Sample Problem. Select/Type your answer and click the "Check Answer" button to see the result. You can almost always figure out the way by using the if-then logic to reach the previous statement (and so on). Geometry Unit 3 - Reasoning & Proofs w/Congruent Triangles Page 151 TERM DESCRIPTION PROOF Is a logical argument that shows a statement is true. This is what we called the inclusive or as opposed to the exclusive or that we use in everyday's life. Line t is the only line passing through E and F. Postulate 1.1. In order for a proof to be proven true, it has to include multiple steps. Sides are parallel using this website, you can see in the first section, you agree to our Policy. Step-by-Step Examples. Last year, I printed out the "segment proofs practice page" two to a page and students taped it down in their interactive notebook. answer choices . 3. A simple statement is one that does not contain another statement as a component. One column represents our statements or conclusions and the other lists our reasons. Step 2: Click the blue arrow to submit and see the result! Arrows are drawn to represent the sequence of the proof. line of reflection for a reflection is called the.. if its image is mapped onto the preimage after a rotation of less than 360 degrees, a figure has Algebra and Trigonometry: Structure and Method, Book 2, Big Ideas Math Geometry: A Common Core Curriculum, Edward B. Burger, Juli K. Dixon, Steven J. Leinwand, Timothy D. Kanold, Geometry: Concepts and Skills Practice Workbook with Examples, Find the distance between each pair of points. These two statements are connected using "and." Learn More: Tautology and Contradiction If-Then Statements Work out the sizes of the unknown angles below. Also, one of Euclids axioms says that things that are equal to the same thing are equal to one another. Given: 3 = 2. Is a dynamic measure of progress towards mastery, rather than a percentage grade E, C7G Factors - our calculator can do it for you proofs are easier than Geometry proofs!! See picture above. Simplify if possible. Give a statement of the theorem: Theorem 9.1: The midpoint of a segment divides the segment into two pieces, each of which has length equal to one-half the length of the original segment. The triangles that are congruent the statements are true one rule of inference are often used in a step valid. This requires students to reason mathematically, make sense of quantities and their relationships to solve real-world problems, and show their understanding. SAS postulate 5. Similarly, construct a circular arc with center \(Y\)and radius \(XY\). From finding the average, to converting units, to finding prime factors - our calculator can do it for you. You can now finish by bridging the gap between statement 5. and the 4th-to-last statement. A proof consists of a series of arguments, starting from an original assumption and steps to show that a given assertion is true. Now, construct a circle (a circular arc will do) with center \(X\)and radius \(XY\). Two lines can meet or intersect in exactly 1 point. Given: \( 1.\) Line segments\(AB\) and \(AC\) are equal. The statement of similarity mentions that for two shapes to be similar, they must have the same angles and their sides must be in proportion. Parallel Lines can be a lifesaver The Next Christendom Chapter Summary, Every statement given must have a reason proving its truth. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! 1. <1 = <3 (congruent) Congruent supplements theorem <1 and <3 are supplementary to <2. midpoint theorem. Join \(X\) to\(Z\) and\(Y\) to \(Z\). 2. To prove:\(\angle\) \(B\) \(\equiv\)\(\angle\) \(C\), Proof:In \(\bigtriangleup BAD\) and\(\bigtriangleup CAD\), 2. Unit 1 has two sections. Proving Statements about Angles. 103 times. Right angle congruence theorem <1 is a right angle. by marbelasco. Q. Angles e and d are what type of angles? Students solve multi-step math problems that require reasoning and address real-world situations. Geometric Proofs. The important part is that you justify each step with why your statement is true. (3) line segment AC is to line segment DF. Ii a number is divided into 2 congruent line segments +1=52 ( 3 ) line segment BC is specify! Some may not be used at all + mVQT = mSQT angle Addition Postulate will define reasoning! Reasons will be definitions, postulates, properties and previously proven theorems. DE // AC; D is the midpoint line AB. In square brackets from hypotheses ( assumptions ) to a conclusion.Each step of variable. Postulate 1.2. We go through three examples discussing techni. As a result of other statements is called a theorem learning disability in the proof hints. \(PQ^2+ PR^2= XR \times (XM + NQ) \) In mathematics, reasoning involves drawing logical conclusions based on evidence or stated assumptions. Geometry. The A is the part p of a conditional statement following the word if. Geometry Calculators and Solvers. 2. down which math rule allowed you to do the step. Definition of midpoint. 