Conditional Statements
Geometric proofs are usually stated in a shorthand that emphasizes rules and relationships.  They use conditional statements, definitions, and postulates in a way that defines the connection between the hypothesis (the “if” portion) and its consequences (the “then” portion).  For example, the statement, “If a figure is a square, then it must be a polygon” is only true if the given figure is both a square and that a square is a type of polygon.

Instances and Counterexamples
While one instance, or even thousands of instances, of a statement being true does not necessarily prove it, only one counterexample will disprove it.  Suppose the example were turned around such that the conditional statement read “If a figure is a polygon, then it must be a square.”  That statement can be disproven with one counterexample.  A triangle is a polygon, but it is not a square.

Definitions and Postulates
Before a conditional statement is made, definitions are given.  For example, the figure is shown to have four sides of equal length that are perpendicular.  In addition, polygons are defined as geometric figures that have three or more sides.  If the reader didn’t know that the figure had four sides (the “if” portion), or what the definition of a polygon (the “then” portion, there wouldn’t be a way to connect the parts of the statement.  Therefore, there wouldn’t be a way to find instances or counterexamples.  Without a definition of the figure, the statement might be true or it might be false.  What if the figure really had three sides and angles whose sum equaled 180⁰, but no one knew it?

Computer Programs
Conditional statements are essential in computer programs.  They state the parameters of the problem in such a way that the computer can reach a conclusion.   Suppose a computer program is set up to calculate the number of diagonals in any polygon that has more than three sides.  To define what the antecedent is, programmers will set up both parts of the conditional statement.  If N ≥ 3, then solve N∙(N-3)/2.

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What Are Parallel Lines?
Parallel lines do not intersect at any point.  If they are in the same plane, parallel lines have the same tilt.  When they are graphed, both lines have the same slope.  If a transversal line crosses both parallel lines, the corresponding angles have the same measurement.  This property of parallel lines allows students to deduce the measurement of one angle formed by one of the parallel lines and the transversal, if the measure of a corresponding angle is known.

Parallel Lines Postulate
As an example, in the figure below, lines a and b are parallel, and line t is a transversal cutting them.  Suppose angle β measures 110⁰ where the transverse line t cuts through line a.  Its corresponding angle where the transverse line t cuts through line b will also measure 110⁰.  The measure of angle β and angle θ is 180⁰, so angle θ (theta) measures 180-110 or 70⁰.

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What  Are Perpendicular Lines?
Perpendicular lines exist in a special relationship to one another, as they intersect at a 90⁰ angle.  They do not have to be horizontal or vertical, but just intersect at right angles.  By extension of the Parallel Lines Theorem, if two lines are perpendicular to the same line, they are parallel to each other.  If one angle measures 90⁰, all the other angles (adjacent, vertical, and corresponding) that are formed by a transverse line cutting another line will also measure 90⁰.  Suppose angle β in the figure above measures 90⁰.  Angle θ also measures 90⁰.  This application is also called the Perpendicular to Parallels Theorem.

Construction of Parallel and Perpendicular Lines
Geometric properties of normal two-dimensional space ensure that parallel and perpendicular lines are readily constructed using nothing but an unmarked straightedge and compass.  By using the endpoints of a line segment, students can construct another line perpendicular to that segment, and then construct a parallel line with the same slope as the first line. Students can use the process to demonstrate how the theorems work, by creating the transverse perpendicular to the first line, then constructing another line parallel to it.

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Translations
Translations are one type of transformation that occur when a figure slides to a different place in the coordinate plane.  The preimage and image are the same size and shape and all the points correspond to each other.  The relationship between the preimage and image can be measured by direction and magnitude.  In order to measure magnitude, measure the distance between a point in the preimage with its corresponding point in the image.  In order to measure direction, measure the ray between a point in the preimage with its corresponding point in the image.

Rotations
Rotations are the type of transformation that occur when the preimage and the image turn around a central point.  The preimage and image are the same size and shape also, and all the points correspond to each other.  However, the preimage and image are at different orientations around a central point.  The magnitude of the rotation is the angle between the central point, a point on the preimage, and its corresponding point on the image.  (Cosine and sine functions can be expressed in terms of rotations.)

Complex Transformations
With translations and rotations the preimage and its resulting image have the same size and shape, but there are more complex transformations that do not necessarily have the same size or exactly the same shape.  Some of these transformations occur when the figure in the preimage is split into parts in the image, or when the preimage is distorted so that the image does not have the same shape.   For example, the representation of an area on a map  is not the same shape as  its corresponding area on a globe, although the locations are the map and the globe are the same.

