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.
Video: Definition of a Parallelogram
The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
Special cases
Quadrilaterals by symmetry
Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles
Rectangle – A parallelogram with four angles of equal size.
Rhombus – A parallelogram with four sides of equal length.
Square – A parallelogram with four sides of equal length and angles of equal size (right angles).
Video: What is a Parallelogram and what are its Special Cases?
Characterizations
A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:
Two pairs of opposite sides are equal in length.
Two pairs of opposite angles are equal in measure.
The diagonals bisect each other.
One pair of opposite sides is parallel and equal in length.
Adjacent angles are supplementary.
Each diagonal divides the quadrilateral into two congruent triangles.
The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.)
It has rotational symmetry of order 2.
The sum of the distances from any interior point to the sides is independent of the location of the point. (This is an extension of Viviani's theorem.)
There is a point X in the plane of the quadrilateral with the property that every straight line through X divides the quadrilateral into two regions of equal area.
Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.
In geometry, an equilateral triangle is a triangle in which all three sides are equal.
In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.
Video: What is an equilateral triangle
Principal properties
An equilateral triangle. It has equal sides (a=b=c), equal angles (), and equal altitudes (ha=hb=hc).
Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that:
The area is
The perimeter is
The radius of the circumscribed circle is
The radius of the inscribed circle is or
The geometric center of the triangle is the center of the circumscribed and inscribed circles
And the altitude (height) from any side is .
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is
The height of the center from each side, or apothem, is
The radius of the circle circumscribing the three vertices is
The radius of the inscribed circle is
In an equilateral triangle, the altitudes, the angle bisectors, the ......
A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).
The relation between the sides and angles of a right triangle is the basis for trigonometry.
The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B.
If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.
Video: Pythagorean theorem 3 | Right triangles and trigonometry
Principal properties
Area
As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area T is
where a and b are the legs of the triangle.
Video: How to find the area of a right angled triangle
Altitudes
Altitude of a right triangle
If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this:
The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse.
Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
Video: How to Find the Altitude of a Right Triangle
In equations,
(this is sometimes known as the right triangle ......
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
Video: Types of angles
Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.
Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
Types of angles
Individual angles
Right angle.
Reflex angle.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
An angle equal to 1/2 turn (180° or π radians) is called a straight angle.
Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, or a perigon.
Angles that are not right angles or a multiple of a right angle are called oblique angles.
The names, intervals, and measured units are shown in a table below:
Name
acute
right angle
obtuse
straight
reflex
perigon
Units
Interval
Turns
(0, 1/4)
1/4
(1/4, 1/2)
1/2
(1/2, 1)
1
Radians
(0, 1/2π)
1/2π
(1/2π, π)
π
(π, 2π)
2π
Degrees
(0, 90)°
90°
(90, 180)°
180°
(180, 360)°
360°
Gons
(0, 100)g
100g
(100, 200)g
200g
(200, 400)g
400g
Vertical and adjacent angle pairs
Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.
A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as ,,,,,,
In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid.
Video: What is an Isosceles Trapezoid?
Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).
Special cases
Special cases of isosceles trapezoids
Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them.
Another special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid or a trisosceles trapezoid. They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.
Video: Quadrilaterals - Trapezoids, Parallelograms, Rectangles, Squares, and Rhombuses
Self-intersections
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length.
Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.
Convex isosceles trapezoid
Crossed isosceles trapezoid
antiparallelogram
Characterizations
If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid (nor, under the definitions given in Wikipedia, is it sufficient, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides); any of the following properties also distinguishes an isosceles trapezoid from other trapezoids:
The diagonals have the same length.
The base angles have the same measure.
The segment that joins the midpoints of the parallel sides is ....
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how "elongated" it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure below). Ellipses have many similarities with the other two forms of conic sections: parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.
An ellipse (red) obtained as the intersection of a cone with an inclined plane
Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is the above-mentioned eccentricity of the ellipse.
An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics.
Video: Equation of an Ellipse, Deriving the formula
Definition of an ellipse as locus of points
Ellipse: Definition
Ellipse: definition with director circle
An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
An ellipse can be defined using two fixed points, , , called the foci and a distance, usually denoted . The ellipse defined with , and is the set of points such that the sum of the distances is constant and equal to . In order to omit the special case of a line segment, one assumeshttps://wikimedia.org/api/rest_v1/media/math/render/svg/8f096d24464c8357..."> More formally, for a given , an ellipse is the set
The midpoint of the line segment joining the foci is called the .....
In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points.
Video: Parallel Lines, Transversals, and Angles
Transversals play a role in establishing whether two other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, corresponding angles, and alternate angles. By Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.
Eight angles of a transversal. (Vertical angles such as and are always congruent.)
Transversal between non-parallel lines. Consecutive angles are not supplementary.
Transversal between parallel lines. Consecutive angles are supplementary.
Angles of a transversal
A transversal produces 8 angles, as shown in the graph at the above left:
4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and
4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior, namely α1, β1, γ and δ.
A transversal that cuts two parallel lines at right angles is called a perpendicular transversal. In this case, all 8 angles are right angles
When the lines are parallel, a case that is often considered, a transversal produces several congruent and several supplementary angles. Some of these angle pairs have specific names and are discussed below:corresponding angles, alternate angles, and consecutive angles.
Video: Figuring out angles between transversal and parallel lines
Corresponding angles
One pair of corresponding angles. With parallel lines, they are congruent.
), and equal altitudes (ha=hb=hc).
or 
.
(this is sometimes known as the right triangle ......
, 
, called the foci and a distance, usually denoted 
. The ellipse defined with 
such that the sum of the distances 
is constant and equal to 
, an ellipse is the set 
of the line segment joining the foci is called the .....
and 
are always congruent.)
