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Geometry Opener(s) 2/24 2/24 It’s National Tortilla Chip Day, National Trading Card Day, Mexican Flag Day and World Spay Day!!! Happy Birthday Floyd Mayweather, Jr., Billy Zane, Michelle Shocked, Steve Jobs, Joe Lieberman, Michel Legrand and Winslow Homer!!! Agenda 1. Opener (8) Emilyoz.com/tedoz.com 2. [Period 7: Slope √ (5)] 3. [All other periods: Discussion: What’s the same? (5)] 4. Notes 1: Finding Slope (10) 5. Models 1: Slope (5) 6. Practice 1: Slope (5) 7. Notes 2: and || Slopes (10) 8. Models 2: and || Slopes (5) 9. Practice 2: and || Slopes (5) 10. Models 3: and || Slopes (5) 11. Practice 3: and || Slopes (5) 12. Models 4: and || Slopes (5) 13. Practice 4: and || Slopes (5) 14. HW: Wksht. 3-2, p. 134 & Wksht. 3-3, p. 138 (12) 15. HW ?s/Catch-up: Text ?s, p. 142, #16-38 even [Honors: #39-46 all]; ICCE Proofs for Text ?s, p. 137, #33 & 35 (5) 16. HW √ (5) 17. Exit Pass (5) Standard(s) CCSS-HSG-CO.C.9: Prove theorems about lines and angles…when a transversal crosses parallel lines, alternate angles are congruent and corresponding angles are congruent. CCSS-HSG-GPE.B.5: Use the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Essential Question(s) How Do I (HDI) find missing measures using ICCE concepts? HDI prove missing measures using ICCE theorems? HDI differentiate between a gradual and steep rise visually? HDI differentiate between a gradual and steep rise arithmetically? Objective(s) Students will be able to (SWBAT) correlate anti-blobbiness with geometry. SWBAT name and identify || lines, transversals and ICCE angles. SWBAT determine missing ICCE angle measures, using postulates/theorems and/or algebra. SWBAT find slope using formulaic methods. SWBAT find slope using linear graphs. 2/24 What to do today: 1. Do the opener. 2. [Period 7: √ HW] 3. Discuss picture similarities. [Not Period 7] 4. Take some slope notes. 5. Record some slope models. 6. Practice finding slope. 7. Take some /|| slope notes. 8. Record some /|| slope models. 9. Practice finding /|| slope. 10. Record some /|| models/practice. 11. Record and work on HW. 12. Ask ?s about HW/Catch up on HW. 13. √ HW. 14. Do the exit pass. TODAY’S OPENER Find x and y. THE LAST OPENER Find x and y. Exit Pass (12/11 – 13/14) For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 2 (D) 5. Ry= -1 (E) 6. Rx = -3 (F) The Last Exit Pass HOMEWORK Period 1 Finish Wksht. 3-2, p. 134 and 3-3, p. 138 HOMEWORK Period 7 Finish Wksht. 3-2, p. 134 and 3-3, p. 138 HOMEWORK Period 3 Finish Wksht. 3-2, p. 134 and 3-3, p. 138 HOMEWORK Period 5 and 8 Finish Wksht. 3-2, p. 134 and 3-3, p. 138 HOMEWORK Period 2A Finish Wksht. 3-2, p. 134 and 3-3, p. 138 Extra Credit Period 1 Period 2A Period 3 Israel H. (5x) Jose C. (6x) Mirian S. (4x) Perla S. (2x) Melissa A. Israel A. Benito E. Amal S. (3x) Stephanie L. (4x) Alexis S. (3x) Evelyn A. (3x) Daniela G. Yesenia M. Yazmin C. (2x) Joe L. Safeer A. Kevin G. Isabel G. Roxana M. 1. Jaime A. (5x) Nadia L. (2x) Anthony P. Griselda Z. (3x) Jaclyn C. (3x) Brandon S. (6x) Mayra C. Jacob L. (2x) Leo G. Alejandra G. Mayra G. Rodrigo F. Anthony C. Sonia T. (2x) Amanda S. (4x) Josue A. (6x) Arslan A. Angie H. Paulina G. (2x) Alicia R. Ricardo D. Rosie R. Ronny V. Gaby O. Period 5 Period 7 Period 8 Antonio B. (5x) Rogelio G. (6x) Eraldy B. (2x) Anthony G. (3x) Alex A. (3x) Brianna T. Jose B. (7x) Carlos L. (3x) Anadelia G. (3x) Jose