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Geometry Notes IGP - 1: Lines and Segments Undefined Terms Point: A location in space; no size. (zero dimensions). Line: A continuous “straight” set of points that extends indefinitely (forever) in two opposite directions (one dimension). Plane: A continuous set of points forming a “flat surface” extending forever in two dimensions. Space: All points. Three dimensions. (Not real important until the end of the course. For now, we will do everything “in a plane.”) S Between: In the diagram, P is between A and B; Q and S are not. P A B Q Definitions A ray is a “half line;” it has one endpoint and extends indefinitely in one direction. B P A A line segment consists of two endpoints and all the points between them. B A The measure (length) of a line segment is the Line segment AB (at right) has measure (length) 8: B 8 A Note: AB (with the bar over it) represents the actual segment (an object); AB (without the bar) represents the length of the segment (a number) Two line segments are congruent () if E 5 C D 5 F Ex: In ABC, AB BC . If AB = 4x – 10, BC = 2x + 5 and AC = 3x – 2, find the perimeter of the triangle. The midpoint of a line segment If M is the midpoint of AB then Conversely, if A postulate is a statement (not a definition) that is accepted without proof. Postulate: Every segment has exactly one midpoint. Ex: M is the midpoint of line segment PQ . If PM = x2 – 8 and MQ = 2x + 27, find the numerical length of PQ . Three or more points are collinear if they are all E A B C D F Ex: Given FLAT , A is the midpoint of LT , FT = 28 and LA is 6 less than twice FL. Find the length of AT . A bisector of a segment is a line, ray or segment that intersects a segment at its midpoint. Therefore, a bisector of a segment Ex: C BD bisects CT at A. If BA = 3(x + 1), AD = x2 – 7, CA = x2 – 2x and AT = 4x – 8, find the length of BD. x2 – 2x x2 – 7 3(x + 1) B A 4x – 8 T D Geometry Notes IGP - 2: Angles Definitions (continued) An angle is the union of two rays with a common endpoint (the vertex). NOTE: Angles may be named in three ways. 1. By three letters, with the middle letter at the vertex, 2. (Sometimes) by a single letter at the vertex (only if there is no chance of confusion), 3. By a number or lower case letter placed inside the angle near the vertex. C Ex: In the diagram at right, 6 A The measure of an angle is the number of degrees in the angle. A 4 2 3 D 1 Note: The measure of an angle is a measure of rotation (turning). It has nothing to do with the “lengths” of the sides. 5 B 50 Acute angle: Right angle: Obtuse angle: Straight angle: Congruent angles: Two angles that have the same measure. 40 F 40 J Perpendicular segments (or lines or rays) B Ex: Given APC and BP , mAPB = 3x + 20 and mCPB = 5(x – 2). Determine if PB APC . C P A Two angles are complementary if their measures Two angles are supplementary if their measures Ex: The measures of two complementary angles are in the ratio 3:5. Find the measure of the larger angle. Two adjacent angles share a common ray but have no interior points in common. B A C O A Angle bisector: A ray that divides an angle into two congruent angles. P Postulate: Every angle has exactly one bisector. B Ex: HO OP , mHOT = 5x + 3 and mPOT = 2x + 28. Does OT bisect HOP? O T P H 5x + 3 O 2x + 28 Geometry Notes IGP - 3: Definitions and Drawing Conclusions Definitions (Review) In math, a precise definition should work “both ways.” (It is a biconditional.) Ex: A triangle is a polygon with exactly three sides. 1. If a polygon is a triangle, then it has exactly three sides. 2. If a polygon has exactly three sides, then it’s a triangle. Ex: A square is a polygon with exactly four sides. 1. If a polygon is a square, then it has exactly four sides. 2. If a polygon has exactly four sides, then it’s a square. Drawing simple conclusions We can use definitions to draw simple conclusions. Ex: Given: M is the midpoint of AB . . A . M .B Conclusion: Reason: Note: Remember, in proofs, a “given” is assumed to be true. Ex: Given: PQR and PQ QR . Conclusion: Reason: Ex: Given: mABC + mXYZ = 180 Conclusion: Reason: . P . Q .R Ex: Given: JKL is a right angle. J Conclusion: Reason: K L Conclusion: Reason: Conclusion: Reason: Ex: Given: BD bisects ABC A B Conclusion: Reason: D C Not: Not: Q Ex: Given: PQ QR Conclusion: Reason: P R Geometry Notes IGP - 4: Basic Postulates Postulates (aka Axioms) A postulate (also called an axiom) is a statement (not a definition) that is accepted without proof. A theorem is a statement that has been proven using definitions, postulates and previously proven theorems. Basic Postulates 1. Reflexive Postulate: 2. Transitive Postulate: If two things both equal the same (third) thing, then they equal each other. Ex: If a = c and b = c then B Ex: If AB BC and BC CA then C A Ex: Given: AEB BEC, CED BEC E Conclusion: A D C B 3. Substitution Postulate: Equal quantities may be substituted for each other in any expression. Ex: 2x + y = 6 y = 3x + 1 Ex: Given: mAOB + mBOC = 90 mAOB = mCOD A B C Conclusion: O 4. Partition Postulate: The whole equals Ex: Ex: . . . . A B C D A B O C D Ex: For each of the following, name the postulate illustrated. a. Amy is the same height at Bob. Bob is the same height as Chris. So Amy is the same height as Chris. b. Amy, Bob, Chris, Don, Emma and Fred are a hockey team. Fred is the goalie. George is another goalie. So Amy, Bob, Chris, Don, Emma and George