Three Angle Measure Introduction: Fill & Download for Free

Learning Objective(s)· Identify properties, including angle measurements, of quadrilaterals.IntroductionQuadrilaterals are a special type of polygon. As with triangles and other polygons, quadrilaterals have special properties and can be classified by characteristics of their angles and sides. Understanding the properties of different quadrilaterals can help you in solving problems that involve this type of polygon.Defining a QuadrilateralPicking apart the name “quadrilateral” helps you understand what it refers to. The prefix “quad-” means “four,” and “lateral” is derived from the Latin word for “side.” So a quadrilateral is a four-sided polygon.Since it is a polygon, you know that it is a two-dimensional figure made up of straight sides. A quadrilateral also has four angles formed by its four sides. Below are some examples of quadrilaterals. Notice that each figure has four straight sides and four angles.Interior Angles of a QuadrilateralThe sum of the interior angles of any quadrilateral is 360°. Consider the two examples below.You could draw many quadrilaterals such as these and carefully measure the four angles. You would find that for every quadrilateral, the sum of the interior angles will always be 360°.You can also use your knowledge of triangles as a way to understand why the sum of the interior angles of any quadrilateral is 360°. Any quadrilateral can be divided into two triangles as shown in the images below.In the first image, the quadrilaterals have each been divided into two triangles. The angle measurements of one triangle are shown for each.These measurements add up to 180º. Now look at the measurements for the other triangles—they also add up to 180º!Since the sum of the interior angles of any triangle is 180° and there are two triangles in a quadrilateral, the sum of the angles for each quadrilateral is 360°.Specific Types of QuadrilateralsLet’s start by examining the group of quadrilaterals that have two pairs of parallel sides. These quadrilaterals are called parallelograms They take a variety of shapes, but one classic example is shown below.Imagine extending the pairs of opposite sides. They would never intersect because they are parallel. Notice, also, that the opposite angles of a parallelogram are congruent, as are the opposite sides. (Remember that “congruent” means “the same size.”) The geometric symbol for congruent is , so you can write and . The parallel sides are also the same length: and . These relationships are true for all parallelograms.There are two special cases of parallelograms that will be familiar to you from your earliest experiences with geometric shapes. The first special case is called a rectangle. By definition, a rectangle is a parallelogram because its pairs of opposite sides are parallel. A rectangle also has the special characteristic that all of its angles are right angles; all four of its angles are congruent.The other special case of a parallelogram is a special type of rectangle, a square. A square is one of the most basic geometric shapes. It is a special case of a parallelogram that has four congruent sides and four right angles.A square is also a rectangle because it has two sets of parallel sides and four right angles. A square is also a parallelogram because its opposite sides are parallel. So, a square can be classified in any of these three ways, with “parallelogram” being the least specific description and “square,” the most descriptive.Another quadrilateral that you might see is called a rhombus. All four sides of a rhombus are congruent. Its properties include that each pair of opposite sides is parallel, also making it a parallelogram.In summary, all squares are rectangles, but not all rectangles are squares. All rectangles are parallelograms, but not all parallelograms are rectangles. And all of these shapes are quadrilaterals.The diagram below illustrates the relationship between the different types of quadrilaterals.You can use the properties of parallelograms to solve problems. Consider the example that follows.ExampleProblemDetermine the measures of and .is oppositeis oppositeIdentify opposite angles.A property of parallelograms is that opposite angles are congruent.= 60°, so = 60°= 120°, so = 120°Use the given angle measurements to determine measures of opposite angles.Answer= 60° and = 120°TrapezoidsThere is another special type of quadrilateral. This quadrilateral has the property of having only one pair of opposite sides that are parallel. Here is one example of a trapezoid.Notice that , and that and are not parallel. You can easily imagine that if you extended sides and , they would intersect above the figure.If the non-parallel sides of a trapezoid are congruent, the trapezoid is called an isosceles trapezoid. Like the similarly named triangle that has two sides of equal length, the isosceles trapezoid has a pair of opposite sides of equal length. The other pair of opposite sides is parallel. Below is an example of an isosceles trapezoid.In this trapezoid ABCD, and .Which of the following statements is true?A) Some trapezoids are parallelograms.B) All trapezoids are quadrilaterals.C) All rectangles are squares.D) A shape cannot be a parallelogram and a quadrilateral.Show/Hide AnswerYou can use the properties of quadrilaterals to solve problems involving trapezoids. Consider the example below.ExampleProblemFind the measure of .= 360°The sum of the measures of the interior angles of a quadrilateral is 360°.= 90°= 90°The square symbol indicates a right angle.60° + + 90° + 90° = 360°Since three of the four angle measures are given, you can find the fourth angle measurement.+ 240° = 360°= 120°Calculate the measurement of .From the image, you can see that it is an obtuse angle, so its measure must be greater than 90°.Answer= 120°The table below summarizes the special types of quadrilaterals and some of their properties.Name of QuadrilateralQuadrilateralDescriptionParallelogram2 pairs of parallel sides.Opposite sides and opposite angles are congruent.Rectangle2 pairs of parallel sides.4 right angles (90°).Opposite sides are parallel and congruent.All angles are congruent.Square4 congruent sides.4 right angles (90°).Opposite sides are parallel.All angles are congruent.TrapezoidOnly one pair of opposite sides is parallel.SummaryA quadrilateral is a mathematical name for a four-sided polygon. Parallelograms, squares, rectangles, and trapezoids are all examples of quadrilaterals. These quadrilaterals earn their distinction based on their properties, including the number of pairs of parallel sides they have and their angle and side measurements.

