Inverse)

If I cannot by some cookies, then I cannot go to the store. (Contrapositive)



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Formal Logic with Geometry

As I have stated before, you can use geometry to help you with formal logic. For example, if you see an object that makes a right angle, then you know that angle is going to measure to 90 degrees. If you see two lines that intersect at 90 degree angles, then you know that these lines are perpendicular. It's not too difficult once you get the hang of it! :)




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Conditions for Inductive and Deductive reasoning

Holt Geometry Book
INDUCTIVE REASONING:


For inductive reasoning, you reason that a rule or statement is true based on the truth of specific cases.


DEDUCTIVE REASONING:For deductive reasoning, you use logic to make conclusions. You may use a proof to make these conclusions.

The first example is an example of deductive reasoning.

The second example is an example of inductive reasoning.

Presenting Effective Argruments and Proofs

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You may be wondering: "How do I solve a proof?" Well, that's a good question....
The first thing you need to know about proofs is that there are two columns. The first column is the "Steps" column. The second column is titled "Reasoning." In the first column you put how you solved or proved the proof, and the second column explains how your steps are possible according to theorems and postulates. There are different types of proofs, such as algebraic and geometric. The example with red dividers is an algebraic proof and the other example is of a geometric proof. This year in geometry we will focus on geometric proofs and building on these proofs. The picture above shows what a proof looks like. You solve a proof by using deductive reasoning, which is when you come up with effective arguments to present your options for the proofs. This is also known as deductive logic. There's a multitude of postulates and theorems that you can use in order to get a better understanding of what could possibly go under the "Reasoning" column.

Geometry: in our world as we know it

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Geometry is all around us. It is used in various ways, and by many different people at multiple occupations. Architects, surveyors, scientists, gardeners, game designers, animators, piano constructors, rowers for crew or on boats in general, swimmers, manufacturers, machinists, engineers, and astronomers are some of the people who use geometry in their everyday lives and occupations. Geometry can be found in ordinary objects or in places you wouldn't expect. For example, a bicycle is a perfect example of geometry. If you wanted to, you could figure out the angle measurements on different parts of a bicycle. You could classify the different triangles on a bicycle too. For example, the bike above has a right angle in the top left corner, therefore, it must be a right triangle.