* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Keys GEO Openers 4-15
John Wallis wikipedia , lookup
Foundations of geometry wikipedia , lookup
Vincent's theorem wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Mathematical proof wikipedia , lookup
Gödel's incompleteness theorems wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
History of trigonometry wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Central limit theorem wikipedia , lookup
Line (geometry) wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Opener(s) 4/15 4/15 It’s Tax Day…AND Jackie Robinson Day!!! Happy Birthday Emma Watson, Seth Rogen, Samantha Fox, Elizabeth Montgomery, Roy Clark, Harold Washington, Sir Neville Marinner, Bessie Smith, Nikita Khrushchev and Leonardo Da Vinci!!! 4/15 What to do today: 1. Do the opener. 2. Ask questions about HW. 3. Practice some Isosceles algebra and proofs. 4. Work on your homework. 5. Do the exit pass. OPENER Agenda Opener (5) HW Questions (5) Classwork: Me - #15-18, p. 219; You - #19-22, p. 219; Me - #23-24, p. 220, You - #25-26, p. 220; Me - # 29, p. 220; You - #30 & 40, p. 220 (20) Individual work 1: Wksht. 4-6, pgs. 215-16 (15) Exit pass (5) Essential Question(s) How do I prove that triangles are congruent? Objective(s) Students will be able to (SWBAT) use the SSS postulate to prove 2 s are . SWBAT use the SAS postulate to prove 2 s are . SWBAT use the ASA postulate to prove 2 s are . SWBAT use the AAS postulate to prove 2 s are SWBAT determine 2 s’s incongruence based on inadequate side or information. GHF is equilateral with mF = 3x + 4, mG = 6y and mH = 19z + 3. Draw a picture of the triangle, label it and find x, y and z. OUR LAST OPENER Exit Pass Our Last Exit Pass Homework Wksht. 4-6, pgs. 215-16 Agenda writer: Opener answerer: ACHIEVE Manual distributor: Period 8 Timekeeper: Presenter: Filer: Tools Distributor: Student Aide: Notes: Some Basic PTCs (Postulates, Theorems and/or Corollaries) Theorem 4-1: The sum of the measures of the s of m1 + m2 + Sum Theorem a is 180. m3 = 180 ( Sum Thm.) Theorem 4-2: If 2 s of one are to 2 s of a 2nd If A F and rd rd 3 Theorem , then the 3 s of both s are . C D, then rd (3 Thm.) B E Theorem 4-3: The measure of an ext. of a = the mYZP = mX Exterior Theorem sum of the measures of the 2 remote + mY (Ext. Thm.) int. s. Corollary 4-1: The acute s of a right are mG + mJ = Acute Right Corollary complementary. 90 Corollary 4-2: A can have, at most, 1 obtuse or 1 Obtuse/Right Corollary right Definition: 2 s are if and only if (iff) their In s CAT and DOG, if Congruent Triangles corresponding parts are . C D, A O (CPCTC) T G and ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ The corresponding parts of 2 s are 𝐂𝐀 ̅̅̅̅ 𝐃𝐎, 𝐀𝐓 𝐎𝐆 ̅̅̅̅ 𝐆𝐃 ̅̅̅̅ then congruent if and only if the s are 𝐓𝐂 congruent. CAT DOG Theorem 4-4: congruence is reflexive. DOG DOG Properties of congruence is If CAT PAW Congruence symmetric. then PAW CAT (Refl. Prop. / Sym. congruence is transitive. If CAT PAW and Prop. / Trans. Prop.) PAW DOG then CAT DOG Postulate 4.1: 2 s are if the sides of one are congruent BUY CAR nd Side-Side-Side to the sides of a 2 . Congruence (SSS) Postulate 4.2: Side-Angle-Side Congruence (SAS) Postulate 4.3: Angle-Side-Angle Congruence (ASA) Postulate 4.5: Angle-Angle-Side Congruence (AAS) 2 s are if 2 sides and the included angle of EAT BUN one are to the corresponding 2 sides and included angle of another . 