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Opener(s) 2/28 2/28 It’s Rare Disease Day!!! Happy Birthday nylon, Ben Hecht, Linus Pauling, Vincente Minnelli, Bugsy Siegel, Zero Mostel, Frank Gehry, Tommy Tune and Robert Sean Leonard, !!! = Agenda Opener and questions (5) Feedback: Check classwork and homework (7) Lecture: You schmooze…you kaffeeklatsch, you lose (3) Cognitive Tutor (25) Individual Work 1: Wksht. 4-2. p. 189 or homework (5) Exit pass (5) 2/28 What to do today: 1. Do opener. 2. Collect classwork from yesterday. 3. Check homework. 4. Discuss some of your sad, d- and f-producing work ethic toward classwork. 5. Cognitive Tutor (I will put in CT grades over the weekend!). 6. Work on homework. 7. Do exit pass. OPENER Find the measure of each numbered angle. Essential Question(s) How do I classify triangles? How do I determine missing s in a ? Objective(s) Students will be able to (SWBAT) give a name to a triangle based on its sides. SWBAT give a name to a triangle based on its angles. SWBAT determine a missing triangle side length or angle measure using algebra and a triangle’s classification. SWBAT determine a missing angle measure based on the “A = 180” theorem. OUR LAST OPENER A has 3 coordinates: (2,2), (3,9) and (-5, 3). Use the distance formula to find the length of each side (leave it in square root form!) then classify the by sides. Exit Pass Look at 6-13 in your homework. Look at every measure that’s OUTSIDE a (an EXTERIOR ) and the 2 ‘far-away’ measures inside the it touches (the REMOTE INTERIOR s). Do you notice a relationship? What is it? Our Last Exit Pass FGH is equilateral with FG = x + 5, GH = 3x – 9 and FH = 2x -2. Find x and the measure of each side of the . Homework Text ?s, p. 190, 28-38 + 40 Period 8 Agenda writer: Alfredo (2x), Jenny (2x) Opener answerer: ACHIEVE Manual distributor: Timekeeper: Demetrius (8x) Presenter: Filer: Demetrius, Alfredo, Edgar, Steven, Jailene, Areli, Jenny, Jessica, Brian, Alejandra Tools Distributor: Steven, Angela, Salina I should see 2 (TWO) calculations!!! I should see 4 (FOUR) calculations!!! I should see 3 (THREE) calculations!!! I should see 1 (ONE) calculations!!! I should see 4 (FOUR) calculations!!! I should see 2 (TWO) calculations!!! I should see 2 (TWO) calculations!!! I should see 4 (FOUR) calculations!!! I should see 6 (SIX) calculations!!! I should see 8 (EIGHT) calculations!!! I should see 4 (FOUR) calculations!!! I should see an algebraic equation!!! 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