Activity #3. How many things are going on in this simple configuration? When your TEAM has 10 or more things, get ALL of your stuff stamped off!

Transcription

1 Activity #3 How many things are going on in this simple configuration? When your TEAM has 10 or more things, get ALL of your stuff stamped off! ID:A EO2 Level 2 Answers ID:B F Mastery Reform Complete MR for Level 3's only 2 available Academic Support Periods Turn it in (staple cherry form on top of your corrections/ work and put the test in the back) If you are 100% correct on the Mastery Reform, you are eligible for the retake. Everyone taking the retake must complete both the Level 3 and Level 4 sections Do better? Lower grade is dropped Do worse? Level 3 scores are averaged and the lower Level 4 score is dropped H 1

2 Math 3 Quarter 3 Overview EO1 Opp #1 Thu, Jan 19th ML EO1 One Variable Inequalities 12% EO2 Two Variable Inequalities 13% EO3 Similarity & Congruence 25% EO4 Polynomials 25% Q3 Midterm EO2 Opp #1 Fri, Jan 27th EO3 Opp #1 Tue, Feb 14th ML EO4 Opp #1 Wed, Mar 8th ML Level 2 Opp #2 Q3 Final Fri, March 10th EO1 Level 3 Mastery Reform Due Mon, Jan 30th EO1 Opp #2 Tue, Jan B211 EO2 Level 3 Mastery Reform Due Tue, Feb 7th EO2 Opp #2 Thu, Feb 9th EO3 Level 3 Mastery Reform Due Fri, Feb 24th EO3 Opp #2 Tue, Feb 28th p163 TATS Inv 311 Inv 311 p164 #1 5 OYO p179 #1 p180 # p167 STM, CYU, EQ Similarity & Proportion Practice Inv 121 p30 #1 3, EQ OYO p40 #1, p43 #9 Inv 312 p168 #1 5, STM, CYU, EQ OYO p41 #3, 5 p45 #13*, p46 #14, 15** Inv 313 p175 #3, 4, 5***, 6 OYO p #4 Write proofs! Properties of Similar Polygons (based on Inv 1 #5) Parallel Lines Cut by a Transversal Foldable EO1 Opp #2 Tue, Jan 31st Properties, Postulates, & Theorems Angle Vocabulary Solving an SAS Triangle Similarity Theorems Tips for Writing Similarity Proofs Signed Progress Report due last Friday! *p45 #13 Angle Addition Postulate **p46 #15 Exterior Angle Theorem for a Triangle ***p176 #5 Midpoint Connector Theorem for Triangles EO2 Mastery Reform due Tuesday 2

3 Proofs Involving p195 TATS Triangle Centers Similar Inv 321 p196 #2 4, (handout) Triangles 6 7, STM, CYU, EQ EO3 Quiz #1 Mon, Feb 6th Inv 322 p202 #4 STM, CYU, EQ (organizer) EO3 Quiz #2 Fri, Feb 10th Triangle Congruence Flip Book Tips for Congruence Writing Proofs Triangle Centers Properties of Quadrilaterals Team Homework Discussion OYO p27 #31, p #30 OYO p179 #1, p180 #3 Similarity & Proportion Practice OYO p40 #1, p43 #9 OYO p41 #3, 5, p45 #13*, p46 #14, 15** Full credit (2 stamps) before team discussion. Partial credit after. Answers in slides just before Toolkit entries. 3

4 p168 Essential Questions What combinations of side or angle measures are sufficient to determine that two triangles are similar? Language Objective Mathematicians will be able to......use AA, SSS, and SAS to prove 2 triangles similar. Activities #1 5, STM, CYU, EQ p168 4

5 TOOLKIT: Solving a SAS triangle Reference: Investigation 2 Sufficient Conditions for Similarity of Triangles p169 #1 Given SAS information Use the Law Of Cosines to find the 3rd side Use Law Of Sines to find the 2nd angle Use the Triangle Sum Property to find the 3rd angle Example: Use the Law Of Cosines to find the 3rd side Use Law Of Sines to find the 2nd angle Use the Triangle Sum Property to find the 3rd angle 5

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8 Side Angle Side similarity theorem! 83 o 2m 83 o 3m 4m 6m Could you say the triangles were similar? If you saw this: 10m 15m 30m 45m 17m 51m Could you say the triangles were similar? 8