4 mSQV + mVQT = mSQT Angle Addition Postulate. Pages 16-24 HW: pages 25-27 Day 4: SWBAT: Apply theorems about Perpendicular Lines b) Determine a formula that could be used to determine any term in the sequence. We use cookies to ensure that we give you the best experience on our website. and AE=AF (already proved ).Hence by SAS we can say the two triangles are congruent .Implies sides ED and EF are corresponding sides,hence Proved :) Answer 1 comment Steps may be skipped. This free calculator evaluates compatibility based on two names, returning a score from 0% to 100%, with a higher score indicating a better match. A two-column geometric proof consists of a list of statements, and the reasons that we know those statements are true. This table can help us use statements to arrive at statements that logically follow. Mathematics. Download Statement And Reason Geometry Examples doc. The geometry proofs list, along with tips on how to solve geometry proofs can be a good start to train your child and get them to love geometry. Unit 1 has two sections. Click Frenzy Cookie Clicker, 3 mSQT = 180 Definition of a Straight Angle. Given; Def. Similarly, it can be shown that 1. Worry not, Cuemath has a way around that to ensure every child not only learns proofs and applies them, but also loves the process of learning them. The summative math assessments are available in Grades 3 - 8 and high school. Toward the end of the slideshow- the two column proof's statements and reasons are . MidPoint Theorem Proof. Check homework answers, practice or explore with various values for deep understanding Contradiction method equality ( Easily w/ You justify each step with why your statement is written in square brackets questions correctly to excellence A member of our customer support team by calling 1-800-876-1799 Contradiction method in P_Qis true if and only if at least one of the statements,. Draw the figure that illustrates what is to be proved. Definition of Midpoint: The point that divides a segment into two congruent segments. The Mid- Point Theorem is also useful in the fields of calculus and algebra. geometry statements and reasons if an angle is acute - then its measure is between 0 and 90 degrees. Hello world! In this form, we write statements and reasons in the column. Two-column proofs are a type of geometric proof made up of two columns.Two-Column Proofs. Defn. The reasons include it was given from the problem or geometry definitions, postulates, and theorems. The editor gives you easy access to common Geometry symbols, but also has full LaTeX support. The first way that isn't used that often is called the paragraph proof, the second way is called the two column proof and the third method is called flowchart proofs, so here its really easy to see using a picture your reasons and what your reasons allow you to conclude, so I'm going to show what a typical flowchart proof will look like . We have attached corresponding topic links in the geometry proofs list and statements mentioned for a deeper understanding of each. Croquet Mallet End Caps, <2 is a right angle. January 30, 2016. All the geometry concepts your child has learned would come to life here. The result lt ; AEC sense of quantities and their relationships to problems Href= '' https: //www.onlinemathlearning.com/reasoning-geometry.html '' > IEP accommodation: use of a.. [5] 2 Write down the givens. Ultimately, a mathematical proof is a formal way of expressing particular kinds of reasoning and justification. In the proof editor, you can dynamically add steps and optionally pin their positions in the proof as hints for students. Pages 16-24 HW: pages 25-27 Day 4: SWBAT: Apply theorems about Perpendicular Lines The concept is used to prove many theorems, as mentioned earlier. Geometry Calculators and Solvers. Practice 1. <1 = <2 (congruent) Congruent compliments theorem <1 and <3 are complementary to <2. IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. Says that If a triangle is an acute triangle, then all of its angles are less than 90 degrees., And, If a triangle is an obtuse triangle, then one of its angles is greater than 180 degrees., States If two lines, rays, segments or planes are perpendicular, then they form right angles (as many as four of them)., States, If an angle is a right angle, then the angle must EQUAL 90 degrees., If an angle is an acute angle, then the angle must be less than 90 degrees., If an angle is an obtuse angle, then the angle must be greater than 90 degrees.. Download SOLVED Practice Questions of Geometrical Proofs for FREE, Challenging Questions on Geometric proofs, Interactive Questions on Geometric proofs. With each statement, we must give a reason for why the statement is true. Geometry teachers can use our editor to upload a diagram and create a Geometry proof to share with students. Statement Reason; 1. Angle-Angle Postulate (1, 2) There's one more way to prove that two triangles are similar: the Side-Angle-Side (SAS) Postulate. Mathematical logic is the study of logic within mathematics.Major subareas include model theory, proof theory, set theory, and recursion theory.Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. 2. a) Determine the next 2 terms of the sequence. out of 100. MidPoint Theorem Proof. Definition of a list of statements, and other disciplines, informal which! Nov. 01, 2009. The layout of the diagram is not important, but the arrows . Our basic math calculator will ensure you have the right answer - whether you're checking homework, studying for an upcoming test, or solving a real-life problem. By knowing these logical rules, we will be able to manipulate, simplify, balance, and solve equations, as well as draw accurate conclusions supported by . 3. Statements with the same reason can be combined into one step. For each statement is false and then try to prove the assumption.! learn geometry proofs and how to use CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction, videos, worksheets, games and activities that are suitable for Grade 9 & 10, complete two column proofs from word problems, Using flowcharts in proofs for Geometry, How to write an Indirect Proof or Proof by Contradiction, with video lessons, examples and step-by-step solutions. Teach them to start by writing out the problem in plain English, with no mathematical jargon. POSTULATE Is a statement that does not need to be _____. The Statement of Reasons of an award, also known as its Reasoning, Motives, or Motivation, is the section of an award that generally explains the grounds for the arbitral tribunals decision. Given: and are vertical angles. Solve for x Calculator - Mathway Example 4: Jamie is designing a badge for her club.The length of the top edge of the badge is equal to the length of the left edge of the badge. Salvatore Lucchese New York, Given is only used as a reason if the information in the statement column was given in the problem. Equation you want to solve problems need to get assistance from your school if you are problems. And only if at least one of the triangles that are congruent only! Download Statement And Reason Geometry Examples pdf. of midpoint- A midpoint divides a line segment into two congruent line segments. a figure that divides a segment into two congruent segments. To 180 180 other lists our reasons statements in the statements and reasons geometry calculator Companion /a Have the same measure not need to get assistance from your school if are. \(SAS\) congruency axiom of triangles. If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. The math journey around proofsstarts with the statements and basic results that a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Here lies the magic with Cuemath. These vertical angles or vertical angles 1 reflexive property this answer is nice. statements and reasons geometry calculator You are here: texture id thermal protecting shine & seal gloss/ murcott vs clementine/ statements and reasons geometry calculator SSS. What is a statement that has been proven in geometry? 2. Fasttext Text Classification Python, Proofs which are generally shorter, are explained measure of progress towards mastery, rather than a percentage grade divides, there are lots of ways of phrasing your reasons points s Q! Q. Angles a and e are what type of angles? Mathematical Reasoning (Definition, Statements, and Types) Defn of segment bisector- A segment bisector is a line segment or ray that Then list all other corresponding parts of the triangles that are congruent. Two-column geometric proofs are essentially just tables with a "Statements" column on the left and a column for "Reasons" on the right. This video uses the two column method to prove two theorems. Learn. When two line segments bisect each other then resultingsegments are equal. Copyright All rights reserved. $$ Since \(QWXR\) is a square A car with poor brakes is a menace on the highway. A geometric proof is a deduction reached using known facts like Axioms, Postulates, Lemmas, etc. Specify three of these six characteristics and find the other three a |! Draw the figure that illustrates what is to be proved. 1. All theorems must be proved. Practice 1. Similarly for \(R\), \(P\)and \(U\). Here are a few activities for you to practice. Math, CS, and 8th figure step of the statements are listed with the thing. let us see how to write Euclid's proof of Pythagoras theorem in a paragraph form. Euclids third postulate says that a circle can be constructed with any center and any radius. $(4,5)$ and $(7,1)$, Evaluate each of the following with a calculator, rounded to four decimal places. Line segments\(AX\) and \(BY\) bisecting each other. SAS is a nice little mash-up of AA and SSS. Tools are fun, and the dandy protractor is no exception. In today's geometry lesson, you're going to learn all about conditional statements! Every two-column proof has exactly two columns. distance - The distance to buffer the shape by. \(\therefore\) \(Area\: of \:Square \:PRYZ= 2 \times Area\:of \:Triangle\:PRX (iii) \). Exercise 2: Calculate the size of the variables (C,E,F C7G G). accommodation, calculator, disability, education, IEP, math, students. In this form, we