Billiards Anyone?To be more precise in the global environment of Quora, when you say we, which country do you mean?To be fair, and while we are at it, which time frame do you also have in mind?For example. In the country that does not exist anymore, the USSR, in the time frame of the past, between the years of 1976 and 1992, I personally received plenty of training in geometric, analytic and computational thinking that was probably as perfectly mixed and balanced as one can hope for a perfectly mixed and balanced combination of training in geometric, analytic and computational thinking in basic mathematical education to be.Based on what I see with my own eyes in public and private schools of the state of New Jersey, USA, and public and private colleges on the Eastern seaboard of the US, the answer unfortunately is a sad yes.The challenge, however, is to not go in the wrong, sacrificial, direction while attempting to fix this problem - the training in geometric thinking should not come at the expense of training in analytic and computational thinking. Put differently, the training in the visual or the spatial thinking should ideally be harmoniously intertwined into a double helix with its analytic and computational counterparts.The idea here is to expose the young minds in their formative years to a well-balanced potion of geometric, analytic and computational thinking in mathematics and to let those kids see and feel the symbiotic relationship across the three for themselves.Why: geometry often serves as or becomes an impetus to analytic developments. The relevant examples are too numerous to list. Take the geometric interpretation of complex numbers by Caspar Wessel and company or Newton’s “Principia”. The reason why it is so difficult to read “Principia” is because it is not set in the language of vectors which make life much easier. Tensors make life even easier and things more generic.Concrete Example.There is this internationally famous and internationally popular academic exercise when the kids are asked to measure a certain amount of water using two buckets or pails of known capacity.Here is one:Using one [math]5-[/math]gallon bucket and one [math]7-[/math]gallon bucket, measure exactly [math]1[/math] gallon of water given its infinite supplyDiscussionThese types of problems are easy to state and formulate. You do not need a lengthy and rigorous formal training in mathematics to comprehend what is being asked of you to accomplish.Yet, these little problems form a tip of a mighty iceberg that is not only and purely mathematical in nature but is also physical as it has something to do with Quantum Mechanics.Most indoctrinated readers will, no doubt, roll their eyes just about now - yeah, yeah, we know the drill. The Euclidean GCD algorithm, coprime integers, the Division Algorithm, yada yada, blah blah blah. The elementary, middle schoolish, theory of numbers.That is correct.This particular problem and its like images can be used as a gentle and hopefully engaging introduction into the topics mentioned above by hinting that [math]1[/math] can be obtained from [math]5[/math] and [math]7[/math] arithmetically as [math]1 = 5\cdot 3 - 7\cdot 2[/math].The rest of the educational story unfolds from there.But.Before you read on, we encourage you to give your wits a mild exercise and see if you can invent or come up with or synthesize a purely geometric solution of not only this problem in particular but of a family of like problems.Geometry (and Physics … in the theory of numbers? Yep. Why not?)Here is a rather mechanical and, you’ve been warned, addictive, way of solving this type of problems that requires very little intellectual involvement on the problem-solver’s part.As these things work in mathematics, it is inventing or discovering or finding such a method that needs a bit of creative juice.As these things work in mathematics, once you settle on a number-theoretic solution of a problem, it is a major challenge to overcome the mental inertia and even think about searching for an alternative way to solve the same problem. But developing the trait of overcoming the intellectual inertia is a part of a basic education in mathematics.Tile a Euclidean plane with a [math]5\times 7[/math] board comprised of [math]70[/math] equilateral triangles numbered just like a chessboard along the horizontal and vertical directions (Fig. 1):The board’s lower and upper edges that are [math]7[/math] triangles wide will be responsible for the watery transitions of the [math]7-[/math]gallon bucket, while the board’s left and right edges that are [math]5[/math] triangles wide will be responsible for the watery transitions of the [math]5-[/math]gallon bucket. The (natural) numbers along the edges show the potential number of gallons in each