Which of the following theorems states that "2 triangles with corresponding angles are equal'?
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b. SAS\n\nSide-Angle-Side... two triangles with congruent angles with SAS relationship are equal triangles.
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Aha, geometry!\n\nDraw a diagram and follow the steps below.\n\nThe trick is to show that the triangles DBE and \ntriangle DAF are similar.\n\nProof That triangles DBE and triangle DAF are similar.\n\nLet <ABC> donote angle ABC.\nWe need to show any two corresponding angles of these traingles are equal.\n\nFirst angle,\n<FDA> = <BDE>  because D,B,F collinear and D,E,A collinear.\n\nSecond angle,\n<DAF> = <DAC> because A,F,C collinear\n      = (pi) - <ACD> - <ADC> angles in triangle ACD\n      = (pi) - <ACD> - <ADC>  ...(1)\n\nNow, separately  manipulate,<ACD> and <ADC>.\n\n<ACD> = <ACB> + <BCD> splitting <ACD> with line CB\n      = <ACB> + <CAB>, tangent property <BCD> = <CAB> \n      = (pi) - <CBA>  angles in triangle ABC\n      = <ABE>  because C,B,E are all colinear\n\nNow, similarly\n\n<ADC> = <ADB> + <BDC> splitting <ADC> with line DB\n      = <ADB> + <BAD>, tangent property <BDC> = <BAD> \n      = (pi) - <ABD>  angles in triangle ABD\n         \nSubstituiting these two info into (1), we get,  \n\n<DAF> = (pi) - <ACD> - <ADC>\n      = (pi) - <ABE> - [ (pi) - <ABD> ]\n      = <ABD> - <ABE>\n      = <BDE> because BE splits <ABD> \n\nDone. In showing the triangle DAF and triangle DBE are similar.\n\nHence, by ratio for similar triangles,\n\n        DA/DF = DB/DE\n\nOr Rewriting we get\n \n      DA * DE = DB * DF\n\nProved.\n\nNot so difficult, was it. hehehe.\n\nRegards
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Thales understood similar triangles and right triangles, and what is more, used that knowledge in practical ways. The story is told in DL (loc. cit.) that he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid’s shadow measured from the center of the pyramid at that moment must have been equal to its height.
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All congruent figures are similar. Congruent means they have the same exact shape, similar means they have the same shape (same angle measures), but different sizes (side lengths). So, if you say two triangles are congruent, then they are also similar.
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Do you mean the Pythagorean theorem?\n\nIn any right triangle, the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.
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alternative angles you mean? It's kinda like the corners of the letter Z. The angles are equal. The bottom and top line is parallel. And both angles are at the opposite sides of the line.
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D.  The angles have to add up to 180.  If 2 of the angles are the same, then the 3rd has the same.  You have to get an enclosed side to be the same for the triangles to be congruent.
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An isosceles triangle is a triangle with (at least) two equal sides and two equal angles.\n\nhttp://mathworld.wolfram.com/IsoscelesTriangle.html
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Pythagorean Theorem\n \nLet's build up squares on the sides of a right triangle. Pythagoras' Theorem then claims that the sum of (the areas of) two small squares equals (the area of) the big one.\n\nIn algebraic terms, a2 + b2 = c2 where c is the hypotenuse while a and b are the legs of the triangle.\n\nThe theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.\n\nBelow is a collection of 69 approaches to proving the theorem. Some of the proofs are accompanied by Java illustrations, but most have been written in plain HTML.\n\nRemark\nThe statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that a triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact.\n\nW. Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968.\n\nIn all likelihood, Loomis drew inspiration from a series of short articles in The American Mathematical Monthly published by B. F. Yanney and J. A. Calderhead in 1896-1899. Counting possible variations in calculations derived from the same geometric configurations, the potential number of proofs there grew into thousands. For example, the authors counted 45 proofs based on the diagram of proof #6 and virtually as many based on the diagram of #19 below. I'll give an example of their approach in proof #56. (In all, there were 100 "shorthand" proofs.)