Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Duality (projective geometry) wikipedia , lookup
Integer triangle wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Noether's theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Line (geometry) wikipedia , lookup
Euclidean geometry wikipedia , lookup
Geometry Opener(s) 4/30 TODAY’S OPENER 4/30 It’s Bugs Bunny Day, Dia de Los Ninos, International Jazz Day, National Adopt a Shelter Pet Day, National Animal Advocacy Day, National Honesty Day, National Raisin Day, Spank Out Day and Walpurgis Night!!! Happy Birthday Carl Gauss, Alice B. Toklas, Eve Arden, Percy Heath, Cloris Leachman, Annie Dillard, Jill Clayburgh, Lars von Trier and Kirsten Dunst!!! What is the value of x? a. Make a conjecture. b. Provide some mathematical evidence supporting or negating your conjecture. c. Write a sentence explaining why your evidence supports or negates your conjecture. Agenda 1. Opener (8) 2. Video: http://www.pbslearningmedia.org/resource/90e224b5-564d-47f0-956658a11370dee9/architecture-trigonometry/ (7) 3. Practice 1/Notes 1: Sohcahtoa PPT completion (10) 4. Practice 2/CW: Trig ratios investigation – Wksht 74, p. 369 (20) 5. HW Assignment: p. 371-2, #1-2 on each (5) 6. Computer Room Instructions (7) 7. Notes 2: Double Entry Journal till ‘Unit Circle’ https://www.mathsisfun.com/algebra/trigonometry.html# (20) 8. Ind. Work: Chart Completion (10) 9. Calculator: http://www.mathworksheetsgo.com/trigonometry- THE LAST OPENER What is the value of x? a. Make a conjecture. b. Provide some mathematical evidence supporting or negating your conjecture. c. Write a sentence explaining why your evidence supports or negates your conjecture. calculators/sine-cosine-calculator-online.php 10. CW: 8 problems/8 solutions https://www.khanacademy.org/math/geometry/right_triangles_topic/ccgeometry-trig/e/trigonometry_0.5 (15?) 11. Exit Pass (5?) Standard(s) Essential Question(s) CCSS-HSG-SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. CCSS-HSG-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. CCSS-HSG-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. How Do I (HDI) use PYT? HDI use the converse of PYT? HDI find trig ratios using right triangles? HDI solve problems using trig ratios? Objective(s) Students will be able to (SWBAT) identify a PYT triple. SWBAT calculate the side lengths of a using PYT. SWBAT use the convers of PYT. SWBAT identify the opp. side, adj. side and hyp. In a right . SWBAT determine the sin, cos and tan ratios for an in a right . SWBAT find values of trig functions for acute angles. SWBAT find angle measures of right triangles. Exit Pass (12/11 – 13/14) The Last Exit Pass 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) HOMEWORK Period 1 Complete Wksht. 7-4, p. 369; Wksht. 7-4, p. 371, #1-2; Wksht. 7-4, p. 372, #1-2; Special Triangles Trig Ratios HOMEWORK Period 7 No homework tonight. HOMEWORK Period 2A Complete Wksht. 7-4, p. 369 HOMEWORK Period 3 Complete Wksht. 7-4, p. 369; Wksht. 7-4, p. 371, #1-2; Wksht. 7-4, p. 372, #1-2; Special Triangles Trig Ratios HOMEWORK Period 5 No homework tonight. HOMEWORK Period 8 No homework tonight. Extra Credit Period 1 Period 2A Period 3 Alex H. (4x) Israel A. (4x) Jocelyn L. (4x) Amal S. (6x) Angel V. Victor C. (4x) Israel H. (3x) Stephanie L. (4x) Yazmin C. Melissa A. (2x) Evelyn A. (2x) Mirian S. Alexis S. Brandon S. (5x) Gabriel M. (2x) Rodrigo F. (6x) Josh P. (5x) Alejandra G. (4x) Jaime A. (5x) Jacob L. (9x) Anthony P. (3x) Enrique G. Abrahan G. Alex A. (2x) Jocelyn J. (2x) Michelle S. Carlos O. Arslan A. Ronny V. (2x) Rosie R. (4x) Javier L. (2x) Gaby O. (3x) Claudia M. (2x) Omar R. (2x) Ricardo D. (2x) Josue A. Alicia R. Javier D. (2x) Catalina Z. (2x) Luis H. Period 5 Period 7 Period 8 Antonio B. (3x) Liz L. (2x) Anadelia G. Jose C. (5x) Jose B. (2x) Jesus H. (3x) Carlos L. (2x) Jose D. Rob C. Briana T. (3x) Jose G. (2x) Eraldy B. Rogelio G. (2x) Aurora G. Jorge L. Kamil L. Alfredo F. (2x) Gustavo C. Ana R. Ruby L. (3x) Danny G. Adriana H. (3x) Gabriela G. (5x) Jackie B. Julian E. (2x) Cristian A Jocelyn C. (4x) Lily F. Cynthia R. (3x) Jessica T. Jorge C. (2x) Santi H. Kevin A. (2x) Andrea N. (3x) Ernesto M. Maria M. (2x) Fernando V. (2x) Alejandra P. Joe C. 