corresponding bucket.Remember that all our numbered points actually have two [math](x, y)[/math] coordinates but to avoid the clutter we only show one such coordinate except for the corner points: for example, the left point marked with a [math]2[/math] has a [math](0,2)[/math] coordinate and a bottom point marked with a [math]5[/math] has a [math](5,0)[/math] coordinate and so on.Each such coordinate tells us how much water does the corresponding bucket have or should have as the solution of the problem unoflds. For example, the point with the coordinates, say, [math](5,0)[/math] tells us that there are [math]5[/math] gallons of water in the (horizontal) [math]7-[/math]gallon bucket and [math]0[/math] gallons of water in the (vertical) [math]5-[/math]gallon bucket, etc.Imagine now that we place a small orange mathematical billiard ball, a geometric point, into the board’s lower left corner marked with a [math]0[/math] (Fig. 2):We now have two choices since we can hit the ball with a cue to roll:rightward along the board’s lower edge toward the [math](7,0)[/math] point orleftward along the board’s left edge toward the [math](0,5)[/math] pointSay, we go with the second choice and we hit the ball to roll leftward toward the upper left corner marked with a [math]5[/math] and a [math]0[/math] (Fig. 3):and we can already record the first step of a solution as the ball reaches the upper edge of the board:step 1: pour [math]5[/math] gallons of water into a [math]5[/math]-gallon bucketOptional: the kids at this point may be asked to invent their own notation to record these steps. Whatever they invent is perfectly fine - so long as they can explain why they chose this particular notation. Note that the invention process creates the emotional investment in a problem. Different notations should be discussed and compared.Depending on how receptive the kids are, you can suggest the idea of compressing the long names of things into a short thingy - abbreviate the word bucket into a [math]B[/math]. Then put the capacity of a bucket into its lower right corner as in [math]B_5[/math]. Then put the number of gallons of water this bucket currently holds right next to it as in [math]B_5(0)[/math] or [math]B_5(5)[/math] and so on.Thus, after the first step we have [math]B_5(5)[/math] and [math]B_7(0)[/math].Next, we explain a little rule according to which our mathematical ball will bounce around the board once launched and which comes from the discipline in physics known as optics:the angle of incidence is equal to the angle of reflectionPhysicists like to measure the angles of incidence relative to a normal but here we will measure the angles of incidence relative to a tangent (technically) or relative to the edges of the board. There is no need to scare the kids with the techno babble of normals and tangents - just show them how to measure one angle of incidence.When our mathematical ball hits the upper edge of the triangle-ruled board it does so under the [math]60-[/math]degree angle as shown in Fig. 4 below because all our triangles are equilateral (Fig. 4):If, according to our bouncy rule, that angle of incidence is equal to the angle of reflection then where will our mathematical billiard ball roll next?You are correct - our mathematical ball will bounce off the upper edge of the board and then it will start rolling toward the lower right corner under the same [math]60-[/math]degree angle measured relative to the board’s upper edge (Fig. 5):and we can already record the second step of a solution:step 2: pour all [math]5[/math] gallons of water from [math]B_5[/math] into [math]B_7[/math] so that we have [math]B_7(5)[/math] and [math]B_5(0)[/math], as the coordinate of the point [math](5,0)[/math] clearly suggestsNow that we understand the rules of the game, we proceed mechanically: our ball will bounce off the lower edge of the board and will roll toward its upper edge (Fig. 6):and we record the next step of a solution:step 3: pour [math]5[/math] gallons of water into [math]B_5[/math] again so that we have [math]B_5(5)[/math] and [math]B_7(5)[/math], as the coordinate of the point [math](5,5)[/math] clearly suggestsWhere will our ball bounce next? We know - it will reflect off the board’s upper edge and roll toward its right counterpart (Fig. 7):and we record the next step of a solution:step 4: pour only [math]2[/math] gallons of water from [math]B_5[/math] into [math]B_7[/math] so that we have [math]B_5(3)[/math] and [math]B_7(7)[/math]Keep going (Fig. 8):This bounce means that we have to empty the bucket [math]B_7[/math] because the horizontal coordinate of the left point [math]3[/math] is equal to [math]0[/math].Remember that all our numbered points actually have two coordinates:step 5: empty [math]B_7[/math] so that we have [math]B_7(0)[/math] and [math]B_5(3)[/math]Keep going (Fig. 9):Excellent:step 6: pour all [math]3[/math] gallons of water from [math]B_5[/math] into [math]B_7[/math] so that we have [math]B_5(0)[/math] and [math]B_7(3)[/math]Keep rollin’ (Fig. 10):recording the next step of a solution and remembering that the upper point [math]3[/math] has two [math](x,y)[/math] coordinates [math](3, 5)[/math]:step 7: pour [math]5[/math] gallons of water into [math]B_5[/math] for the third time so that we have [math]B_5(5)[/math] and [math]B_7(3)[/math]Bounce once more (Fig. 11):and … bingo! We have a winner:step 8: pour [math]4[/math] gallons of water from [math]B_5[/math] into [math]B_7[/math] so that we have [math]B_5(1)[/math] and [math]B_7(7)[/math]which effectively terminates our process as it measures off - geometrically - exactly [math]1[/math] gallon of water using one [math]5-[/math]gallon bucket and one [math]7-[/math]gallon bucket given the water’s infinite supply, as requested.Is that the only solution?Nope.What is the other one?The other solution is generated when we initially hit the ball rolling rightward along the board’s lower edge - when we first fill the [math]7-[/math]gallon bucket with all [math]7[/math] gallons of water that it can accommodate .However, that solution will cost us [math]12[/math] steps - not [math]8[/math] as in the first case since another way to record a [math]1[/math] is as [math]1 = 7\cdot 3 - 5\cdot 4[/math] and it looks like we will be filling the [math]7-[/math]gallon bucket three times.The reader is encouraged to generate and trace that solution on her/his own as we will not be as thorough:step 1: pour [math]7[/math] gallons of water into [math]B_7[/math] for [math]B_5(0), B_7(7)[/math]step 2: pour [math]5[/math] gallons from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(5), B_7(2)[/math]step 3: empty [math]B_5[/math] for [math]B_5(0), B_7(2)[/math]step 4: pour [math]2[/math] gallons from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(2), B_7(0)[/math]step 5: fill [math]B_7[/math] the second time for [math]B_5(2), B_7(7)[/math]step 6: pour [math]3[/math] gallons from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(5), B_7(4)[/math]step 7: empty [math]B_5[/math] for [math]B_5(0), B_7(4)[/math]step 8: pour [math]4[/math] gallons from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(4), B_7(0)[/math]step 9: fill [math]B_7[/math] for the third time for [math]B_5(4), B_7(7)[/math]step 10: pour [math]1[/math] gallon from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(5), B_7(6)[/math]step 11: empty [math]B_5[/math] for [math]B_5(0), B_7(6)[/math]step 12: pour [math]6[/math] gallons from [math]B_7[/math] into [math]B_5[/math] for [math]B_5(6), B_7(1)[/math]The above steps were an addictive piece of cake to generate - just play the game of mathematical billiards (officially - in [math]2-[/math]space) according to the rules described.Please do not blame me if your husband or your wife, your kids or your boss start complaining that you keep your nose in a piece of paper all day long drawing lines and murmuring something. Unless you are in a classroom.There is a lot going on here with such a simple device.As a teacher you know that the concept of an algorithm is touched upon; that the concept of optimality is touched upon - some solutions are shorter than others; that the concept of generalization is touched upon - different billiards configurations may solve different 3-bucket and even 4-bucket problems if we step out into 3-space of a tetrahedral billiards; that the concept of improvisation is touched upon - some kids may suggest alternative setups of, say, making the edges of a board out of mirrors and instead of a billiard ball shining a laser light into the board; that the concept of a trajectory is touched upon …If you understood how the game of mathematical billiards is played then you should be able to solve the following problem geometrically:Using one [math]7-[/math]gallon bucket and one [math]11-[/math]gallon bucket, measure exactly [math]2[/math] gallons of water given its infinite supplyGood luck.Extra for experts.The layer of knowledge that you are inquiring about is just too enormous to even delineate in one answer. Here are a few more samples to keep us honest.In this Quora answer we find a [math]3-[/math]space configuration that demonstrates the following algebraic identity:[math]a^2(b+c) +b^2(c+a) +c^2(a+b) +2abc = (a+b) (b+c) (c+a) \tag*{}[/math]and in this Quora space (named By::Analogy) we make an attempt to demonstrate various computer science algorithms using geometric and physical models. For example, our currently most recent find shows how to swap the contents of two arrays using rotations of a plane honoring the [math]O(1)[/math] space complexity. Etc.