\n\nPythagorean Theorem generalizes to spaces of higher dimensions. Some of the generalizations are far from obvious.\n\nLarry Hoehn came up with a plane generalization which is related to the law of cosines but is shorter and looks nicer.\n\nThe Theorem whose formulation leads to the notion of Euclidean distance and Euclidean and Hilbert spaces, plays an important role in Mathematics as a whole. I began collecting math facts whose proof may be based on the Pythagorean Theorem.\n\nWherever all three sides of a right triangle are integers, their lengths form a Pythagorean triple (or Pythagorean numbers). There is a general formula for obtaining all such numbers.\n\nMy first math droodle was also related to the Pythagorean theorem. Unlike a proof without words, a droodle may suggest a statement, not just a proof.\n\nSeveral false proofs of the theorem have also been published. I shall collect some in a separate page. It is better to learn from mistakes of others than to commit one's own.\n\nIt is known that the Pythagorean Theorem is Equivalent to Parallel Postulate.\n\nThe Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular. Besides the statement of the Pythagorean theorem, Bride's chair has many interesting properties, many quite elementary.\n\nProfessor Edsger W. Dijkstra found an absolutely stunning generalization of the Pythagorean theorem. If, in a triangle, angles a, b, g lie opposite the sides of length a, b, c, then\n\n(EWD) sign(a + b - g) = sign(a2 + b2 - c2),  \n\nwhere sign(t) is the signum function:\n\n  sign(t) = -1, for t < 0, \nsign(0) =  0, \nsign(t) =  1, for t > 0. \n \n\nThe theorem this page is devoted to is trea
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equal number of sides and equal measure of angles\n\n\nor at least thats what i remember from geometry!
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in a right triangle (any triangle with a 90deg angle), the hypotenuse is opposite the 90 angle.  The other two are proportional:  the smaller side is opposit the smaller angle, the (next) larger side is opposite the (next) larger angle.\n\nThe sum of the squares of both sides will equal the square of the hypotenuse.\n\nFor any integer n which is greater than 2, then a^n + b^n = c^n has no solutions in non-zero integers a, b, and c\n\nsin of an angle = opposite side/hypotenuse\ncos of an angle = adjacent side/hypotenuse\ntan of an angle = opposite side/adjacent side\n\nSOHCAHTOA\n\nthat's all you ever need to know to get through basic trigonometry.
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A _______________ B\n... &#92;.................. ...............&#92;\n..... &#92;.................. ...............&#92;\n....... &#92;_______________&#92;\n...... D.................. ...............C\n\nLet ABCD be a quadrilateral with AB || DC and AB = DC\n\nDraw diagonal AC\n\nNow in Δs ABC, CDA\nAB = DC (Data)\n<BAC = <DCA ( Equal alternate angles, AB||DC)\nAC is common\nTherefore ΔABC ≡ ΔCDA (SAS property)\n\nTherefore < BCA = < DAC (Corresponding angles of congruent triangles are equal)\n\nTherefore BC||AD (alternate angles are equal)\n\nSo ABCD is a parallelogram as opposite sides are parallel
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It is possible only when the given triangle is isosceles or equilateral. It's also possible for right-angled triangles but only when the given angle isn't 90 degrees.
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The previous answers are either wrong/incomplete/mischievious.\nFirstly there is nothing called parallel angles! You probably meant to ask what happens if the angles of two triangles are same. In that case there is no congruence/similarity between the two triangles.\nBut if two angles are equal betwen them, they are SIMILAR (same shape) and not congruent.\nIf in addition they have a side equal, they become congruent!
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When two angles are supplementary, their sum is 180 degrees.  For example:\n\na + b = c\n100 + 80 = 180, therefore a and b are supplementary.\n\nIf a + b = 90, the angles are complementary.
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http://en.wikipedia.org/wiki/Pythagorean_theorem#Proofs\n\nCheck below for the "proof of converse".  It's the most straightforward there is.\n\nBasically, you try to construct a triangle with x,y, and z, only to discover that it is congruent to a right triangle!
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If the sides are parallel then these angles will be equal.