The Problems of Perspective
Geometric transformations are a mathematical expression of the way artists distort figures in perspective in order to represent three-dimensional objects on a two-dimensional flat canvas.  Lines and angles are distorted and repeated in order to fool the eye into seeing distance where the only distances are an optical illusion.  The difference between a two dimensional square and a box drawn to look like it is in three dimensions involve translations , rotations, and complex transformations.

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Definitions
A quadrilateral is a parallelogram if both pairs of its opposite sides are parallel.  A rhombus is a type of quadrilateral with all sides the same length.  A rectangle has 4 right angles, and a square has four equal sides and four right angles.  These definitions exist in a hierarchy of relationships.  For example, every square is also a parallelogram, because both pairs of its opposite sides are equal, but every parallelogram is not a square.  Similarly, every square is also a rhombus, because all four sides are the same length, but every rhombus is not a square.  Likewise, every square is also a rectangle, because a rectangle has 4 right angles, but every rectangle is not a square.

Kites
Kites are a special type of quadrilateral with two distinct pairs of consecutive sides the same length.  Because rhombi and squares also have sides the same length, they are also kites, but the reverse is not true.  Every kite is not a rhombus, because all sides of a kite are not equal.  Similarly, every kite is not a parallelogram, because the opposite sides of a kite are not necessarily parallel.

Trapezoids
Trapezoids are quadrilaterals that have one pair of parallel sides.   The parallel sides are called bases.  If the base angles are equal, the trapezoid is a special type called an isosceles trapezoid.  Rectangles are a special type of isosceles trapezoid with opposite sides parallel and equal angles.

Diagonals
Quadrilaterals have two diagonal lines.  In  a kite, the diagonal joining the ends forms a line of symmetry.  In a rhombus, both diagonals form lines of symmetry.  Lines of symmetry hold powerful properties that help measure angles, area, and relationships between geometric figures.

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Carbohydrates
Carbohydrates are composed of carbon, hydrogen, and oxygen in the same ratio as in water, so that scientists first thought that carbon simply bonded with water.  In other words, carbon was hydrated, later shortened to carbohydrate.  Sugars and starches are common forms of carbohydrates, used for energy by living organisms.  Atoms are arranged so that the carbon atoms are arranged in a long chain and the hydrogen and hydrogen-oxygen pairs are joined to carbon on either side of the chain.  The simplest sugars are found in fruits and honey.  When simple sugar monomers (called monosaccharides) are joined by the thousands in long chains, they produce starches.

Lipids
Lipids also contain  carbon, hydrogen ,and  oxygen, but the molecules are more complex.  They make up fats, oils, and waxes, and do not dissolve in water.  Three carbon molecules joined in a ring form a backbone for the chain.  Fatty acids are joined to the glycerol chain, in complex molecules.  They are used by living organisms to store energy and also form cell membranes.

Proteins
All proteins are made up of amino acids.  They contain carbon, hydrogen, and oxygen, as well as other elements.  Amino acids have a distinct chemical structure, and combine in specific ways to form the different types of proteins.  Long chains of proteins bond in specific patterns depending on their purpose.  For example, the proteins that make up egg white are different from the proteins that make up hemoglobin.  Some proteins are enzymes that speed up chemical reactions, producing even more energy to sustain life.

Nucleic Acids
Nucleic acids are special organic molecules that contain and carry out cellular function.  There are two forms that are produced:  DNA (deoxyribonucleic acid), and RNA (ribonucleic acid).  While DNA contains the instructions for cellular function and passes those instructions along during cellular reproduction, RNA carries them out.  Both DNA and RNA are essential to genes.

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Ionic Chemical Bonds
Ionic bonds are formed by electronic attraction.  The simplest and most common example is the attraction between sodium (chemical name Na), and chlorine (Cl), to form NaCl, sodium chloride, common table salt.  The outermost level of a sodium atom only has one electron, and can easily give it up in a  chemical reaction.  The outermost level of a chlorine atom, by contrast, has 7 electrons, and can easily take an electron from another atom.   When the sodium atom gives up its electron, it is slightly positively charged, and when the chlorine atom gains an electron, it is slightly negatively charged.  The positively charged sodium atom and the negatively charged  chlorine atom are held together in a compound be an ionic bond.