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 Degrees 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360° Special Angle Trig Ratios Sine Cosine Tangent 0 1 0 Some Of Hancock’s Classes Are Hard To Obtain A’s (in) I know about PYT… If you have the measures of 2 sides of a right triangle, you can find the 3 rd by using: 12 a 2 + b2 = c 2 15 but what happens if I only know an angle measure and 1 side? Essential Question(s) How do I (HDI) find the measure of ‘immeasurable’ distances in a triangle? Objective(s) Students will be able to (SWBAT) name the three trigonometric ratios and their corresponding fractions. SWBAT determine numerical trigonometric ratios in a triangle. SWBAT find missing side lengths in a triangle using trigonometric ratios. If you have the measure of 1 side and 1 angle of a right triangle, you can find the other sides by using: 12 SOHCAHTOA 15° Some Of Hancock’s SOH Classes Are Hard CAH To Obtain A’s (in) TOA Sine (Sin) Cosine (Cos) Tangent (Tan) Opposite Hypotenuse 𝑶𝑷𝑷 𝑯𝒀𝑷 Adjacent Hypotenuse 𝑨𝑫𝑱 𝑯𝒀𝑷 Opposite Adjacent 𝑶𝑷𝑷 𝑨𝑫𝑱 The HYPOTENUSE is the looooooooooongest side of the triangle. The OPPOSITE side is the side across from the known angle. The ADJACENT side is the side touching the known angle. JENNY’S FISHTAIL Step 1. Draw an arrow from the known (NON-90°) to the side across from it. That’s the OPP (opposite side). Step 2. Draw an arrow from that arrowhead around the right box. That’s the ADJ (adjacent side). Step 3. Draw an arrow from that arrowhead to the longest side. That’s the HYP (hypotenuse). Solving Δ Problems with SOHCAHTOA Let’s recap… 1st, remember we are working with RIGHT triangles!!! YOU HAVE: 2 sides of a Δ YOU WANT: The 3rd side of the Δ 2 sides of a Δ A non-right of the Δ Another side of the Δ 1 side & 1 of a Δ 1 side & 1 of a Δ To solve the Δ…find ALL the missing parts YOU USE: PYT (a + b = c ) SOHCAHTOA SOHCAHTOA SOHCAHTOA then PYT 2 2 2 SOHCAHTOA So what sort of problems will I encounter? Some Of Hancock’s SOH Classes Are Hard CAH To Obtain A’s (in) TOA A Sine (Sin) Cosine ( os) Objective(s) Students will be able to (SWBAT) name the three trigonometric ratios and their corresponding fractions. 𝑨𝑫𝑱 𝑯𝒀𝑷 Adjacent Hypotenuse Tangent (Tan) 𝑶𝑷𝑷 𝑨𝑫𝑱 Opposite Adjacent A 52° Essential Question(s) How do I (HDI) find the measure of ‘immeasurable’ distances in a triangle? 𝑶𝑷𝑷 𝑯𝒀𝑷 Opposite Hypotenuse A 52° X° ? X 8 9 ?° C 6 ?° B You know an and a side… You want to find a side… 1. Draw your arrows from the and label OAH. 2. Match OAH with SOHCAHTOA. 3. Set up your equation. 4. Find decimal for trig ratio. 5. If X is in denominator…DIVIDE side by trig decimal. 12 C X You know an and a side… You want to find a side… 1. Draw your arrows from the and label OAH. 2. Match OAH with SOHCAHTOA. 3. Set up your equation. 4. Find decimal for trig ratio. 5. If X is in numerator…MULTIPLY side by trig decimal. ?° B C You know two sides… You want to find an … 1. Draw your arrows from the and label OAH. 2. Match OAH with SOHCAHTOA. 3. Set up your equation. 4. Find decimal for trig ratio. 5. On your calculator, hit BLUE BUTTON, hit trig ratio and type in decimal. 6. Hit ENTER. B Practice / Homework Pits, Hips, Legs and ABCs Where do you find pits, hips, legs and ABCs? In a… RIGHT TRIANGLE Let’s label the parts: Essential Question(s) How do I (HDI) use PYT? or Objective(s) Students will be able to (SWBAT) or name the parts of a right triangle. SWBAT label the HYP of a right triangle. SWBAT label or the LEGs of a right triangle. SWBAT label What’s the relationship between the LEGs and the HYP? the parts of a right triangle with ABC. The Pythagorean Theorem SWBAT use PYT to find a which is, in formulaic form, … right triangle’s missing length. + = Classwork/Homework 7. 8. http://www.youtube.com/watch?v=5l_0BqSmXwA http://www.youtube.com/watch?v=pSgUvclFnrQ https://www.youtube.com/watch?v=_OJ1yVs0aQE 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.