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Geometry Opener(s) 4/3 4/3 It’s National Chocolate Mousse Day, National Find-ARainbow Day, Don’t Go To Work Unless It’s Fun Day, Corrupt Society Day, Tweed Day, Pony Express Day, American Circus Day and Hospital Admitting Clerks Day!!! Happy Birthday Picabo Street, Eddie Murphy, David Hyde Pierce, Alec Baldwin, Sandra Boynton, Tony Orlando, Jane Goodall, Doris Day, Marlon Brando, Henry Luce, Leslie Howard, Dooley Wilson and Washington Irving!!! 4/3 What to do today: 1. Do the opener. 2. Work on Cognitive Tutor. 3. Do the exit pass. TODAY’S OPENER Find x AND the indicated sides. Agenda 1. Opener (5) 2. Individual Work: Cognitive Tutor (41) 3. Exit Pass (5) Standard(s) CCSS-M-G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures Essential Question(s) How do I use similarity and congruence to find numerical relationships among triangle parts? Objective(s) Students will be able to (SWBAT) establish the congruence or non-congruence of two geometric figures. SWBAT establish the similarity or non-similarity of two geometric figures. SWBAT find missing angle measures using congruence or similarity. SWBAT find missing side measures using congruence or similarity The Last Opener Find x so that l || m. Exit Pass The Last Exit Pass HOMEWORK Period 1 Text ?s, p. 311, #6-8, 12. HOMEWORK Period 6 Text ?s, p. 311, #6-8, 12. Extra Credit Period 1 Period 6 Marisol Mireya Valerie Imelda 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 Theorem 2.10 Right Congruence Theorem Theorem 2.11 Adjacent Right s Theorem Theorem 4-6 Leg-Leg (LL) Congruence Theorem 4-8 Leg-Angle (LA) Congruence Perpendicular lines intersect to form 4 right s. All right s are . Theorem 3-4 Perpendicular Transversal Theorem Postulate 3.2 Slope of Lines Perpendicular lines form adjacent s. 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. Postulate 3.2 and || Lines Postulate Theorem 4-7 If the 2 legs of one right are to Hypotenuse-Angle the corresponding parts of another (HA) Congruence right , then both s are . Postulate 4-4 If the leg and acute of one right Hypotenuse-Leg (HL) are to the corresponding parts of Congruence another right , then both s are . If the hypotenuse and acute of one right are to the corresponding parts of another right , then both s are . If the hypotenuse and one leg of one right are to the corresponding parts of another right , then both s are . 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 If 2 lines are cut by a transversal so that each pair of CO s is , then the lines are ||. If 2 lines are cut by a transversal so that each pair of AE s is , then the lines are ||. 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. Linear Equation in Standard Form 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. 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. AE CO AI CO CI AE AI/ CI Notes: Parallel Postulates 1 3-21 2 3 4 5 6 7 CO s || lines Postulate AE s || lines Postulate CI s || lines Postulate AI s || lines Postulate lines || lines Postulate Parallel Postulate 8 If corresponding s are , then lines are ||. If alternate exterior s are , then lines are ||. If consecutive interior s are supplementary, then lines are ||. If alternate interior s are , then lines are ||. If 2 lines are to the same line, then lines are ||. You are given a line and a point not on the line. There is ONLY ONE line through that point that’s || to the given line.