Covalent Chemical Bonds
Covalent chemical bonds are stronger than ionic bonds, because more than one element shares electrons.  The water molecule, or H₂O, is one of the most common examples.  The hydrogen atoms share electrons with the oxygen atoms, and the oxygen atoms also share electrons with the hydrogen atoms, so the bond works both ways.  Methane is another common compound.  Carbon (C) shares electrons with 4 hydrogen atoms (H₄), and the hydrogen atoms share electrons with the carbon atoms in covalent bonds.

Common Elements for Life
Elements such as hydrogen (H), carbon (C), nitrogen (N), and oxygen (O),are the most common elements found in life, making up over 99 % of living things.  Sodium (S), magnesium (Mg), potassium (K), calcium (Ca), phosphorus (P), sulfur (S), and chlorine (Cl) make up less than 0.7%, although they are crucial to certain chemical reactions.    Other trace elements include vanadium (V), chromium (Cr), manganese (Mn), molybdenum (Mo), iron (Fe), cobalt (Co), copper (Cu), zinc (Zn), silicon (Si), tin (Sn), selenium (Se), and iodine (I).

Chemical Reactions
Chemical reactions occur when the bonds between atoms in a compound are broken down, so that new bonds between atoms may be formed.  The most common way is through heat energy.  For example, if sugar is heated past a certain point it will turn black, forming carbon, and release water vapor  and carbon dioxide.  Because most chemical reactions in living organisms use up more energy than they create, living organisms need food for most processes.

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Solutions
One common type of mixture is a solution, in which molecules of one substance are dissolved in another substance, but neither substance changes what it is.  For example, a simple solution of sugar water that is safe to use in hummingbird feeders contains 4 parts water to every part of sugar.  The mixture is stirred and boiled to remove impurities, but neither the sugar or water are changed to form the solution.  In the same way, the oxygen dissolved in water is available for fish to use.

Suspensions
Suspensions are another common mixture found in living things.  Unlike solutions, particles that are mixed with liquid are larger than individual molecules.  For example, pond water that is full of silt looks cloudy, and the silt will eventually fall to the bottom of the pond.  Red and white blood cells are suspended in blood plasma.

Colloids
Colloids are a special type of suspension that combine some of the features of a solution with some of the more typical features of a suspension.  That is because the particles suspended in the solvent are too small to sink to the bottom of the suspension but too large to dissolve and make a solution.  The best example of the colloid in biology is the cell interior.  The structures within the cell remain suspended, as do food particles within the cell.  This allows cells to use food and their interior parts to remain in the same relationship.

Compounds
Compounds differ from mixtures in that a new substance is formed by a chemical reaction.  For example, water is a compound of hydrogen and oxygen.  Two colorless gases combine to form a substance that is necessary to life .  Even though both hydrogen and oxygen are highly flammable in their pure state as separate elements, their compound water is used to put out fires.  Compounds are formed from elements when their electrons are bonded in chemical reactions, and those chemical reactions release the energy needed to support life.

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Cell Theory
By the middle of the 1800′s, three parts of cell theory were in place.  First, all organisms are composed of cells.  Second, cells are the smallest, most basic unit of structure and function in organisms, whether they are one-celled amoeba, or many-celled whales.  Third, all cells come from other cells.  It is characteristic of life that it reproduces itself.  Viruses and such forms may be an exception, but they do not function as life until they invade living cells and reproduce.

Size and Shape of Cells
The smallest of cells are a form of bacteria that can only be seen with an electron microscope, and are 0.1 micrometers or 1 ten-millionth of a meter long.  The largest cell is an ostrich egg, which has a diameter of about 100 millimeters.  Most cells are between 2 and 200 micrometers long, so that most of them can be seem with a regular light microscope.  The shapes of cells vary according to their function.  For example, the cells in the human body vary from the flat, thin cells of the skin and other membranes, to long, branched nerve fibers.  Although different types of cells vary a great deal, they have three basic parts; the cell nucleus, the cell membrane and cell wall, and the cytoplasm.

The Cell Nucleus
The cell nucleus is the control center for most of the living cell’s activities.  For example, proteins are built according to specialized instructions.  The nucleus also contains chromosomes, made up of DNA, that contains the genetic information that is transmitted when each cell reproduces.  Nucleoli within the cell nucleus contain DNA, RNA, and proteins.  The nucleus is separated from the rest of the cell by the nuclear envelope, which has a double membrane.

Cell Membrane and Cell Wall
The cell membrane is the outer boundary of the cell, and it is composed of proteins and lipids.  It gives the cell shape and regulates food that goes in and wastes that go out.  Plants, algae, fungi, and some bacteria also have a more rigid cell wall in addition to the cell membrane.  It allows water and dissolved substances to contact the membrane and also gives the cell its shape.