9 Law of Cosines 9

10 Side Side Side similarity theorem! 10m 15m 30m 45m 17m 51m Could you say the triangles were similar? 10

11 TOOLKIT: Similarity Theorems #1 SAS Similarity Theorem! SAS ~ Thm #2 SSS Similarity Theorem! SSS ~ Thm #3 AA Similarity Theorem! AA ~ Thm NOTE: Why are ASA and SAA not included as similarity theorems??? ANSWER: When considering "sufficient conditions" we are looking for the minimum criteria that consistently proves similarity. Since we have proved AA as a similarity theorem, that's the minimum criteria. Both ASA and SAA add an additional criteria of a S (side). Can you see that both ASA and SAA have AA in them? What is sufficient information? Correspondence patterns that work for similarity: p172 SAS SSS AA 11

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14 p o 38 o 57 o 95 o 28 o 57 o Similar? N 14

15 15 35 o o 16 Similar? Y Similar? Y 15

16 Similar? N 83 o 34 o o 83 o 34 o o Similar? Y 16

17 2. Fill in the 3 parts that are given to you. Make sure they go in the right spot! c is always the longest side. If equal, then it's a right triangle. If NOT equal, then it's NOT a right triangle 2. Fill in the 3 parts that are given to you. Make sure they go in the right spot! c is always the longest side. If equal, then it's a right triangle. If NOT equal, then it's NOT a right triangle temp.notebook Math 1 TOOLKITS TOOLKIT: Pythagorean Theorem & Its Converse The Pythagorean Theorem: For a right triangle, the sum of the two leg lengths squared is equal to the length of the hypotenuse squared. a 2 + b 2 = c 2 a b c The Pythagorean Theorem can be used to find any missing side of a RIGHT triangle so long as two sides of the triangle are known. Examples: Find the hypotenuse: Find the leg: The Converse of the Pythagorean Theorem: For a any triangle with sides a, b, c, if a 2 + b 2 = c 2, then it's a right triangle? 1. Write a 2 + b 2 = c 2 2. Fill in the 3 parts that are given to you. Make sure they go in the right spot! c is always the longest side. 3. Solve each side of the equation. If equal, then it's a right triangle. If NOT equal, then it's NOT a right triangle 4. Write your answer in a complete sentence Asking the question "Is it true?" Example #1? 1. Write a2 + b2 = c2 Example #2? 1. Write a2 + b2 = c2 13in 12in 3. Solve each side of the equation. 5cm 7cm 3cm 3. Solve each side of the equation. 5in 4. Write your answer in a complete sentence 4. Write your answer in a complete sentence 17

18 TOOLKIT: Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. Examples: a c b Activity #2: Can these side lengths form a triangle? Quadrilateral? Explain. 1) 5, 7, 13 4) 3, 5, 7, 13 2) 6, 12, 9 5) 12, 6, 9, 25 3) 25, 10, 15 6) 24, 2, 14, 8 TOOLKIT: Quadrilateral Inequality Theorem The sum of any three side lengths of a quadrilateral is greater than the fourth side length. Examples: Can form a quadrilateral 5 2 a 5 b d 12 Cannot form a quadrilateral c 3 18

19 TOOLKIT: Triangle Congruency Theorems Triangles have 3 angles and 3 side lengths. To prove that triangles are congruent, you must show that there are 3 congruent parts. A C B F D E Y Z X CONGRUENT FIGURES All corresponding sides and all corresponding angles are congruent (equal)! ΔABC ΔDEF ΔXYZ AB DE XY AC DF XZ BC EF YZ <A <D <X <B <E <Y <C <F <Z So, how many combinations of angles and sides are possible? Six! S.A.S. S.S.S. A.A.S. A.S.A. H.L So, it must be a right triangle to have an hypotenuse! But two of them can NEVER be used to prove congruence. S.S.A A.A.A Be careful to recognize that the order of these letters represents the shortest consecutive path around the triangle. Do you understand??? GIVEN: Could be in list, a sentence, or a diagram. PROVE: Could be either that triangles are congruent or that parts are congruent. STATEMENT triangles are congruent Like parts are congruent Must prove that triangles are congruent first, then parts Like sides, or angles TOOLKIT: Proofs CPCTC REASON The body of the proof must prove that the three pairs of corresponding parts are congruent (and leads to the congruence theorem that you're using). SSS, SAS, AAS, or ASA Congruence Theorm (No SSA!!!) Corresponding Parts of Congruent Triangles are Congruent 19