I dont know about any other countries but I believe that Indian mathematics curriculum is most vast in comparison to other countries. To get admission in any higher educational institute students must be well versed with all the following concepts --Class 11th curriculumUnit-I: Sets and FunctionsChapter 1: SetsSets and their representationsEmpty setFinite and Infinite setsEqual sets. SubsetsSubsets of a set of real numbers especially intervals (with notations)Power setUniversal setVenn diagramsUnion and Intersection of setsDifference of setsComplement of a setProperties of Complement SetsPractical Problems based on setsChapter 2: Relations & FunctionsOrdered pairsCartesian product of setsNumber of elements in the cartesian product of two finite setsCartesian product of the sets of real (up to R × R)Definition of −RelationPictorial diagramsDomainCo-domainRange of a relationFunction as a special kind of relation from one set to anotherPictorial representation of a function, domain, co-domain and range of a functionReal valued functions, domain and range of these functions −ConstantIdentityPolynomialRationalModulusSignumExponentialLogarithmicGreatest integer functions (with their graphs)Sum, difference, product and quotients of functions.Chapter 3: Trigonometric FunctionsPositive and negative anglesMeasuring angles in radians and in degrees and conversion of one into otherDefinition of trigonometric functions with the help of unit circleTruth of the sin2x + cos2x = 1, for all xSigns of trigonometric functionsDomain and range of trigonometric functions and their graphsExpressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applicationIdentities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3xGeneral solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.Unit-II: AlgebraChapter 1: Principle of Mathematical InductionProcess of the proof by induction −Motivating the application of the method by looking at natural numbers as the least inductive subset of real numbersThe principle of mathematical induction and simple applicationsChapter 2: Complex Numbers and Quadratic EquationsNeed for complex numbers, especially √1, to be motivated by inability to solve some of the quadratic equationsAlgebraic properties of complex numbersArgand plane and polar representation of complex numbersStatement of Fundamental Theorem of AlgebraSolution of quadratic equations in the complex number systemSquare root of a complex numberChapter 3: Linear InequalitiesLinear inequalitiesAlgebraic solutions of linear inequalities in one variable and their representation on the number lineGraphical solution of linear inequalities in two variablesGraphical solution of system of linear inequalities in two variablesChapter 4: Permutations and CombinationsFundamental principle of countingFactorial n(n!) Permutations and combinationsDerivation of formulae and their connectionsSimple applications.Chapter 5: Binomial TheoremHistoryStatement and proof of the binomial theorem for positive integral indicesPascal's triangleGeneral and middle term in binomial expansionSimple applicationsChapter 6: Sequence and SeriesSequence and SeriesArithmetic Progression (A.P.)Arithmetic Mean (A.M.)Geometric Progression (G.P.)General term of a G.P.Sum of n terms of a G.P.Arithmetic and Geometric series infinite G.P. and its sumGeometric mean (G.M.)Relation between A.M. and G.M.Unit-III: Coordinate GeometryChapter 1: Straight LinesBrief recall of two dimensional geometries from earlier classesShifting of originSlope of a line and angle between two linesVarious forms of equations of a line −Parallel to axisPoint-slope formSlope-intercept formTwo-point formIntercept formNormal formGeneral equation of a lineEquation of family of lines passing through the point of intersection of two linesDistance of a point from a lineChapter 2: Conic SectionsSections of a cone −CirclesEllipseParabolaHyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section.Standard equations and simple properties of −ParabolaEllipseHyperbolaStandard equation of a circleChapter 3. Introduction to Three–dimensional GeometryCoordinate axes and coordinate planes in three dimensionsCoordinates of a pointDistance between two points and section formulaUnit-IV: CalculusChapter 1: Limits and DerivativesDerivative introduced as rate of change both as that of distance function and geometricallyIntuitive idea of limitLimits of −Polynomials and rational functionsTrigonometric, exponential and logarithmic functionsDefinition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functionsThe derivative of polynomial and trigonometric functionsUnit-V: Mathematical ReasoningChapter 1: Mathematical ReasoningMathematically acceptable statementsConnecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and MathematicsValidating the statements involving the connecting words difference between contradiction, converse and contrapositiveUnit-VI: Statistics and ProbabilityChapter 1: StatisticsMeasures of dispersion −RangeMean deviationVarianceStandard deviation of ungrouped/grouped dataAnalysis of frequency distributions with equal means but different variances.Chapter 2: ProbabilityRandom experiments −OutcomesSample spaces (set representation)Events −Occurrence of events, 'not', 'and' and 'or' eventsExhaustive eventsMutually exclusive eventsAxiomatic (set theoretic) probabilityConnections with the theories of earlier classesProbability of −An eventprobability of 'not', 'and' and 'or' eventsClass 12th curriculumUnit I: Relations and FunctionsChapter 1: Relations and FunctionsTypes of relations −ReflexiveSymmetrictransitive and equivalence relationsOne to one and onto functionscomposite functionsinverse of a functionBinary operationsChapter 2: Inverse Trigonometric FunctionsDefinition, range, domain, principal value branchGraphs of inverse trigonometric functionsElementary properties of inverse trigonometric functionsUnit II: AlgebraChapter 1: MatricesConcept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices.Operation on matrices: Addition and multiplication and multiplication with a scalarSimple properties of addition, multiplication and scalar multiplicationNoncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2)Concept of elementary row and column operationsInvertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).Chapter 2: DeterminantsDeterminant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangleAd joint and inverse of a square matrixConsistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrixUnit III: CalculusChapter 1: Continuity and DifferentiabilityContinuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functionsConcept of exponential and logarithmic functions.Derivatives of logarithmic and exponential functionsLogarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivativesRolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretationChapter 2: Applications of DerivativesApplications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool)Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)Chapter 3: IntegralsIntegration as inverse process of differentiationIntegration of a variety of functions by substitution, by partial fractions and by partsEvaluation of simple integrals of the following types and problems based on them$\int \frac{dx}{x^2\pm {a^2}'}$, $\int \frac{dx}{\sqrt{x^2\pm {a^2}'}}$, $\int \frac{dx}{\sqrt{a^2-x^2}}$, $\int \frac{dx}{ax^2+bx+c} \int \frac{dx}{\sqrt{ax^2+bx+c}}$$\int \frac{px+q}{ax^2+bx+c}dx$, $\int \frac{px+q}{\sqrt{ax^2+bx+c}}dx$, $\int \sqrt{a^2\pm x^2}dx$, $\int \sqrt{x^2-a^2}dx$$\int \sqrt{ax^2+bx+c}dx$, $\int \left ( px+q \right )\sqrt{ax^2+bx+c}dx$Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof)Basic properties of definite integrals and evaluation of definite integralsChapter 4: Applications of the IntegralsApplications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only)Area between any of the two above said curves (the region should be clearly identifiable)Chapter 5: Differential EquationsDefinition, order and degree, general and particular solutions of a differential equationFormation of differential equation whose general solution is givenSolution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degreeSolutions of linear differential equation of the type −dy/dx + py = q, where p and q are functions of x or constantsdx/dy + px = q, where p and q are functions of y or constantsUnit IV: Vectors and Three-Dimensional GeometryChapter 1: VectorsVectors and scalars, magnitude and direction of a vectorDirection cosines and direction ratios of a vectorTypes of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratioDefinition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectorsChapter 2: Three - dimensional GeometryDirection cosines and direction ratios of a line joining two pointsCartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two linesCartesian and vector equation of a planeAngle between −Two linesTwo planesA line and a planeDistance of a point from a planeUnit V: Linear ProgrammingChapter 1: Linear ProgrammingIntroductionRelated terminology such as −ConstraintsObjective functionOptimizationDifferent types of linear programming (L.P.) ProblemsMathematical formulation of L.P. ProblemsGraphical method of solution for problems in two variablesFeasible and infeasible regions (bounded and unbounded)Feasible and infeasible solutionsOptimal feasible solutions (up to three non-trivial constraints)Unit VI: ProbabilityChapter 1: ProbabilityConditional probabilityMultiplication theorem on probabilityIndependent events, total probabilityBaye's theoremRandom variable and its probability distributionMean and variance of random variableRepeated independent (Bernoulli) trials and Binomial distributionSo, in my opinion the Indian mathematics would be the hardest curriculum in the world.