Cytoplasm
The cytoplasm is everything in the cell that is between the nuclear membrane and the outer cell membrane.  Channels within the cytoplasm transport materials that are formed in the nucleus and other structures within the cell, such as proteins,  energy produced within mitochondria, and nutrients.

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Simple Light Microscopes
The simplest microscope is a magnifying glass that works by focusing light to magnify very small objects.  Scientists may have used them as early as the 15th century, although the  finest lenses were made by van Leeuwenhoek in the late 17th century. He was able to study  and draw organisms in pond water, a very familiar activity in science classes from elementary school on.

Compound Light Microscopes
The earliest compound light microscope had a lens on each end of a long tube.  They worked by using one lens as a magnifying glass, then magnifying that image with another lens.  Scientists tended to prefer single lenses for magnification, because the images from early compound light microscopes were distorted.  Modern compound light microscopes minimize some of the distortion by the way the lenses are arranged.

Magnification
The most common microscopes used in science classes are compound light microscopes.  One lens is near the eyepiece of the microscope, and the other lens is positioned near the specimen slide.  The light source is under the slide, so that light actually travels through the slide and specimen and then is focused through the lenses.  The increase in the specimen’s size is measured by magnification.  The power of the lenses is multiplied as they work together, so an ocular lens in the eyepiece that magnifies 10 times and an objective lens that magnifies  the size of the object 43 times will magnify the entire specimen 43 X 10, or 430 times its size.  At higher magnification than 2000 times, images become too distorted and fuzzy and lose resolution.

Preparing Specimens
Many specimens must be specially prepared for viewing under a microscope.  Tiny slices of matter must be thin and small enough to fit on the microscope slide.  A microtome is a tool that cuts very thin slices from specimen material.  In addition, specimens must often be stained in order to show up details.  Common stains color red or blue, and special dyes have been developed that do not kill or distort cell structures.  Specimens for the electron microscope must be embedded in plastic, cut even thinner, and coated with a very thin metallic layer.

Electron Microscopes
Electron microscopes produce a source with a wavelength smaller than visible light, so objects can be magnified much more than 2000 times their size.  However, specimens are prepared differently and are viewed in a vacuum.  The transmission electron microscope magnifies specimens more than 200,000 times their size, and a scanning electron microscope produces highly-detailed, three-dimensional images.  Electron microscopes have been developed that combine the best features  of the transmission electron microscope and the scanning electron microscope.

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An Early Series of Experiments
Spontaneous generation, known also as abiogenesis, was the theory believed by most philosophers and scientists of the day, as there was no way to test any alternative ideas.  Some of the earliest experiments to challenge abiogenesis were performed during the Italian Renaissance in the 1600s.  Redi hypothesized that flies laid eggs on the meat as it decayed and that the meat was nothing more than a source of food for the maggots that hatched. In the first set of experiments he performed, he put pieces of meat in jars, left some of them open , and closed some others.  The jars that were open attracted flies, and maggots formed in the meat as it decayed.  In the closed jars, meat decayed, but maggots did not form in the decaying meat.  Redi concluded that maggots weren’t spontaneously generated in the decaying meat, but came from the eggs laid by flies in the decaying meat.

Scientific Method Contradicts Spontaneous Generation
The experiments that Redi conducted were among the earliest to have an experimental group and a control group.  In the experimental group, jars were sealed so that meat decayed in them, and in the control group, jars were unsealed so that the flies could lay eggs on the decaying meat.  In a second set of experiments, jars in the experimental group were covered with wire mesh so that air could get through, but flies could not.  Careful control of experimental conditions showed most scientists that maggots did not develop through spontaneous generation.

Microscopes and Microorganisms
By the 1700s, many scientists were using early microscopes to study organisms that they hadn’t been able to detect before.  They were able to devise experiments that further contradicted earlier ideas of spontaneous generation.  For example, scientists were able to show  small organisms were already living in what appeared to be lifeless water or broth.  They varied the amount of time liquid in laboratory flasks were boiled in order to destroy life.

From Spontaneous Generation to Biogenesis
Scientists such as Louis Pasteur in the late 1800s were able to show that living organisms develop from other living organisms, the concept of biogenesis.  Small organisms, such as bacteria found on dust particles, contaminate materials by reproducing more organisms rather than spontaneously generating where no life was found before.

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