D. (3x) Cesar H. (2x) Saul R. Alex A. Jose C. Adriana H. (6x) Jackie B. (4x) Jose R. (6x) Julian E. (4x) Jocelyn C. (9x) Jenny Q. Ruby L. (3x) Ana R. (4x) V. Limon Zelexus R. Kamil L. Diego P. Alfredo F. Lilliana F. (3x) Christian A. (2x) Liz A. (2x) Jessica T. (2x) Jorge C. (2x) Gerardo L. (2x) Val R. Gerardo L. Esme V. (2x) Alejandra P. Cynthia R. Xavier G. (3x) Maria M. (2x) Fernando V. (2x) Watch a portion of RHYTHmetric video (“What are the 3 ‘laymen’s’ terms for 3 of our 4 transformations?”). http://www.youtube.com/watch?v=NKtJd1hkI9k http://www.youtube.com/watch?v=V9uYcnjlAks http://www.youtube.com/watch?v=X1xPZjItmDk%20 http://www.shodor.org/interactivate/activities/Transmographer/ Your Name Your Period Parallel Lines and Transversals 2/17/15 FIGURE OBJECT My Grandfather’s Stool ICCE s Alternate Interior s: 4 & 6 2 1 3 4 Consecutive Interior s: 4 & 5 Corresponding s: 2 & 6 6 5 Alternate Exterior s: 1 & 7 8 7 Statements 1. Reasons 1. Given 2. Def. of TR 3. Def. of Cor. s 4. Cor. s Post. Prove: x = 16° 5. Def. of 6. Sub. Prop. 7. Subt. Prop. 8. Sub. Prop. 9. Add. Prop. 10. Sub. Prop. Statements 1. Reasons 1. Given 2. Def. of TR 3. Def. of Alt. Int. s 4. Alt. Int. s Thm. 5. Def. of Prove: m1 = 67° 6. Sub. Prop. 7. Def. of Alt. Int. s 8. Alt. Int. s Thm. 9. Def. of 10. Sub. Prop. 11. Add. Post. 12. Sub. Prop. 13. Sub. Prop. Prove: x = 34° Use Def. of Linear Pair, Supplement Thm. and Angle Add. Post. Given: m6 = 23° Prove: m3 = 67° Prove: m1 = 113° Honors 1 2 4 5 6 7 9 8 3 Fingernail Car hood Streetlamp Mount Fuji Airplane wing High heel Slanting Roof Brad Pitt’s nose Horizon B A Everything You Always Wanted To Know About Lines But Were Afraid To Ask What’s slope? a. It’s the steepness of a road or a mountain side OR A LINE. b. It’s the ratio of change between 2 y-coordinates and 2 xcoordinates. c. In formulas and graphs, it’s the variable ‘m’. You need 2 points first. Then you can choose 1 of 2 methods: So how do I find slope? The Formula Methods The Counting Method 1. Call one point (x1, y1) 2. Call the other (x2, y2) 3. Plug them into 𝒚 −𝒚 m = 𝒙𝟏 − 𝒙 𝟐 1. Plot the points in a graph. 2. Connect the points. 3. From 1 point to the other, count the up and down squares. Put this number on the top of your ratio. 4. From the other point, count the right to left squares. Put it on the bottom. 5. Simplify and change the sign to negative if the line FALLS to the right. 𝟏 𝟐 4. Calculate 5. OR put one point on top of the other and subtract: (x1, y1) -(x2, y2) 6. Your y value goes on top; Your x value goes on bottom. Slope-Intercept Form Point-Slope Form If you know the slope… If you know the slope… m and you know the y-intercept (where the line crosses the y-axis)… (0, b) you end up with the equation… y = mx + b m and you know one point on the line… (x1, y1) you end up with the equation… y – y1 = m(x – x1) m = 3/2 m = -2 m = 5/4 Equations of b=4 b=6 (4,6) Lines? lines have opposite sign/reciprocal slopes that multiplied together = -1. WHEN YOU ARE GIVEN a point and a equation… 1. Find the slope in the given equation. 2. FLIP it…and change the sign. 3. Plug the slope and point in the POINT-SLOPE formula. 4. Simplify or change to slope-intercept form. (3,1) y = -3x + 2 (4,7) (3, 6) (6, 2) m = -13/5 m=3 m=1 (5, -7) (1, 6) (2, 5) Equations of || Lines? || lines have the SAME slope. WHEN YOU ARE GIVEN a point and a || equation… 1. Find the slope in the given equation. 