2 s are if 2 angles and the included side of FAT FLY 1 are to 2 corresponding angles and the included side of the other . 2 s are if 2 angles and a non- included side PUR PLE of 1 are to the 2 corresponding angles and non-included side of the other ISOSCELES TRIANGLES Isosceles s Thm. If 2 sides of a (Thm. 4.9) are , then the s opposite those sides are . RIGHT TRIANGLES Theorem 4-6: 2 RIGHT s are Right Leg-Leg if the legs of one Congruence are to the (LL) legs of the other . Isosceles Sides Thm. (Thm. 4.10) If 2 s of a are , then the sides opposite those s are . Theorem 4-7: Right HypotenuseAngle Congruence (HA) Equilateral / Equiangular Corollary (Cor. 4.3) A is equilateral if Theorem 4-8: Right Leg-Angle it is equiangular Congruence AND (LA) A is equiangular if it is equilateral. Equilateral Measures Corollary (Cor. 4.4) Each angle of an equilateral measures 60. Postulate 4.4: Right HypotenuseLeg Congruence (HL) 2 RIGHT s are if the hyp. and an acute of one are to the hyp. and corresponding acute of the other . 2 RIGHT s are if 1 leg and 1 acute of one are to the corresponding leg and corresponding acute of the other . 2 right s are if the hyp. and a leg of one are to the hyp. and corresponding leg of the other . 13 Must-Follow Proof Steps 1. If you have givens, 2. If you have tick marks or arcs marked in the figure, 3. WHATEVER… 4. If you see single or double arrows in the figure, 5. If you have a || GIVEN, 6. If you have a bisect GIVEN, 7. If you have a median GIVEN, 8. If you have a midpoint GIVEN/STATEMENT, 9. If you are asked to prove 2 s are , 10. If you are asked to prove 2 parts are , 11. If you have a postulate Reason that says ASA, SSS, SAS or AAS, 12. If you have segments that are part of larger segments, 13. If you have angles that are part of linear pairs, Write them EXACTLY the same in the Statements column and say GIVEN in the Reasons column. Write them as Statements in the Statements column and GIVENS in the Reasons column. As you list GIVENS, place tick and arc marks in your figure!!! Write it as a PARALLEL Statement and GIVEN. Write it as a Statement with an AI or CO Postulate Reason. Write it as a Statement with a Def. of Bisect Reason. Write it as a midpoint Statement with a Def. of Median Reason. Write it as a Segments Statement with a Def. of Midpoint Reason. Your last Reason must be ASA, SSS, SAS or AAS. Your last Reason must be CPCTC. You MUST have 3 Statements in your proof that match the 3 letters of your postulate Reason!!! Write a segment GIVEN Insert a Reflexive statement/reason Add the Reflexive segment to both sides Use the Segment Addition Postulate Write a angle GIVEN Insert a supplementary s statement Follow it with a Definition reason Use the ‘s Supp. to s are ’ Thm. A Sample Coordinate Plane Large Groups Alfredo Angela Mildred Lesly Lucia Areli Demetrius Rolando Janene Salina Angelo Brian Group 1 David Mani Jailene Group 2 Anarely & Marco Tony & Jasmine Mag & Marcella Steven Group 3 Josefina Edgar Jessica Group 4 Jen & Susana Nataly & Cruz Gab & Alejandra Group 5 Javier Sandra Elizabeth Groups of Three Group 1 Lesly Rolando Mildred Group 2 Maggie Steve Natalie Group 3 Jasmine Mani Anarely Group 4 Angela Lucia Brian Group 5 Salina Alfredo Josefina Group 6 Alejandra Jessica David Group 7 Group 9 Cruz Edgar Elizabeth Group 8 Jailene Angelo Marcela Group 10 Sandra Javier