The Pythagorean Theorem: For a right triangle, the sum of the two leg lengths squared is equal to the length of the hypotenuse squared.

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1 Math 1 TOOLKITS

2 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 Asking = c 2 the question "Is it true?" 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 Example #1? 1. Write a 2 + b 2 = c 2 Example #2? 1. Write a 2 + b 2 = c 2 13in 12in 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 5cm 7cm 3cm 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 5in 4. Write your answer in a complete sentence 4. Write your answer in a complete sentence

3 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

4 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

5 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???

6 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

7 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.

8 Postulate (or axiom) Theorem Reflexive Property understood to be true with proof statement that has been proved using deductive reasoning from definitions, accepted facts, and relations Congruent to itself (applies to both segments lengths and angles) D A shared line C B shared angle D C A E B Vertical Angles Theorem Substitution Property of Equality Triangle Sum Property Subtraction Property of Equality 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 Subtracting the same number from each side of an equation gives us an equivalent equation. C A B

9 Math 2 TOOLKITS

10 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

11 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

12 TOOLKIT: Trig Functions: Sine, Cosine, and Tangent Hypotenuse Ө x Adjacent P(x, y) y Opposite Soh Cah Toa y Opposite sin Ө = = r Hypotenuse cos Ө = x r Adjacent Hypotenuse tan Ө = y x Opposite Adjacent Memorize! Note to Students: This Toolkit is sufficient by itself, but you can also add the next 3 slides if you want.

13 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

14 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.

15 TOOLKIT: Angles Elevation & Depression

16 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!

17 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

18 TOOLKIT: Law of Cosines (LoC) Solving for the 3rd side when given SAS Solving for the 1st angle when given SSS Equal, Equivalent Forms Solving a SAS Triangle: Given SAS information 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

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21 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?

22 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

23 Attachments M3 211 Organizer.docx M3 311 Parallel Lines & Transversal Foldable.pdf SelectorTools.exe For each of the triangles below.docx