20 TOOLKIT: Corresponding Parts of Congruent Triangles are Congruent (CPCTC) When developing a proof for triangle parts, you must first prove that the triangles are congruent, and then you can use CPCTC as the reason the triangle's other parts are congruent. CPCTC: Corresponding Parts of Congruent Triangles are Congruent Be sure that you understand each part of this statement. Reflexive Property Congruent to itself (applies to both segments lengths and angles) D A shared line C B shared angle D A E B C Vertical Angles Theorem Substitution Property of Equality Triangle Sum Property When two lines intersect, opposite angles that share the same vertex or corner point are congruent. If the value of 2 quantities are known to be equal, then the value of one quantity can be replaced by the value of the other. The sum of all angles of a triangle is equal to 180 o C A B Subtraction Property of Equality Subtracting the same number from each side of an equation gives us an equivalent equation. 20

21 TOOLKIT: Law of Sines or "I use the Law of Sines when I know SAA or SSA." Set Up Tips 1. Begin with the fraction bars & equal sign 2. Put your unknown in the first numerator 3. Put the opposite term in the denominator 4. Put the known angle & opposite side pair on the right Side, Angle, Angle Side Angle Side, Side, Angle In both cases I know 1 angle & Side Angle opposite side pair! Angle Side Math 2 TOOLKITS 21

22 B a c TOOLKIT: Pythagorean Theorem C b A When solving for the hypotenuse: When solving for one of the legs: Pythagorean Theorem Converse If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Pythagorean Triples: There are certain sets of numbers that have a very special property in connection to the Pythagorean Theorem. Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem. 3, 4, 5 5, 12, 13 8, 15, 17 y 66 o Terminal Side Ө Initial Side TOOLKIT: Standard Position Angle x An angle formed by rotating (counterclockwise is positive, clockwise is negative) a ray from its initial position (vertex at the origin and laying on the x axis) to the terminal position. For any standard position angle whose terminal side passes through the point (x, y), we know the 3 side lengths are... Ө 66 o x P(x, y) y 22

23 TOOLKIT: Trig Functions: Sine, Cosine, and Tangent Hypotenuse Ө x Adjacent P(x, y) y Opposite Soh Cah Toa y sin Ө = = r x r y Opposite Hypotenuse Adjacent cos Ө = = x Hypotenuse Opposite tan Ө = = Adjacent Memorize! Note to Students: This Toolkit is sufficient by itself, but you can also add the next 3 slides if you want. TOOLKIT: Inverse Trig Functions Any time we're solving for the ANGLE we use the INVERSE trigonometry function. Take the inverse trig function of each side of the equation Any value times its inverse equals 1 Use your calculator (in DEGREE mode) to find the answer 23

24 TOOLKIT: "Special" Right Triangles There are two "special" right triangles with which you need to be familiar; the 45º 45º 90º triangle and the 30º 60º 90º triangle. The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions. When you then rationalize the denominator, you'll get This chart shows the values with rationalized denominators as well as the and trick to help those students who like memorizing tables. TOOLKIT: Angles Elevation & Depression 24

25 TOOLKIT: Law of Sines, Part 1 or "I use the Law of Sines when I know SAA or SSA." In both cases I know 1 angle & opposite side pair! TOOLKIT: Law of Sines, Part 2 or "I use the Law of Sines when I know SAA or SSA." Set Up Tips 1. Begin with the fraction bars & equal sign 2. Put your unknown in the first numerator 3. Put the opposite term in the denominator 4. Put the known angle & opposite side pair on the right Side, Angle, Angle Side Angle Side, Side, Angle In both cases I know 1 angle & Side Angle opposite side pair! Angle Side 25

26 TOOLKIT: Law of Cosines (LoC) Solving for the 3rd side when given SAS Solving for the 1st angle when given SSS Given SAS information Equal, Equivalent Forms Solving a SAS Triangle: Use LOC to find the 3rd side Use LOS to find the 2nd angle Solving a SSS Triangle: Given SSS information Use the Triangle Sum Property to find the 3rd angle Use LoC to find one angle Use LoS to find a 2nd angle Use the Triangle Sum Property to find the 3rd angle 26