Demetrius Group 11 Jenny Marco Anthony Areli Gabino Janeen Group 12 Groups of Three and Four Group 1 Lesly Rolando Mildred Anthony Group 2 Maggie Marcela Natalie Demetrius Group 3 Jasmine Mani Anarely Angela Group 4 Areli Gabino Brian Group 5 Salina Cruz Josefina Group 6 Alejandra Jessica David Lucia Group 7 Sandra Javier Jenny Marco Group 8 Jailene Angelo Edgar Elizabeth Group 9 Alejandro Janeen Steve Alfredo YOUR PROOF CHEAT SHEET IF YOU NEED TO WRITE A PROOF ABOUT ALGEBRAIC EQUATIONS…LOOK AT THESE: Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property Distributive Property IF YOU NEED TO WRITE A PROOF ABOUT LINES, SEGMENTS, RAYS…LOOK AT THESE: For every number a, a = a. Postulate 2.1 For all numbers a & b, if a = b, then b = a. For all numbers a, b & c, if a = b and b = c, then a = c. For all numbers a, b & c, if a = b, then a + c = b + c & a – c = b – c. For all numbers a, b & c, if a = b, then a * c = b * c & a ÷ c = b ÷ c. For all numbers a & b, if a = b, then a may be replaced by b in any equation or expression. For all numbers a, b & c, a(b + c) = ab + ac Postulatd 2.2 Postulate 2.3 Postulate 2.4 Postulate 2.5 Postulate 2.6 Postulate 2.7 The Midpoint Theorem IF YOU NEED TO WRITE A PROOF ABOUT THE LENGTH OF LINES, SEGMENTS, RAYS…LOOK AT THESE: Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property Segment Addition Postulate Through any two points, there is exactly ONE LINE. Through any three points not on the same line, there is exactly ONE PLANE. A line contains at least TWO POINTS. A plane contains at least THREE POINTS not on the same line. If two points lie in a plane, then the entire line containing those points LIE IN THE PLANE. If two lines intersect, then their intersection is exactly ONE POINT. It two planes intersect, then their intersection is a LINE. If M is the midpoint of segment PQ, then segment PM is congruent to segment MQ. IF YOU NEED TO WRITE A PROOF ABOUT THE MEASURE OF ANGLES…LOOK AT THESE: AB = AB (Congruence?) If AB = CD, then CD = AB If AB = CD and CD = EF, then AB = EF If AB = CD, then AB EF = CD EF If AB = CD, then AB */ EF = CD */ EF If AB = CD, then AB may be replaced by CD If B is between A and C, then AB + BC = AC If AB + BC = AC, then B is between A and C Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property m1 = m1 (Congruence?) If m1 = m2, then m2 = m1 If m1 = m2 and m2 = m3, then m1 = m3 If m1 = m2, then m1 m3 = m2 m3 DEFINITION OF CONGRUENCE Whenever you change from to = or from = to . If m1 = m2, then m1 */ m3 = m2 */ m3 If m1 = m2, then m1 may be replaced by m2 IF YOU NEED TO WRITE A PROOF ABOUT ANGLES IN GENERAL…LOOK AT THESE: Postulate 2.11 The Addition Postulate Theorem 2.5 The Equalities Theorem If R is in the interior of PQS, then mPQR + mRQS = mPQS. THE CONVERSE IS ALSO TRUE!!!!!! Q Congruence of s is Reflexive, Symmetric & Transitive P R S Theorem 2.8 Vertical s Theorem If 2 s are vertical, then they are . (1 3 and 2 4) IF YOU NEED TO WRITE A PROOF ABOUT COMPLEMENTARY or SUPPLEMENTARY ANGLES …LOOK AT THESE: Theorem 2.3 Supplement Theorem If 2 s form a linear pair, then they are supplementary s. Theorem 2.4 Complement Theorem If the non-common sides of 2 adjacent s form a right , then they are complementary s. Theorem 2.12 Supplementary Right s Therorem Theorem 2.6 R The Supplements Theorem S P Q Q P If 2 s are and supplementary, then each is a right . Theorem 2.7 The