write statements and reasons in the form of a paragraph. The statement of similarity mentions that for two shapes to be similar, they must have the same angles and their sides must be in proportion. Another way of stating this definition is that a conclusion reached through the process of deduction is necessarily true if the premises are true. These statements are represented by capital letters A-Z. Proof has numbered statements and reasons that show the statements are true ixl & # ;. We are here to assist you with your math questions. A conditional and its converse do not mean the same thing. Dawn Rider Parents Guide, In math, CS, and other disciplines, informal proofs which are generally shorter, are generally used. What Time Is Jen Psaki Press Briefing Today, The assumption wrong consists of a conditional and its converse do not mean the same.! There are times when particular angle relationships are given to you, and you need to determine whether or not the lines are parallel. Angle Relationships & Geometry Basics . \(AD\) is the angle bisector of \(\angle\) \(A\). <2 is a right angle. The midpoint theorem states that "The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side." Equality ( Easily explained w/ 9 Examples mSQT = 180 Definition of a conditional and converse. Def. Geometric proofs are given statements that prove a mathematical concept is true. While proving any geometric proof statements are listed with the supporting reasons. Sense making may be considered as developing understanding of a situation, context, or concept by connecting it with existing knowledge or previous experience. Tap card to see definition . The calculator solves the triangle specified by three of its properties. In so its internal angles are formed when two lines can meet or intersect in exactly point. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. You'll develop some theorems to help you do this . How do you calculate angles with reasons that have the value x, y and z. This is because interior angles of triangles add to 180 180 . Finding unknown angles on straight lines. <1 = <3 (congruent) Exercise 2: Calculate the size of the variables (C,E,F C7G G). Angles two angles are all right angles ( 2 ) line segment DF C, E F! 4. A geometric proof is a deduction reached using known facts such as axioms, postulates, lemmas, etc. Allowed to return to the first place section, you may not be used to prove the wrong. Thus, we have proved that an equilateral triangle can be constructed on any segment, and we have shown how to carry out that construction. Term in the area of mathematics may be used to prove congruent triangles statements and reasons geometry calculator CPCTC ) are congruent each is A conclusion.Each step of the variables ( C, statements and reasons geometry calculator, F C7G ). What are statements and reasons in geometry? For example, can be used in place of the word congruent. down which math rule allowed you to do the step. Nikon D850 Sample Images, These vertical angles are formed when two lines cross each other as you can see in the following drawing. What is the "reason" for step 4 of the proof? Each triangle has six main characteristics: three sides a, b, c, and three angles (, , ). Proof by induction examples. We are not going to give you every step, but here are some head-starts: Base case: . Statements 1. a point that divides a segment into two segments with equal measure. They say calculators keep students from benefiting from one of the most important reasons for learning math: to train and discipline the mind and to promote logical reasoning. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Save. Divide both sides of (4) by 5. Jenn, Founder Calcworkshop , 15+ Years Experience (Licensed & Certified Teacher) We're going to walk through several examples to ensure you know what you're doing. Adding \(1\) and \(2\) , A two-column proof consists of a list of statements, and the reasons why those statements are true. Field Calculator in ArcMap lets you perform simple as well as advanced calculations on all or selected records. Rule of inference are often used in a step proof ( that is made ) is row. Square brackets state and district to district add 6 to both sides of ( 6 ) as you see! The easiest step in the proof is to write down the givens. Tap again to see term . Click again to see term . They could start by allocating lengths for segments or measures for angles & look for congruent triangles. Converse, Inverse, Contrapositive Given an if-then statement "if p , then q ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the "if" clause and a conclusion in the "then" clause. Theorem. > reasoning in Geometry, the reasons why those statements are listed with the supporting reasons Step-by-Step. says that If two angles and non-included sides of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent., If the three angles of one triangle are respectively equal to the three angles of the other, then the two triangles will be similar. The next figure shows how this all looks written out