are a hockey team. c. Amy, Bob, Chris, Don, Emma and Fred are a hockey team. Fred is the goalie. Herb is baseball pitcher. So Amy, Bob, Chris, Don, Emma and Herb are a hockey team. d. A soccer team is made up three forwards, four midfielders, three fullbacks and a goalkeeper. e. A basketball team is made up a center, two forwards, two guards and a goalkeeper. Ex: Which of the following is an example of the reflexive postulate? (1) Amy looks in the mirror. (2) Amy is the same height as Amy. (3) Amy is the same height as Bob. (4) Amy is taller than Bob. Bob is taller than Chris. So Amy is taller Chris. (5) None of these. Ex: Equality is transitive: If a = b and b = c then a = c. Which of the following are also transitive? a. not equal to () b. greater than (>) c. parallel (||) d. perpendicular () e. “lives in the same town as” f. “lives next door to” g. “goes to the same school as” h. “is related to (by blood)” Geometry Notes IGP - 5: Addition and Subtraction Postulates 5. Addition Postulate: Equal quantities may be added to both sides of an equation. Ex: If and a=b x=y Note: In the Addition Postulate, we always add two equations to get a new equation. then Ex: 2x + 3y = 9 x – 3y = 3 Note: Always line up the equal signs and add vertically on each side. A Ex: Given: ADB , AEC AD AE , DB EC D E F B C Note: For addition of line segments to make sense, a) They must share an endpoint. . A . B . C . D .C b) They must be collinear. . A c) They must not overlap. . A AB BC . B . B . D . C A Ex: Given: ABC , FED AB ED , BC FE F AB CD AC BD B E C D Ex: Given: AFB DCE, BFE ECB B A F C D E Note: For addition of angles to make sense, the angles must be adjacent (and non-overlapping). P X R Y XOZ + XOY = Z O XOY + YOZ = YOZ+ PQR = Q M Ex: Given: YDM NDO O Y N D 6. Subtraction Postulate: Equal quantities may be subtracted from both sides of an equation. Ex: If and a=b x=y Note: In the Subtraction Postulate, we always subtract two equations to get a new equation. then P Ex: ABCD , AC BD A B C D Note: For subtraction of line segments to make sense, a) They must share an endpoint. AC AB . B . A b) They must be collinear. .. C AC BC c) They must overlap. AB BC Ex: NRT , NGL, NT NL, RT GL N R G T L B A Ex: ABC ADC, ABD CDB D C Note: For subtraction of angles to make sense, the angles must X a) share a ray and XOZ – XOY = Y b) overlap XOZ – YOZ = YOZ – XOY = O Z P Ex: Given: QPS TPR Q R B D S T C Geometry Notes IGP - 6: Multiplication and Division Postulates 7. Multiplication Postulate: Both sides of an equation may be multiplied by equal (non-zero) quantities. Ex: If and a=b x=y then x2 2 x 1 1 6 3 2 Ex: 8. Division Postulate: Both sides of an equation may be divided by equal (non-zero) quantities. Ex: If and a=b x=y then Ex: 4y = 3x + 20 Variation: “Halves* of equal quantities are equal.” (* or thirds or fourths, etc) If a = b, then E A Ex: Ex: ABC ADC DE bisects ADC BF bisects ABC 2 D 1 F C B Geometry Notes IGP - 7: Statement-Reason Proofs Proofs A formal geometry proof is a series of statements in logical order. Each statement is justified by a reason. Statements 1. Should start with one or more givens 2. Are facts/true that are relevant to the problem 3. Should follow a logical order Each new statement should either a. Be a direct conclusion from one or more previous statements or b. Go together with one or more previous statements to lead to a conclusion 4. The final statement is whatever was to be proved. Reasons 1. Should explain why the statement is true, often buy referring to previous statements 2. Acceptable reasons are a. Given (but only if the statement really was given!) b. Definitions: write them out. c. Postulates: by name for the few that have a name; otherwise write them out. d. Previously proven theorems: write them out. J Ex: Given: KJM NJL Prove: KJL MJN 1. Mark the givens on the diagram. (See what you know.) 2. Work backwards. (Find out what you need to prove.) 3. Try to have a plan. (Figure out how to get from what you know to where you need to go.) 4. Write the proof. K L B D M N C Ex: Given: AMPL , AM EX , EX PL Prove: AP ML E A X M P L Geometry Notes IGP - 8: Simple Angle Theorems A theorem is a statement that has been proven using definitions, postulates and/or previously proven theorems. Theorem: All right angles are congruent. Given: A and B are right angles Prove: A B Theorem: All straight angles are congruent. Theorem: If two adjacent angles form a straight line, they are supplementary. Given: AOC and BOC, AOB Prove: AOC and BOC are supplementary C A O B Theorem: If two adjacent angles form a right angle, then they are complementary. Theorem: If two angles are congruent, then their supplements are also congruent. Given: 1 4, 2 supp. to 1, 3 supp. to 4 Prove: 2 3 2 1 3 4 Theorem: If two angles are supplementary to the same angle, then they are congruent. Note: The previous two theorems are still true if the words “supplements” and “supplementary” are replaced by “complements” and “complementary”. Definition: Vertical angles are non-adjacent angles formed by two intersecting lines. 1 4 2 3 Theorem: Vertical angles are congruent. (Prove for HW.) Ex: Given: ABCD , ABP DCP Prove: CBP BCP P A Statement B C Reason N Ex: Given: MOR , LOQ , NO LO , PO OR Prove: MON QOP Reason P Q M L Statement D O R Geometry Notes IGP – 9/10: Proofs Practice Ex: Given: AB AC , AE AF Prove: BAE FAC D C E A B F T Ex: Given: PIW , GIN , IT bisects PIG Prove: NIT WIT P G I N W