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In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance.
Parallel is a term in geometry and in everyday life that refers to a property of lines or planes. Parallel lines or planes are next to each other, but never touch each other. This means they never intersect at any point.
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
Parallel: is parallel to If A||B then line A will never touch line B, thus both lines are rotated in the same angle. x||(x+1)
- Etymology and Trapezium Versus Trapezoid
- Inclusive Versus Exclusive Definition
- Special Cases
- Condition of Existence
- Characterizations
- Midsegment and Height
- Area
- Diagonals
- Other Properties
- Applications
The ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (trapezia literally 'table', itself from τετράς (tetrás) 'four' + πέζα (péza) 'foot; end, border, edg...
There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Some sources use the term proper trapezoid to describe trapezoids under the exclu...
A right trapezoid (also called right-angled trapezoid) has two adjacent right angles. Right trapezoids are used in the trapezoidal rulefor estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer baseedge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base. An isosceles trapez...
Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram trapezoid with a and bparallel only when 1. | d − c | < | b − a | < d + c . {\displaystyle \displaystyle |d-c|<|b-a|
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: 1. It has two adjacent angles that are supplementary, that is, they add up to 180 degrees. 2. The angle between a side and a diagonalis equal to the angle between the opposite side and the same diagonal. 3. The diagonals cu...
The midsegment (also called the median or midline) of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and bof the trapezoid, 1. m = a + b 2 . {\displaystyle m={\frac {a+b}{2}}.} The midsegment of a trapezoid is one of the two bimedians(t...
The area Kof a trapezoid is given by 1. K = a + b 2 ⋅ h = m h {\displaystyle K={\frac {a+b}{2}}\cdot h=mh} where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a great mathematician-astronomer...
The lengths of the diagonals are 1. p = a b 2 − a 2 b − a c 2 + b d 2 b − a , {\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},} 2. q = a b 2 − a 2 b − a d 2 + b c 2 b − a {\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}} where a is the short base, b is the long base, and c and dare the trapezoid legs. If the t...
The center of area (center of mass for a uniform lamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side bgiven by 1. x = h 3 ( 2 a + b a + b ) . {\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).} The center of area divides this segment in the ratio (when tak...
Architecture
In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids. This was the standard style for the doors and windows of the Inca.
Geometry
The crossed ladders problemis the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.
Biology
In morphology, taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziformcommonly are useful in descriptions of particular organs or forms.
A parallelogram is a polygon with four sides (a quadrilateral). It has two pairs of parallel sides (line segments which never meet if the lines were allowed to extend beyond their end points). The opposite sides of a parallelogram have the same length (they are equally long).
The symbol for parallel lines is \(\parallel,\) so we can say that \(\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}\) in that figure. According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often.
