Lesson by Mr. Erlin, 2015
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BFF 01I can correctly identify relations between and among points lines and planes. Use terms including collinear, coplanar, parallel and skew.
To the left are the class notes explaining these terms by Mr. Erlin.
Here is a link to a video of me explaining the same concepts: BFF 01 Explanation Also see the Vocabulary page for short definitions of terms used in my explanation, along with a better explanation of dimensions. |
Image courtesy of sparknotes.com
Image courtesy of clarku.edu
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I can explain the Parallel Postulate: through a point not on a given line there is exactly one parallel line.
BFF 02See the first picture to the left as a representation of the parallel postulate as worded above. Essentially it says that you can define two unique parallel lines with two collinear points and one noncollinear point.
Euclid stated this same postulate in an interesting (albeit confusing) way in his book, the Elements: "That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." This statement can be reworded in many different ways. For instance: Parallel lines are the same distance from each other at any two points along each line perpendicular with each other. While this postulate (originally Euclid's fifth postulate) was for centuries considered inferior and not used in many proofs, it was instrumental in the discovery of Non-Euclidean geometry. Extra Resources
Euclid's Parallel Postulate- TED-Ed |
Image courtesy of http://ncetm.org.uk
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BFF 03I can, using Examples and Counterexamples, answer questions regarding points, lines and planes using “Always”, “Sometimes” or “Never”.
A statement is ALWAYS true when it has examples but no counterexamples.
A statement is SOMETIMES true when it has examples and counterexamples. A statement is NEVER true when it has no examples but at least one counterexample. With this in mind, check out my document with statements and try to fill in the blanks with ALWAYS, SOMETIMES, or NEVER. Click here for the document here for the answer key. Refer back to BFF 01 explanations above if you have trouble finding examples and counterexamples. Extra Resources
Math Plane - ASN |
BFF 04I can identify the differences among segments, rays, and lines (use the term endpoint).
Lines, as explained on the Vocabulary page, are infinitely long in the one dimension we call "length". Because they are infinite, they have no endpoints, or points at the very end of them.
Rays, however, do have an endpoint. They are basically half a line- they have one endpoint, unlike lines, however they do extend infinitely, if only in one direction. Segments do not extend infinitely like lines and rays, but are finite and end in two endpoints. |
Lesson by Mr. Erlin, 2015
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BFF 05I can correctly name points, segments, rays, lines, and planes (using letters and symbols).
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Image courtesy of coreknowledge.org.uk
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Bff 06To find the distance between two points on a number line, I can first state the Ruler Postulate using variables, then use substitution to find the distance between the two points, SHOW WORK.
We all know how to measure something with a ruler. Usually, we just line up the 0 mark with the first end of something, and count up the number line until we get to the mark that lines up with the end of the object. However, we don't usually know how to explicitly state why this works. This can be done using the Ruler Postulate:
The distance between A and B can be measured by taking the absolute value of the difference between their coordinates on a number line. AB=|a-b| A problem using this postulate may look like this: a=-2, b=2 Find AB AB=|a-b| AB=|(-2)-(2)| AB=|-4| AB=4 Extra Resources Geometry Center Worksheet by Mr. Erlin |
Image courtesy of www.utdanacenter.org
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Bff 07I can state the Segment Addition Postulate using variables, then use substitution to evaluate the lengths for a given problem. Use correct notation.
The Segment Addition Postulate is pretty intuitive. Simply put, a part plus a part equals a whole. Looking at the second diagram at left, and calling the measure of the first part AB, the measure of the second part BC, and the measure of the whole thing AC, we can state the postulate as follows:
A problem using this postulate may look like this: AB=3x, BC=30, AC=8x Find AC State the postulate: AB+BC=AC Substitute: 3x+30=8x Solve: 5x=30 x=6 AC=48 Extra Resources Worksheet by Mr. Erlin |
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Bff 08I can demonstrate which notations for segments can be used in congruencies and which can be used in equalities.
Concruency (≅) is in a completely different family than equality (=). If a statement has symbols from only one family, then it's valid. If there are symbols from different families, it is invalid.
See the table to the left for a list of segment symbols in each family (when we get to angles, this table will grow a bit). Extra Resources Video by MinuteMath |
Bff 08.5h1I can derive the Segment Overlap Theorem from previously known Postulates and the Algebraic Properties of Equality.
This is the first theorem you are asked to derive. It is essentially explained in the Google Drawing I created to the left. If you are not confident deriving this formula, see the video walk-through below: Deriving the Segment Overlap Theorem Postulates you may need to justify your logic can be found on the page Theorems and Postulates. Extra Resources Video by Mr. Erlin |
BFF 08.5H2I can state the Segment Overlap Theorem (include a sketch of the corresponding diagram) using variables, then apply the theorem to find lengths of segments. SHOW WORK.
See the diagram at the left.
First, write the theorem in letters: If AC=BD, then AB=CD Extending the theorem (using Angle Addition Postulate), we can say: AB+BC=AC and therefore: AB+BC=BD substituting, we get: (52)+(20)=BD solving: 70=BD Extra Resources |
Midpoint formula can also be used on a 2D graph
photo courtesy of hotmath.com |
Bff 09I can state then apply the Midpoint Formula to find the midpoint of a segment on a number line.
To find the midpoint of a segment, we take the average the endpoints' values on a number line, usually represented by single lower-case letters.
Using the image at left, the midpoint formula can be stated as follows: m=(a+b)/2 Applying: If a=10 and b=20, then: m=(10+20)/2 m=30/2 m=15 Extra Resources Video by virtualnerd |
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Bff 10I can state three valid conclusions given that C is the midpoint of segment AB. (Consider both congruence and equality).
Refer to the diagram to the left and the BFF 08 table to check the validity of statements (remember, valid statements have symbols on only one side of the table).
If C is indeed the midpoint of AB, then: 1. AC=CB 2. segment AC≅segment CB We know from the Segment Addition Postulate that: AC+CB=AB Replacing either CB with AC or vice versa (since they are equal), we can say that: 3. AC+AC=AB 2(AC)=AB 4. CB+CB=AB 2(CB)=AB Extra Resources Additional Explanation by Mr. Erlin |
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Bff 11Given a segment, I can sketch a bisector correctly showing congruent parts.
Lines and rays cannot be bisected.
However, segments can be bisected with points, lines, and planes. To sketch a bisected segment that correctly shows congruent parts, draw the same number of slashes on each side of the bisector, as shown at left. Extra Resources Perpendicular Bisector- Khan Academy |