27 TOOLKIT: Similarity Theorems #1 SAS Similarity Theorem! #2 SSS Similarity Theorem! #3 AA Similarity Theorem! NOTE: Why are ASA and SAA not included as similarity theorems??? ANSWER: When considering "sufficient conditions" we are looking for the minimum criteria that consistently proves similarity. Since we have proved AA as a similarity theorem, that's the minimum criteria. Both ASA and SAA add an additional criteria of a S (side). Can you see that both ASA and SAA have AA in them? 27

28 TOOLKIT: Midpoint Connector Theorem for Triangles. If a line segment joins the midpoints of two sides of a triangle, then it is half the length of the third side of the triangle and parallel to the third side. 2 MN = AC MN AC Math 3 EO3 TOOLKITS Similarity Calculating Scale Factors Properties of Similar Polygons (see Inv 1 #5 slides) Parallel Lines Cut by a Transversal Foldable Parallel Lines Cut by a Transversal Properties Properties, Postulates, & Theorems Angle Definitions Solving a SAS Triangle Similarity Theorems Tips for Writing Similarity Proofs 28

29 TOOLKIT: Calculating Scale Factors Similar ~ triangles have: 1. Congruent angles. 2. Proporonal sides (same scale factor). = Scale Factor > 1 Smaller to larger polygon side length of the larger polygon side length of the smaller polygon side length of the smaller polygon Larger to smaller polygon = side length of the larger polygon Scale Factor < 1 Examples: Scale Factors Large to small Small to large Solve for x: Scale Factors Large to small Small to large Solve for x: Scale Factors Large to small Small to large Solve for x: Scale Factors Large to small Small to large Solve for x: TOOLKIT: Properties of Similar Polygons See p166 Investigation 311 #5 All Isosceles right triangles are similar. All equilateral triangles are similar. All squares are similar. All rhombuses are NOT similar. All regular (all sides congruent) hexagons are similar...all regular n agons are similar. 29

30 temp.notebook Parallel Lines Cut by a Transversal FOLDABLE TOOLKIT: TOOLKIT: Properties, Postulates, & Theorems Name Description Linear Pair Postulate p31 Vertical Angles Theorem p32 Angle Addition Postulate Diagram Two adjacent angles whose unshared sides form a straight angle are supplementary (equal to 180o) 180o When two lines intersect, opposite angles that share the same vertex or corner point are congruent. A If P is a point in the interior of then P B See p45 #13 C Exterior Angle An exterior angle of a triangle Theorem for a is equal to the sum of the two Triangle remote interior angles See p46 #15 Midpoint Connector Theorem for Triangles If the two midpoints of a triangle are connected, then the midline is parallel to and half the length of the third side. See p176 #5 Substitution Property of Equality Addition, p31 Subtraction, Multiplication or Division Property of Equality* If the value of 2 quantities are know to be equal, then the value of one quantity can be replaced by the value of the other. Adding, subtracting, multiplying, or dividing* the same number from each side of an equation gives us an equivalent equation. *Pick just one operation! Corresponding CAs Angles Postulate are Congruent Converse of the Corresponding Angles Postulate If two lines are intersected by a transversal and corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Theorem AIAs are Congruent Converse of the Alternate Interior If two lines are intersected by a transversal and alternate Angles Theorem interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Theorem AEAs are Congruent Converse of the Alternate Exterior If two lines are intersected by a transversal and alternate exterior angles are congruent, then the lines are parallel. Angles Theorem Same side Interior Angles Theorem Converse of the Same side Interior Angles Theorem Same side Exterior Angles Theorem Converse of the Same side Exterior Angles Theorem Reflexive property SSIAs are Supplementary If two lines are intersected by a transversal and same side interior angles are congruent, then the lines are parallel. SSEAs are Supplementary If two lines are intersected by a transversal and same side exterior angles are congruent, then the lines are parallel. Congruent to itself or NOTE: We will add to this Toolkit as necessary. 30