2. Use that same slope in your new equation. 3. Plug the slope and point in the POINT-SLOPE formula. 4. Simplify or change to slope-intercept form. 𝟏 (-4, 2) y=𝟐x+5 (4, 1) (2, -3) (1, -1) What’s a Line? Y = mx + b or y - y1 = m(x – x1) D – Denominators don’t contain variables. O – Operations with variables don’t include ÷. V – Variables don’t get multiplied together. E – Exponents don’t include 2 or 3 or 4 or a square root…√)…ONLY 1!!!!!!!!!!!!! 3yx = 7 y = 10 – 5x2 6y = -3x – 15 𝟐 y=𝒙 Yes or No Yes or No Yes or No Yes or No How Do I Graph? If you see an x and a y… 1. Get y on one side by itself. 2. Divide EVERYTHING by the y coefficient. 3. Plot the solitary number on the y axis. 4. Turn the x coefficient into a fraction. 5. From the plotted point, count up the numerator…count right the denominator. (If the fraction is NEGATIVE, count up the numerator…count left the denominator.) 6. Connect the points with a double-arrowed line! If you only see an x or only see a y… 1. Divide both sides by the coefficient…plot a vertical line if x…plot a horizontal line if y. Let’s try graphing… (0, 3) (5, 0) (2, 3) (5, 7) (2, 8) (2, -8) (1.5, -1), (3, 1.5) Let’s try graphing… Passes through (8, -6) || to 2x – y = 4 Passes through (6, 1) || to line with x-intercept -3 and y-intercept 5 Passes through (2, -2) to x + 5y = 6 Passes through (-2, 1) to y = 4x – 11 Let’s try graphing… || to 4x - 3y = -6 and passing through (0, -1) to 5x + y = 10 and passes through (5,1) || to y = -2x – 5 and passing through (0, 3) to 4x = 3 Let’s try graphing… || to y = 2x + 4 and passing to 2x + 7y = 14 through (0, -2) || to y = -2x – 4 and passing through (0, 2) to 6x + 2y = 6 Let’s try graphing… (1, 3), m = 2 (2, -4), m = -1 Passes through (0, 1) Passes through (4, -5) with m = 1 3 with m = 2 What’s Slope? What’s So Special About || & Lines? What’s Slope? Recap What’s Special About || Lines? Recap PARALLEL LINES HAVE THE SAME If you have 2 points… (x1, y1) and (x2, y2) SLOPE!!! Then put one point on top of the other and subtract: Are these lines || ? (x1, y1) -(x2, y2) (4,3) (1,-3) (1,2) (-1,3) You end up with a slope ‘fraction’. (2,8) (-2,2) (0,9) (6,0) The first difference goes on the bottom (3,9) (-2,-1) y = 2x* and the second difference goes on the top: Make an x/y table and find values 𝑇ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑦1 − 𝑦2) for x = 0 and x = 1. Then determine the slope. 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑥1 − 𝑥2) What’s the slope? (1, 5) (-1, -3) (8, -5) (4, -2) (4, 5) (2, 7) What’s Special About Lines? Recap PERPENDICULAR LINES HAVE OPPOSITE SIGNS/RECIPROCAL SLOPES!!!! When Their Slopes Are Multiplied, They Equal -1!!!! Are these lines ? (-2,0) (0, 5) (2,0) (0,-5) (1,0) (0,3) (3,0) (0,1) (-2,-3) (2,5) x + 2y = 10* Do the one hand on the x; one hand on the y method, graph & determine slope…or change to y = x + # and determine slope. What About Graphing || & Recap Lines? 1. Plot the point it passes through. 2. If ||, use the slope of the line it’s parallel to…in fraction form. 3. If , use the slope of the line it’s perpendicular to…in fraction form… FLIP it, then change its sign. 4. From the plotted point, count up the numerator…count right the denominator. 5. (If the fraction is NEGATIVE, count up the numerator…count left the denominator.) 6. Connect the points with a doublearrowed line! TRANSFORMATION GRAPHIC ORGANIZER REFLECTIONS TRANSLATIONS ROTATIONS METHODS 1. 