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# Geometry Theorems List

Geometry Theorems List:

Here are the List of Geometry Theorems , given below

1. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
2. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other
3. If two angles form a linear pair, then they are supplementary angles.-Supplementary theorem.
4. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
5. If two sides of two adjacent acute angles are perpendicular,then the angles are complementary.
6. There is  no common sides of two adjacent angles, then the angles are called complementary angles. - Complementary angles:
7. If two lines are perpendicular, then they intersect to form four right angles.
8. f two lines are parallel to the same line, then they are parallel to each other
9. f two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.
10. Alternate Interior Angles:If two parallel lines are cut by a transversal, then the pairs of  alternate interior angles are congruent.
11. Consecutive Interior Angles: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary
12. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel
13. The sum of the measures of the interior angles of a triangle is180o -  Triangle Sum Theorem
14. The acute angles of a right triangle are complementary. -Corollary Theorem
15. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles-Exterior angle Theorem
16. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.-third angle theorem
17. If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. -Angle Bisector Theorem
18. f a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.- Perpendicular Bisector Theorem.
19. If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. - Angle Angle Side Congruence Theorem
20. If two sides of a triangle are congruent, then the angles opposite them are congruent Corollary: If a triangle is equilateral, then it is equiangular. -Base Angle Theorem
21. If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent. - Right angle Theorem
22. If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
23. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle
24. If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.- Hinge Theorem.

26. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
27. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram.
28. If a quadrilateral is a parallelogram, then its opposite angles are congruent.
29. If a quadrilateral is a parallelogram, then its opposite sides are congruent.
30. The sum of the measures of the interior angles of a quadrilateral is 360º.
31. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
32. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
33. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
34. If a quadrilateral is a parallelogram, then its diagonals bisect each other.
35. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. - Mid Segment Theorem
36. The mid segment of a trapezoid is parallel to each base, and its length isone half the sum of the lengths of the bases.

From Wikipedia

Algebraic geometry and analytic geometry

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Background Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere. Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Riemann's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field. The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic. This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. Chow's theorem Chow's theorem, proved by W. L. Chow. is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry. Serre's GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was GÃ©ometrie AlgÃ©brique et GÃ©omÃ©trie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves. Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings. Formal statement of GAGA Let (X,\mathcal O_X) be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map Î»X: Xan â†’ X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). Suppose Ï†: X â†’ Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map Ï†an: Xan â†’ Yan such Î»Y Â° Ï†an = Ï† Â° Î»Y. There is a sheaf \mathcal O_X^{an} on Xan such that (X^{an}, \mathcal O_X^{an}) is a ringed space and Î»X: Xan â†’ X becomes a map of ringed spaces. The space (X^{an}, \mathcal O_X^{an}) is called the "analytifiction" of (X,\mathcal O_X) and is an analytic space. For every Ï†: X â†’ Y the map Ï†an defined above is a mapping of analytic spaces. Furthermore, the map Ï† â†¦ Ï†an maps open immersions into open immersions. If X = C[x1,...,xn] then Xan = Cn and \mathcal O_X^{an}(U) for every polydisc U is a suitable quotient of the space of holomorphic functions on U. For every sheaf \mathcal F on X (called algebraic sheaf) there is a sheaf \mathcal F^{an} on Xan (called analytic sheaf) and a map of sheaves of \mathcal O_X -modules \lambda_X^*: \mathcal F\rightarrow (\lambda_X)_* \mathcal F^{an} . The sheaf \mathcal F^{an} is defined as \lambda_X^{-1} \mathcal F \otimes_{\lambda_X^{-1} \mathcal O_X} \mathcal O_X^{an} . The correspondence \mathcal F \mapsto \mathcal F^{an} defines an exact functor from the category of sheaves over (X, \mathcal O_X) to the category of sheaves of (X^{an}, \mathcal O_X^{an}) . The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) If f: X â†’ Y is an arbitrary morphism of schemes of finite type over C and \mathcal F is coherent then the natural map (f_* \mathcal F)^{an}\rightarrow f_*^{an} \mathcal F^{an} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves (R^i f_* \mathcal F)^{an} \cong R^i f_*^{an} \mathcal F^{an} in this case. Now assume that Xan is hausdorff and compact. If \mathcal F, \mathcal G are two coherent algebraic sheaves on (X, \mathcal O_X) and if f: \mathcal F^{an} \rightarrow \mathcal G^{an} is a map of sheaves of \mathcal O_X^{an} modules then there exists a unique map of sheaves of \mathcal O_X modules \varphi: \mathcal F\rightarrow \mathcal G with f = Ï†an. If \mathcal R is a coherent analytic sheaf of \mathcal O_X^{an} modules over Xan then there exists a coherent algebraic sheaf \mathcal F of \mathcal O_X -modules and an isomorphism \mathcal F^{an} \cong \mathcal R . Moishezon manifolds A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a KÃ¤hler metric.

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:

a^2 + b^2 = c^2\!\,

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally by Descartes in his work La GÃ©omÃ©trie, and extending today into other branches of mathematics.

The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including extension to many-dimensional Euclidean spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.