31 TOOLKIT: Angle Vocabulary Define the following terms. Diagrams would be helpful. Point identifies a position, has no dimension, labeled with a single capital letter Line determined by two points, infinite length, no thickness or width, labeled with two capital letters with a line above or one single lower case letter Plane a two-dimensional serface determined by 3 points, infinite length and width but no thickness, labeled with 3 capital letters or one italicized capital letter Angle two rays with a common starting point Angles can be named with numbers Angles can be named with letters. V U W Y X Note that is not specific. It could be referring to any of 4 angles. Right Angle equals 90 degrees Obtuse Angle greater than 90 degrees Acute Angle less than 90 degrees Straight Angle equals 180 degrees Adjacent Angles two angles with a common vertex and shared side (ray) Linear Pair adjacent angles whose measures add up to 180 degrees (supplementary) Vertical Angles two congruent angles formed by intersecting lines Complementary Angles 2 angles whose measures add up to 90 degrees Supplementary Angles 2 angles whose measures add up to 180 degrees Perpendicular Lines lines in a plane that intersect at 90 degree angles Parallel Lines lines in a plane do not intersect Transversal line that intersects parallel lines Postulate (or axiom) Theorem NOTE: We will add to this Toolkit as necessary. TOOLKIT: Solving a SAS triangle Reference: Investigation 2 Sufficient Conditions for Similarity of Triangles p169 #1 Given SAS information Use the Law Of Cosines to find the 3rd side Use Law Of Sines to find the 2nd angle Use the Triangle Sum Property to find the 3rd angle Example: Use the Law Of Cosines to find the 3rd side Use Law Of Sines to find the 2nd angle Use the Triangle Sum Property to find the 3rd angle 31

32 TOOLKIT: Similarity Theorems #1 SAS Similarity Theorem! SAS ~ Thm #2 SSS Similarity Theorem! SSS ~ Thm #3 AA Similarity Theorem! AA ~ Thm NOTE: Why are ASA and SAA not included as similarity theorems??? ANSWER: When considering "sufficient conditions" we are looking for the minimum criteria that consistently proves similarity. Since we have proved AA as a similarity theorem, that's the minimum criteria. Both ASA and SAA add an additional criteria of a S (side). Can you see that both ASA and SAA have AA in them? TOOLKIT: Tips for Writing Similarity Proofs GIVEN: Could be in list, a sentence, or a diagram. PROVE: Typically four options Triangles are similar ~ Corresponding angles are congruent Sides are related by the same scale factor Lines are parallel STATEMENT 1. Proofs typically begin by stating the given information 1. Given REASON The body of the proof must lead to the similarity theorem that you're using: AA ~ Thm: Write statements & reasons that show that the two pairs of corresponding angles are congruent. SSS ~ Thm: Write statements & reasons that show that all corresponding sides are related by the same scale factor. SAS ~ Thm: Write statements and reasons that show that the two pairs of corresponding sides are related by the same scale factore and that the included angles are congruent. Your final statement must be exactly the same as what you're asked to prove in the prompt. If proving that triangles are similar, then your last statement is a similarity statement AA ~ Thm or SSS ~ Thm or SAS ~ Thm If proving that Prove that corresponding angles are congruent, then your last statement is an angle NOTE: You must have shown that the triangles congurence statement. are similar first, then you can use this reason! CASTC* (Corresponding Angles of Similar Triangles are Congruent) If proving that sides are related by the same scale factor, then your last statement is a scale factor statement where k NOTE: You must have shown that the triangles is the scale factor. are similar first, then you can use this reason! CSSTSSSF* (Corresponding Sides of Similar Triangles Share the Same Scale Factor) If proving that lines are parallel, then your last statement is a parallel lines statement. Pick the reason that fits the proof { If......corresponding angles are congruent......alternate interior angles are congruent......alternate exterior angles are congruent......same side interior angles are supplementary......same side exterior angles are supplementary......then the two lines cut by the transversal are parallel. Converse of the Corresponding Angles Postulate Converse of the Alternate Interior Angles Theorem Converse of the Alternate Exterior Angles Theorem Converse of the Same side Interior Angles Theorem Converse of the Same side Exterior Angles Theorem *CASTC and CSSTSSSF are the acronyms for the phrases in parenthesis. If you can recall the acronym correctly, then use it. If not, you need to write out a similarly worded reason. 32

33 Attachments M3 211 Organizer.docx M3 311 Parallel Lines & Transversal Foldable.pdf SelectorTools.exe