2. 3. 4. Mira Formulas Tracing Paper Fold on reflection line and hold up to the light 5. Ruler and measuring 6. Compass and bisector (1 big arc on reflection line; 2 little arcs from intersections) METHODS 1. Formulas 2. Counting units on graph Flips Orientation changes Isometric; an isometry Lengths, angles, betweenness and collinearity preserved Pre-image Image Formula…over the x-axis: CHARACTERISTICS (x, y) (x, -y) Notation: R(x-axis)(ABC)=A’B’C’ 1. Formulas (around origin) 2. Tracing Paper (draw axes) 3. Spinning (always turn paper 3. Input/output tables (to figure paper in opposite direction out the formula) of CW or CCW arrow) 4. Tracing paper 4. SFA (Subtract spin point /formula/add spin point) 5. Compass and protractor (line to spin point; measure angle; line away from spin point) CHARACTERISTICS METHODS Slides Orientation the same Isometric; an isometry Lengths, angles, betweenness and collinearity preserved Pre-image Image 2 reflections over || lines = 1 translation Formula…3 right and 3 down: (x, y) (x+3, y-3) Notation: T<3, -3>(ABC)=A’B’C’ CHARACTERISTICS Turns/Spins Orientation changes Isometric; an isometry Lengths, angles, betweenness and collinearity preserved Pre-image Image A rotation of 270 around the origin = 1 reflection Formula…180° CW (clockwise) around origin: (x, y) (-x,-y) Notation: r(180°, O)(ABC)=A’B’C’ FORMULAS (x, y) (x, -y): over x axis (x, y) (-x, y): over yaxis (x, y) (-x, -y): over the origin (x, y) (y, x): over x=y (x, y) (-y, -x): over -x=y FORMULAS x -2 3 -4 y 3 -6 3 A flip over the origin (x, y) (-x, -y) +x is right -x is left +y is up -y is down r(-270,O)(x, y) = (y, -x) {CCW} EXAMPLES EXAMPLES A slide 3 right and 3 down… (x, y) (x + 3, y – 3) x’ y’ x” y” r(90,O)(x, y) = (y, -x) {CW} r(180,O)(x, y) = (-x, -y) {CW} r(270,O)(x, y) = (-y, x) {CW} r(-90,O)(x, y) = (-y, x) {CCW} r(-180,O)(x, y) = (-x, -y) {CCW} EXAMPLES A flip over the x-axis… (x, y) (x, -y) FORMULAS x -4 -1 -4 y 5 5 0 A slide 1 left and 2 up (x, y) (x - 1, y + 2) x’ y’ x” y” A spin 180° CW… (x, y) (-x, -y) x y 1 4 5 2 3 -1 A 90° CCW (counterclockwise) spin around the origin (x, y) (-y, x) x’ y’ x” y” Today’s Worksheets Rubrics Line of Symmetry Wksht. #2. 2 pts. 50 % at least 3 pts. #3. 2 pts. 64% at least 6 pts. #4. 2 pts. (Correctly identifying 5 letters with 76% at least 9 pts. lines of symmetry; correctly identifying 5 88% at least 12 pts. letters with NO lines of symmetry) 93% at least 15 #5. 6 pts. (1 for each image) 100% 18 #6. 4 pts. (1 for each image WITH the specified # of lines of symmetry) #7. 2 pts. (Yes with an example to support ‘yes’ or no with counterexample to support ‘no’) Reflections Wksht. #1. 8 pts. (1 for each ? or gap fill) #2. 7 pts. (1 for each instruction or request to write a statement) #3. 4 pts. (1 for each image) #4. 5 pts. (1 for each instruction; 2 for the entire path) 50 % at least 4 pts. 64% at least 8 pts. 76% at least 12 pts. 88% at least 16 pts. 93% at least 20 100% 24 Reflections on a Coordinate Plane Wksht. #2. 8 pts. (1 for each table entry and ?) 50 % at least 6 pts. #3. 7 pts. (1 for each table entry and ?) 64% at least 12 pts. #4. 9 pts. (1 for each table entry, ? and 76% at least 18 pts. labeling the image) 88% at least 24 pts. #5. 12 pts. (1 for each set of pre-image & 93% at least 30 image coordinates; 2 for each set of 100% 36 formulaic coordinates [in the boxes!]) 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