Geometry Notes, Chapter 1-2

Transcription

1 Geometry Notes, hapter 1- In these sections, if you know the vocabulary, many questions are common sense. Use your text to review any vocab listed below that you don t know in each section. These notes are list of material that you are responsible for in each section. If I make changes from the book, those changes are specified here. You NEVER have to memorize Postulate or Theorem numbers (eg. Theorem 1-, etc.) If you forget the name of a postulate or theorem, it is always OK to state the theorem or postulate. Section 1-: vocab: point plane space coplanar points bisector line undefined terms collinear points intersection Section 1-3: segment opposite rays coordinate postulate congruent segments ray number line length congruent rules: Ruler Postulate (just understand concept) Segment ddition Postulate (know inside out) can be abbreviated to SP or Seg. dd Post can only be used to add TWO segments at a time! Section 1-4: angle vertex right angle straight angle adjacent angles sides of an angle acute angle obtuse angle congruent angles bisector of an angle Section 1-5: Postulate 6 : Through any two points there exists exactly one line - used to justify adding a line to a diagram by connecting two points The other postulates are rules that you should be able to visualize, but not necessarily "use" in a problem. rules : Protractor Postulate (understand concept) ngle ddition Postulate know both parts completely!! can be abbreviated to P or dd. Post. can only be used to add TWO angles at a time!

2 Section -1: if-then statements (a.k.a. conditional statements or conditionals) hypothesis conclusion counter-example biconditional (if and only if) : true only if both conditionals are true ( If and only if is often abbreviated to iff by mathematicians but read as if and only if ) an definitions always be written as biconditionals? nswer: Yes an postulates and theorems always be written as biconditionals? nswer: No, only sometimes keep this in mind as you learn new theorems throughout the year Section -: With the following properties, you should know both the technical definitions written in your text and the translations (even more important) below. These should be memorized. addition property : "you can add equal things to both sides of an equation" subtraction property : "you can subtract equal things from both sides of an equation" multiplication property : "you can multiply both sides of an equation by the same number" division property : "you can divide both sides of an equation by the same number (except 0)" substitution property : "the eraser principal : in an equation or inequality, you may erase one quantity and replace it by a quantity which has been shown to be equal" reflexive property : "anything equals itself" symmetric property : "you can switch what is on the left and right side of the = sign" transitive property : "if one thing equals a second and a second equals a third, then the first equals the third" (You may extend this further: if a=b and b=c and c=d and y=z, then a=z) distributive property (as defined in the text, familiar to you as factoring out a common term) Questions to ask yourself: If you use substitution, ask What is the original equation? What thing is being replaced by an equal thing? If you use something like the addition property, ask yourself What is the original equation? What equal things are being added to both sides of the equation? For substitution, you MUST refer to the original equation and the equation/s that you use to substitute from. Example: substitution (5,3,) where statement 5 is the original equation. For addition and subtraction, also refer to the original equation and the line of the equality that gets added or subtracted to both sides. Note: you cannot add or take away the same thing from both sides, just equal things. You may need the reflexive property to make equal. One exception: you may add or subtract the same number to both sides of the equation. addition of = (5, 7) On pp, 38-39, you should read the 3 examples, and see how each reason justifies each statement. Review the Segment ddition Postulate and the ngle ddition Postulate. MEMORIZE exactly what each postulate justifies. Note that the ngle ddition Postulate justifies two different rules.

3 Section -3: Midpoint Theorem (Midpt Thm): know the theorem, how to use it in a proof, and understand the proof of the theorem (in text). Recall that definitions are biconditionals! ngle isector Theorem ( is Thm): know the theorem and understand the proof Know the difference between the Midpoint Theorem and the Midpoint Definition!!!: the definition says that the point splits the segment into two equal parts and can be used like a biconditional (if it's a midpoint then the two segment are equal; if the two segments are equal, then it's a midpoint). The theorem only deals with doubling and halving. Likewise, know the difference between the ngle isector Theorem and Definition (just like the midpoint ones). Note: when referring to theorems, NEVER refer to the theorem number in the text; either call it by the name in the text (if there is one), or write it out in words. In our book, VT is Theorem - 3 because it s the third theorem in chapter. In another book, it might be the 4 th theorem in chapter 5. The theorem numbers have no real meaning so it s not worth knowing them! Section -4: complementary angles (complement) supplementary angles (supplement) vertical angles rule : Vertical ngle Theorem (a.k.a. VT) - know the theorem and how to prove it! Section -6: Know the two theorems, -7 (called The Supplementary ngle Theorem) and -8 (The omplementary ngle Theorem), and be able to prove them (the proof of -7 is in the text) Section -5: Perpendicular lines (know the conclusions the definition justifies - see p.56) and remember definitions are biconditionals! Know the three theorems on page 56. You must write these out in words. Using if and then can be helpful. Using the diagrams on page 56, examples are : if line JK is perp. to line MN, then angle 1 is congruent to angle (reason: if lines are perp, they form congr, adjacent angles) if angle 1 is congruent to angle, then the lines are perpendicular (reason: if lines form congr, adjacent angles, then the lines are perp) if O is perp. to O, then angle O and angle O are complementary (reason: if the ext sides of adjacent angles are perp, then the angles are complementary) The symbol can always be used in place of writing the word perpendicular (eg. is perpendicular to can be written as simply )

4 Making Geometry flashcards (optional) If flashcards help you remember rules better, the best way to do this for many rules is to draw a diagram of a diagram with the if on one side (in pictures) and the then along with the rule on the other side. Examples: Side 1 of card (the if part of a rule) side of card (the then part of the rule) m 1+ m =180 by P (ngle ddition Postulate) 1 The lines are perpendicular by the theorem that says: if lines form congruent adjacent s, then 1 is the midpt of by the Def of Midpt (midpoint: a point that breaks a segment into congruent segments) is the midpoint of = by Def of Midpt. (midpoint: a point that breaks a segment into congruent segments) = 1 by Midpt. Thm or = by Midpt. Def

5 is the midpoint of Some rules don t get triggered by a diagram, so you might do this: Substitution property If you have an equation, then you can replace one thing with something known to be equal to it. e sure to refer to the original equation, and the equations that show the equal things being switched, like: Substitution (5, 3) addition property of equality If you have an equation, you may add equal things to both sides. Hint: identify the original equation, and the equals that are added to both sides