The Pythagorean theorem is named after the GreekmathematicianPythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. (There is much evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they fitted it into a mathematical framework.) "[To the Egyptians and Babylonians] mathematics provided practical tools in the form of 'recipes' designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right."

The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. Popular references to Pythagoras' theorem in literature, plays, musicals, songs, stamps and cartoons abound.

## Other forms

As pointed out in the introduction, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, Pythagoras' theorem can be expressed as the Pythagorean equation:

a^2 + b^2 = c^2\,

or, solved for c:

c = \sqrt{a^2 + b^2}. \,

If c is known, and the length of one of the legs must be found, the following equations can be used:

b = \sqrt{c^2 - a^2}. \,

or

a = \sqrt{c^2 - b^2}. \,

The Pythagorean equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle, the law of cosines reduces to the Pythagorean equation.

## Proofs

This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.

### Proof using similar triangles

This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as Î¸ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:

\frac{a}{c}=\frac{e}{a} \mbox{ and } \frac{b}{c}=\frac{d}{b}.\,

The first result equates the cosine of each angle Î¸ and the second result equates the sines.

These ratios can be written as:

a^2=c\times e \mbox{ and }b^2=c\times d. \,

Summing these two equalities, we obtain

a^2+b^2=c\times e+c\times d=c\times(d+e)=c^2 ,\,\!

which, tidying up, is the Pythagorean theorem:

a^2+b^2=c^2 \ .\,\!

This is a metric proof in the sense of Dantzig, one that depends on lengths, not areas. The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development

Question:Note that this is NOT CHEATING. My teacher doesn't exactly explain everything so well so I need this for my notes. I need every postulate and theorem up to section 3.3 in the McDougal Littel Geometry textbook.

Answers:Maybe I'm missing something here. If I were wondering "where can I find a listing of every postulate and theorem up to section 3.3 in the McDougal Littel Geometry textbook", I would think the best place to look would be in the pages of the McDougal Littel Geometry textbook! I don't have a copy, but you do! Different geometry books might develop the subject in different orders. A postulate in one book might be a theorem in another, and vice versa. You really need to stick to the development of the material as presented in your book. I am sure the postulates and theorems are all well marked and set off in boxes or colored fonts. Just go through the book and copy them down! The exercise of going through your book this way will be a great way to review and possibly to help clarify what your teacher doesn't explain.

Question:can anyone help me reword these theorems? to make them sound more understandable? theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent. theorem 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid theorem 6.16 A trapezoid is isosceles if and only if its diagonals are congruent. theorem 6.18 If a quadrilateral is a kite, then is diagonals are perpendicular. theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. oh wow thanks a lot wow thank you, sv!!!

Answers:theorem 6.14 Base angles of every isosceles trapezoid are congruent. theorem 6.15 Base angles are congruent in every isosceles trapezoid. theorem 6.16 Diagonals are congruent in every isosceles trapezoid. theorem 6.18 Diagonals are mutually perpendicular in every kite. theorem 6.19 One pair of opposite angles are congruent in every kite.

Question:activities involving proving of theorems in geometry..need help!!asap..pls...

Answers:Hi! I saw that no one else had tried to answer your question, but I'm going to try for your 10 pts... If you're looking for what the basic theorems actually are, then they are listed here: http://library.thinkquest.org/2647/geometry/intro/p&t.htm But if you're asking how you'd typically go about proving theorems in general, then that's more complex. So, in general, always start with one or more Postulates which point to facts from which you can deduce Conclusions. Refering back to other Postulates, you can then build up a step by step set fo Conclusions which will be the Proof of the Theorem. You can use two kinds of proving: Direct and Indirect. In direct, you prove step by step until you come up with the desired statement. On the other hand, indirect proving is when you prove the opposite of what you want to prove. Understanding Proofs: (Very well done) http://library.thinkquest.org/2647/geometry/intro/proof.htm

Question:Please give a brief (yet concise) explanation/definition of the following postulates and theorems: 1)Segment Addition Postulate 2)Angle Addition Postulate 3)Linear Pair Postulate 4)Corresponding Angle Postulate 5)Alternate Interior Angle Theorem 6)Alternate Exterior Angle Theorem 7)Consecutive Interior Angle Theorem 8)Corresponding Angle Converse 9)Alternate Interior Angle Converse 10)Alternate Exterior Angle Converse 11)Consecutive Interior Angle Converse 12)Congruent Supplements Theorem 13)Congruent Complements Theorem 14)Vertical Angle Theorem 15)Right Angle Theorem 16)Definition of supplementary angles 17)Definition of complementary angles 18)Definition of right angle 19)Definition of perpendicular lines 20)Definition of congruent angles 21)Definition of congruent segments 22)Exterior Angle Theorem 23)Corollary to the Triangle Sum Theorem 24)Third Angle Theorem 25)SSS Postulate 26)SAS Postulate 27)ASA Postulate 28)AAS Theorem 29)HL Theorem 30)Base Angle Theorem Err, yeah... >_>" It's a lot, isn't it?

Answers:Not to be a snot, but seriously, just crack open any geometry text book and you'll get the answers you need.