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Geometry Opener(s) 4/29 4/29 It’s Biological Clock Day, Great Poetry Reading Day, International Guide Dog Day, Kiss Your Mate Day, National Blueberry Pie Day, Workers Memorial Day and World Day for Safety and Health at Work!!! Happy Birthday Jessica Alba, Penelope Cruz, Jay Leno, AnnMargret, Harper Lee, Ferruccio Lamborghini, Oskar Schindler, Kurt Godel and James Monroe!!! 4/29 What to do today: 1. Do the opener. 2. Ask HW ?s. 3. Watch some videos about ancient Indian geometry. 4. Listen to task instructions. 5. Complete a worksheet about ancient Indian geometry. 6. Present notebook for checking. 7. Do the exit pass. TODAY’S OPENER Find x if ABC JKL. Agenda 1. Opener (5) 2. Homework ?s: Wksht. 6-1, p. 297 (5) 3. Video and Photo Discussion: The Hopewell Indians (15) 4. Intro Lecture: Hopewell Geometry Rubric (5) 5. Pairwork Pairshare: Hopewell Geometry Wksht. (20) 6. Complete Notebook Check [Period 6: Complete Standards Tracking Sheet] (10?) 7. 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 WED HIM. (Same shapes; different sizes) ̅̅̅̅̅ 𝑾𝑬 ̅̅̅̅ 𝑯𝑰 and ̅̅̅̅̅ 𝑾𝑬 = 3x – 2 and ̅̅̅̅ 𝑯𝑰 = 2x. ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅̅ 𝑾𝑫 𝑯𝑴 and 𝑾𝑫 = 25 and 𝑯𝑴 = 20. Find x. Exit Pass The Last Exit Pass HOMEWORK Period 1 Finish Wksht 6-1, p. 297 HOMEWORK Period 6 Finish Wksht 6-1, p. 297 Extra Credit Period 1 Period 6 Marisol Mireya Saul Prisma Refugio Vianey Michael (2x) Edgar Tanya (2x) Jesus Jonatan Valerie (4x) Imelda (3x) Sandra (2x) Lily (3x) Cynthia Jocelyn Denise (2x) 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 Perpendicular lines Theorem 3-4 If a line is to the 1st of two || lines, Perpendicular Transversal Theorem 4 Right s Theorem intersect to form 4 right s. then it is also to the 2nd line. Theorem 2.10 Postulate 3.2 All right s are . 2 non-vertical lines are if and only if the PRODUCT of their Right Congruence Theorem Slope of Lines slopes is -1. (In other words, the 2nd line’s slope is the 1st line’s slope flipped (reciprocal) with changed sign.) Theorem 2.11 Perpendicular lines Postulate 3.2 If 2 lines are to the same 3rd line, then thhose 2 Adjacent Right s Theorem form adjacent s. and || Lines Postulate lines are || to each other. Theorem 4-6 Theorem 4-7 If the 2 legs of one right are to If the hypotenuse and acute of one right Leg-Leg (LL) Congruence Hypotenuse-Angle the corresponding parts of another are to the corresponding parts of (HA) Congruence right , then both s are . another right , then both s are . Theorem 4-8 Postulate 4-4 If the hypotenuse and one leg of one right If the leg and acute of one right are Leg-Angle (LA) to the corresponding parts of another Hypotenuse-Leg (HL) are to the corresponding parts of Congruence Congruence another right , then both s are . right , then both s are . IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT THESE: Postulate 3.1 If 2 || lines are cut by a Postulate 3.4 If 2 lines are cut by a transversal Corresponding Angles transversal, then each pair of CO Corresponding Angles/|| Lines so that each pair of CO s is , Postulate (CO s Post.) s is . Postulate (CO s/|| Lines Post.) then the lines are ||. Theorem 3.1 If 2 || lines are cut by a Theorem 3.5 If 2 lines are cut by a transversal so Alternate Interior Angles transversal, then each pair Alternate Exterior Angles/|| Lines that each pair of AE s is , then the Theorem (AI s Thm.) of AI s is . Theorem (AE s/|| Lines Thm.) lines are ||. Theorem 3.2 If 2 || lines are cut by a Theorem 3.6 If 2 lines are cut by a transversal Consecutive Interior Angles transversal, then each pair Consecutive Interior Angles/|| Lines so that each pair of CI s is Theorem (CI s Thm.) of CI s is supplementary. Theorem (CI s/|| Lines Thm.) supplementary, the lines are ||. Theorem 3.3 If 2 || lines are cut by a Theorem 3.7 If 2 lines are cut by a transversal so Alternate Exterior Angles transversal, then each pair Alternate Interior Angles/|| Lines that each pair of AI s is , then the Theorem (AE s Thm.) of AE s is . Theorem (AI s/|| Lines Thm.) lines are ||. Postulate 3.2 2 non-vertical lines have the same Postulate 3.5 If you have 1 line and 1 point NOT on that Slope of || Lines slope if and only if they are ||. || Postulate line, ONE and only ONE line goes through that point that’s || to the 1st line. Theorem 6.6 Theorem 6.4 A midsegment of a is || to one In ACE with ̅̅̅̅̅ 𝑩𝑫 || ̅̅̅̅ 𝑨𝑬 and Midsegment Thm. Proportionality Thm. intersecting the other 2 sides in distinct side of the , and its length is ½ ̅̅̅̅ 𝑩𝑨 ̅̅̅̅ 𝑫𝑬 the length of that side. points, = . ̅̅̅̅ 𝑪𝑩 ̅̅̅̅ 𝑪𝑫 Postulates and Theorems to IDENTIFY CONGRUENT TRIANGLES: SSS, ASA, SAS or AAS Postulates and Theorems to IDENTIFY SIMILAR TRIANGLES: AA, SSS or SAS Linear Equation in Slope-Intercept Form Linear Equation in Point-Slope Form y = mx + b m = slope, b = yintercept y – y1 = m(x – x1) m = slope, (x1, y1) = 1 point on the 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 4/4 MODEL PRACTICE 4/7 MODEL A(-2,-4), B(4,8), C(6,-2) Show that Show that . . Explanation PRACTICE The vertices of XYZ are X(-1, 8), Y(9, 2) and Z(3, -4). M and N are the midpoints of XZ and YZ. Show that MN || XY and MN = ½ XY. http://www.youtube.com/watch?v=tG6ekcGnDFU http://www.youtube.com/watch?v=5l_0BqSmXwA http://www.youtube.com/watch?v=pSgUvclFnrQ 2 3 2 3 12 Imagine a 9th Hopewell Triangle…Triangle I with hypotenuse of length 15. What would be the length of the shortest leg if Triangle I and Triangle A are similar? Give your answer correct to one decimal place. Show your calculations.