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.
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
Video: Learning about Area
Area formulas
Polygon formulas
Video: Perimeter and area of a non-standard polygon | Perimeter, area, and volume
For a non-self-intersecting (simple) polygon, the Cartesian coordinates (i=0, 1, ..., n-1) of whose n vertices are known, the area is given by the surveyor's formula:
where when i=n-1, then i+1 is expressed as modulus n and so refers to 0.
Rectangles
The area of this rectangle is lw.
The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length l and width w, the formula for the area is:
A = lw (rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula:
A = s2 (square).
Video: Area of a Rectangle
The formula for the area of a rectangle follows directly from the ........
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
Video: Volume
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
Units
Volume measurements from the 1914 The New Student's Reference Work.
Approximate conversion to metric (mL)
Imp.
U.S.
Liquid
Dry
Gill
142
118
138
Pint
568
473
551
Quart
1137
946
1101
Gallon
4546
3785
4405
Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm3) is the volume of a cube whose sides are one centimetre (1 cm) in length.
In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus
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 ......
Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.
A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
Video: Probability for Beginners : Solving Math Problems
What are the different types of probability?
There are two types of probability, namely Theoretical and Experimental probability.
Theoretical probability = Number of favorable outcomes / Total number of outcomes
Experimental probability = Number of favorable outcomes / Total number of trials
Video: Types of probability
Define sample space, power set and events
Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a dice can produce six possible results. One collection of possible results gives an odd number on the dice. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the dice falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.
Video: Experiment, Outcome, Sample space, and event
How do you assign probability to an event?
A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.
Video: Simple probability
How to denote the probability of an event?
The probability of an event A is written as .......
Let's Review
____ is the measure of the likelihood that an event will occur.
Probability is quantified as a number between ____ and ____.
There are two types of probability, namely ____ and ____ probability.
Theoretical probability = Number of favorable outcomes / Total number of ____.
Experimental probability = Number of favorable outcomes / Total number of ____.
The collection of all possible results is called the ____ ____ of the experiment.
The ____ ____ of the sample space is formed by considering all different collections of possible results.
The probability of an event A is written as ____ , ____ , or ____ .
If two events A and B occur on a single performance of an experiment, this is called the ____ or ____ ____ of A and B, denoted as
If two events, A and B are independent then the ____ probability is for example, if two coins are flipped the chance of both being heads is.
If two events are mutually exclusive then the probability of ____ occurring is denoted as .
If two events are mutually exclusive then the probability of ____ occurring is denoted as . For example, the chance of rolling a 1 or 2 on a six-sided die is
Answer
Probability is the measure of the likelihood that an event will occur.
Probability is quantified as a number between 0 and 1.
There are two types of probability, namely Theoretical and Experimental probability.
Theoretical probability = Number of favorable outcomes / Total number of outcomes.
Experimental probability = Number of favorable outcomes / Total number of trials.
The collection of all possible results is called the sample space of the experiment.
The power set of the sample space is formed by considering all different collections of possible results.
The probability of an event A is written as , , or .
If two events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as
If two events, A and B are independent then the joint probability is for example, if two coins are flipped the chance of both being heads is.
If two events are mutually exclusive then the probability of both occurring is denoted as . .
If two events are mutually exclusive then the probability of either occurring is denoted as . For example, the chance of rolling a 1 or 2 on a six-sided die is
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 geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry.
The cube is the only convex polyhedron whose faces are all squares. Without the convex restriction, there's one more such figure, which is made out of 7 cubes; it can be formed by putting one cube in the center and attaching a congruent cube to each of its faces.
(i=0, 1, ..., n-1) of whose n vertices are known, the area is given by the surveyor's formula:
Volume measurements from the 1914 The New Student's Reference Work.
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(this is sometimes known as the right triangle ......
for example, if two coins are flipped the chance of both being heads is
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. For example, the chance of rolling a 1 or 2 on a six-sided die is 
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