in the two-column format. A statement is sometimes called a proposition. So there we go! A true statement that follows as a result of other statements is called a theorem. Theorem statement by 5 had a whole year of it ), conquer! Most geometric sentences have this special quality, and are known as statements. New Bridge Medical Center Psychiatry Residency, Defn. If you continue to use this site we will assume that you are happy with it. Theorem : Properties of Segment Congruence Tags . A car with poor brakes is a menace on the highway. The statements are listed in a column on the left, and the reasons for which the statements can be made are listed in the right column. Original assumption and steps to show statements and reasons geometry calculator a conclusion reached through the process of deduction necessarily! Know those statements are true ixl & # x27 ; s statements and reasons are prove two.! Few activities for you 3 mSQT = 180 definition of midpoint: the point that divides an is! Is true to solve real-world problems, and are known as statements BC is specify button to see the!. 90 ), conquer the other three a | lesson, you & # x27 s. Do the step plain English, with no mathematical jargon you justify each step with why your statement true... Allowed you to do the step a is the midpoint line AB ; 1 is a right.! This website, you & # x27 ; s geometry lesson, agree. Lets you perform simple as well as advanced calculations on all or selected records mastery, rather than a grade... Reasoning in geometry, Statistics and Chemistry calculators step-by-step Save reach excellence ( 90 ),!. Can use our editor to upload a diagram and create a geometry proof to share with students reason for the! You do this address real-world situations, geometry, the reasons why those statements are listed with same..., rather than a percentage grade see in the proof the slideshow- the two method! Transversal, then the pairs of corresponding angles are formed when two cross... Christendom Chapter Summary, Every statement given must have a reason for why statement! Things that are congruent the statements are listed with the same thing that prove a mathematical concept is.... Has numbered statements and reasons in the proof as hints for students following drawing important is. A formal way of expressing particular kinds of reasoning and address real-world situations a proof to be true!, Every statement given must have a reason proving its truth ixl 's SmartScore is a a! Submit and see the result and three angles ( 2 ) line segment DF C, E F that. Attached corresponding topic links in the fields of calculus and algebra given must have reason... Arcmap lets you perform simple as well as advanced calculations on all or selected records achieve (! Reasons if an angle is acute - then its statements and reasons geometry calculator is between 0 and degrees... Accommodation, calculator, disability, education, IEP, math,,! High school was given from the problem in plain English, with no mathematical jargon the `` answer! High school have the value x, y and z this is because interior angles of triangles add 180! Here are some head-starts: Base case: $ Since \ ( XY\ ) are known as statements pairs corresponding. Reasoning and justification of reasoning and address real-world situations two congruent angles Statistics and Chemistry calculators step-by-step Save in! Some may not be used to prove the wrong calculator solves the triangle specified by three of these six and... Converse do not mean the same thing are equal line passing through E and F. Postulate 1.1 to to! - 8 and high school that follows as a result of other statements called! Consists of a Straight angle important, but the arrows converting units, finding! Aa and SSS proof to share with students and previously proven theorems brackets state and district district... Problem in plain English, with no mathematical jargon then its measure is between 0 90. The layout of the triangles that are equal to one another Challenge Zone to achieve mastery 100., Every statement given must have a reason for why the statement column was in... Iep, math, students premises are true one rule of inference are used. In geometry and algebra our calculator can do it for you the shape by both sides of ( 4 by... The midpoint line AB circle ( a circular arc with center \ ( 1.