Deductive and Inductive Reasoning
Inductive and Deductive reasoning was at first a very hard idea to grasp in my mind. But after receiving great teaching from Kristy Good at Animas, I learned to grasp it easily in my mind. Deductive reasoning is simply the idea of taking several statistics or conjectures, and creating broader ideas from building off of those base ideas. Inductive reasoning, on the other hand, is simply "inducing" ideas based off one or more simple observations. For example, if you were to take 5 shirts out of some random wardrobe of about 50 shirts, you can use Inductive reasoning to suggest that this is a wardrobe of red shirts.
Recognizing Patterns and Creating Equations based off of them
One of the first things we have learned how to do in Geometry this year, is taking several numbers with hidden patterns, identifying those patterns, and creating equations out of them. For example, if you have a table like the one below, you would use the equation: y=DX+C. D would be the difference between 2 of the Y values, and to find C, you would subtract the difference times the X-value from any of the Y-values. In this case, the equation would be: Y=4X+2.
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Special Angle Relationships
Another subject we have already learned about is "Special Angle Relationships". These relationships include: corresponding angles, alternate interior angles, and alternate exterior angles. I have listed and defined each of these below.
-corresponding angles: the two angles, one on top of another, one exterior, while the other interior, formed when a pair of parallel lines are cut by a transversal.
-alternate interior angles: the two congruent interior angles, one to the left or right and above the other, formed when two parallel lines are cut by a transversal.
-alternate exterior angles: the two congruent exter
-corresponding angles: the two angles, one on top of another, one exterior, while the other interior, formed when a pair of parallel lines are cut by a transversal.
-alternate interior angles: the two congruent interior angles, one to the left or right and above the other, formed when two parallel lines are cut by a transversal.
-alternate exterior angles: the two congruent exter
Triangles' points of concurrency
Each triangle has one of each of the following: a centroid, an incenter, an orthocenter, and a circumcenter. A centroid is the point of concurrency in any triangle, where the medians of the triangle all meet at the same point. An incenter is the point of concurrency in any triangle, where all the angle bisectors meet at the same point. A circumcenter is the point of concurrency in any triangle, where all the perpendicular bisecors of the triangle's segments meet at the same point. An orthocenter is the point of concurrency in any triangle, where all the altitudes of the triangle meet at the same point. |
Creating Constructions (Compass and Straight Edge)
As a part of classwork, we learned to create certain constructions using only a compass and straight edge. To do this, we followed several steps for each of the following, retaining compass settings, and more: congruent line segments, congruent angles, perpendicular bisectors, parallel lines, and points of concurrency to fit into it all. So, I have prepared the steps to constructing 3 of these below for you.
- congruent segments: to create a segment congruent to a given segment, we used our compasses to measure the length of the given segment, created a point on a separate piece of paper, then place the point of the compass on that point, retaining the setting for the compass, then drawing a small arc with it. Once we have drawn that small arc, we connect it to the point with our straight edge.
- congruent angles: to create congruent angles without the use of a protractor, we first created a base ray with a point where the other ray would extend out to make a congruent angle to the original angle. then, we would use our compass on the original angle, making an arc that creates two points on its rays by holding the compass at the angle's point, then drawing the arc. We then angled our compasses to cover the distance from the point to the first arc point. Once we did that, we measured that distance out on our new rays, and made a big arc, just like the original. We then angled our compasses to cover the distance from the two new points on the original angle, and put a new arc from the new point on the new angle. After angling our compasses again to the original angle's point-to-top-arc distance, we made an X from the new angle's newest arc, and connected the point to the new point created by the X, and it became congruent to the original angle.
- Perpendicular Bisectors: to create perpendicular bisectors on lines or segments without the use of a protractor, we started with a random point on that line/segment. At that point, we put the point of our compasses, then extended their angles over the line at short distances, creating an arc at either side of the point. Then, from each of those 2 arcs, we stretched our compasses to other random distances, creating arcs above and below the random point on the original segment, using the vertexes of each point on the original line/segment. Once that is done, there will be 4 new arcs, creating 2 X's, one above the line and one below the line. once these X's are made, it is time to connect the points they make to construct a perpendicular to the original line!