Complements R Theorem S Theorem 2.13 Linear Pair Right s Therorem s supplementary to the same or to s are . (If m1 + m2 = 180 and m2 + m3 = 180, then 1 3.) s complementary to the same or to s are . (If m1 + m2 = 90 and m2 + m3 = 90, then 1 3.) If 2 s form a linear pair, then they are right s. YOUR PROOF CHEAT SHEET (continued) IF YOU NEED TO WRITE A PROOF ABOUT RIGHT ANGLES or PERPENDICULAR LINES…LOOK AT THESE: Theorem 2.9 4 Right s Theorem Perpendicular lines intersect to form 4 right s. Theorem 2.10 Right Congruence Theorem All right s are . Theorem 2.11 Adjacent Right s Theorem Perpendicular lines form adjacent s. Theorem 3-4 Perpendicular Transversal Theorem Postulate 3.2 Slope of Lines Postulate 3.2 and || Lines Postulate If a line is to the 1st of two || lines, then it is also to the 2nd line. 2 non-vertical lines are if and only if the PRODUCT of their slopes is -1. (In other words, the 2nd line’s slope is the 1st line’s slope flipped (reciprocal) with changed sign.) If 2 lines are to the same 3rd line, then those 2 lines are || to each other. IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT THESE: Postulate 3.1 Corresponding Angles Postulate (CO s Post.) If 2 || lines are cut by a transversal, then each pair of CO s is . Theorem 3.1 Alternate Interior Angles Theorem (AI s Thm.) Theorem 3.2 Consecutive Interior Angles Theorem (CI s Thm.) Theorem 3.3 Alternate Exterior Angles Theorem (AE s Thm.) Postulate 3.2 Slope of || Lines If 2 || lines are cut by a transversal, then each pair of AI s is . If 2 || lines are cut by a transversal, then each pair of CI s is supplementary. If 2 || lines are cut by a transversal, then each pair of AE s is . 2 non-vertical lines have the same slope if and only if they are ||. Postulate 3.4 Corresponding Angles/|| Lines Postulate (CO s/|| Lines Post.) Theorem 3.5 Alternate Exterior Angles/|| Lines Theorem (AE s/|| Lines Thm.) Theorem 3.6 Consecutive Interior Angles/|| Lines Theorem (CI s/|| Lines Thm.) Theorem 3.7 Alternate Interior Angles/|| Lines Theorem (AI s/|| Lines Thm.) Postulate 3.5 || Postulate Linear Equation in SlopeIntercept Form Linear Equation in PointSlope Form y = mx + b m = slope, b = y-intercept y – y1 = m(x – x1) m = slope, (x1, y1) = 1 point on the line CI s: 2 inside || lines on SAME side of transversal. CO s: 1 inside || lines & 1 outside || lines, on OPPOSITE sides of transversal. AI s: 2 inside || lines on OPPOSITE sides of transversal. AE s: 2 outside || lines on OPPOSITE sides of transversal. If 2 lines are cut by a transversal so that each pair of CI s is supplementary, then the lines are ||. If 2 lines are cut by a transversal so that each pair of AI s is , then the lines are ||. If you have 1 line and 1 point NOT on that line, ONE and only ONE line goes through that point that’s || to the 1st line. Ax + By = C I – Numbers and coefficients can only be Integers. (No fractions or decimals.) P – The x coefficient must be Positive. (A > 0) O – Zero can only appear beside a variable Once. (If A = 0, then B ≠ 0) D – Numbers and coefficients can only be Divisible by 1. (GCF = 1) S – Variables can only be on the Same side of the equal sign. AE CO CI AE If 2 lines are cut by a transversal so that each pair of AE s is , then the lines are ||. Linear Equation in Standard Form AI CO If 2 lines are cut by a transversal so that each pair of CO s is , then the lines are ||. AI/ CI