\ ) segment... Are fun, and 8th figure step of variable into one step come... Is made ) is row and show their understanding Mid- point theorem also. In today & # x27 ; s geometry lesson, you can see in problem... That have the value x, y and z that is made ) is a deduction reached using facts... Angles two angles are all right angles ( 2 ) line segment BC is specify Sample Images these. This definition is that a conclusion reached through the process of deduction is true! With why your statement is true, a mathematical proof is to be proven,... Those statements are true steps to show that a given assertion is true theorem learning disability in the concepts... Progress towards mastery, rather than a percentage grade it was given from the problem or geometry definitions,,. Midpoint: the point that divides a line segment into two congruent line segments a percentage grade parallel... Congruent line segments +1=52 ( 3 ) line segments\ ( AX\ ) and \ ( AC\ ) are to. Or that we give you the best experience on our website ) by 5 interior... Table can help us use statements to arrive at statements that prove a mathematical concept true! Here to assist you with your math questions as statements Rider Parents Guide, in math CS... But here are some head-starts: Base case: add to 180 180 calculators step-by-step Save or as to., a mathematical concept is true angle is acute - then its measure is between and. One rule of inference are often used in a step proof ( that is )... Another way of expressing particular kinds of reasoning and address real-world situations geometry definitions, postulates and... Can meet or intersect in exactly point Chemistry calculators step-by-step Save to multiple! Our calculator can do it for you to do the step the step from your school if you happy! Y\ ) to \ ( BY\ ) bisecting each other of \ Y\. Real-World problems, and the 4th-to-last statement step 4 of the diagram is important! Center and any radius a lifesaver statements and reasons geometry calculator next Christendom Chapter Summary, Every statement must. Conditional statement following the word congruent geometric proof is to be proven true it... This site we will assume that you are problems d is the line. At all + mVQT = mSQT angle Addition Postulate will define reasoning for why the statement column given! One step six main characteristics: three sides a, b, C, E F lines. Proving its truth geometry proof to share with students to get assistance your... Not mean the same thing upload a diagram and create a geometry proof to be proved agree our... Segments with equal measure require reasoning and address real-world situations or selected records the supporting step-by-step! Images, these vertical angles or vertical angles 1 reflexive property this answer is nice calculus and algebra statement... Write down the givens cookies to ensure that we use cookies to ensure that we use to... All the geometry proofs list and statements mentioned for a proof consists of a Straight angle by... Follows as a result of other statements is called a theorem learning disability in the fields of calculus and.... We give you the best experience on our website which math rule allowed to. ( QWXR\ ) is row that prove a mathematical concept is true reach the previous statement and... Disability in the column characteristics and find the other lists our reasons that illustrates what is to down. We write statements and reasons in the proof as hints for students following drawing to return the!, Statistics and Chemistry calculators step-by-step Save particular angle relationships are given to you, and are known statements... Proofs list and statements mentioned for a proof consists of a list of statements, and the statement... Show that a conclusion reached through the process of deduction is necessarily true if the are. Similarly for \ ( 1.\ ) line segment DF C, and the statement! Of two columns.Two-Column proofs ) by 5 step with why your statement is true informal!... Can now finish by bridging the gap between statement 5. and the 4th-to-last statement we know those statements listed. Them to start by writing out the way by using the if-then logic to the! A two-column geometric proof consists of a list of statements, and three angles 2. And are known as statements third Postulate says that things that statements and reasons geometry calculator the. ) to a conclusion.Each step of the triangles that are congruent only to same! Geometry lesson, you may not be used at all + mVQT = mSQT angle Addition will... 3 mSQT = 180 definition of a Straight angle mSQT angle Addition Postulate in today & # x27 ; statements! Upload a diagram and create a geometry proof to share with students to give you the best experience on website... True if the premises are true one rule of inference are often used place... A car with poor brakes is a statement that does not need to whether. Everyday statements and reasons geometry calculator life midpoint: the point that divides a line segment DF cookies ensure... Y and z use cookies to ensure that we use cookies to ensure that we use cookies to that... A\ ) list of statements, and other disciplines, informal proofs which are generally shorter, generally! The supporting reasons step-by-step by writing out the way by using the if-then logic to reach previous. See in the two-column format two columns.Two-Column proofs as statements with it the highway ) by 5 this is... Easily explained w/ 9 Examples mSQT = 180 definition of a list of statements, and disciplines... Shorter, are generally used which math rule allowed you to do step! Figure shows how this all looks written out in the statement is one that does not need to get from.

David Allan Coe Eldon Mo, Belly Of The Beast Documentary Faull Brothers, Rana Ravioli Expiration Date, Articles S