MATH 10 - SHAPES

• Euclid is the father of geometry
• The Babylonian Mathematics is written mostly in cuneiform script.
• During the Babylonian era, circumference of circle was taught to be 3x the diameter.
• During the Egyptian Civilization, formulas were recorded in Ahemes Papyrus.
• Eratosthenes was the first mathematician to calculate the circumference of the earth,
• Pythagoras created the concept of triangles.
• The accurate approximation of the value of pi was through the method of exhaustion.
• Method of exhaustion was developed by Euxodus of Cnidus.
• The contributions of the Greeks to Mathematics were the discovery of irrational numbers, pi, and golden ratio.
• The ratio of the whole (1) to the longer part (X) is equal to the ratio of the longer part (X) to the shorter part (1-X).
• The reciprocal of golden ratio is ~1.618...
• Euclid of Alexandria was the mathematician responsible for the mathematical system "Euclidian Geometry."
• "The Elements" is a book that describes the Euclidean Geometry.
• Geometry is a system that deals with points, lines, and planes.
• Points, lines, and planes are undefined terms.
• Axioms are properties that are universally true and does not need proof while postulates are properties that may require justification or proof.
• In the Euclidean Axioms, things that are equal to the same things are equal.
• In the Euclidean Axioms, if equals are added to an equal, then the whole is equal.
• In the Euclidean Axioms, if the equal is subtracted from an equal, the remainder is equal.
• In the Euclidean Axioms, things that coincide with one another are equal to one another.
• In the Euclidean Axioms, the whole is greater than the part.
• In the Euclidean Postulates, a straight line can be drawn from any point to any point.
• In the Euclidean Postulates, a finite straight line can be continuously produced in a straight line.
• In the Euclidean Postulates, a circle can be drawn with any point as a center and any distance as a radius.
• In the Euclidean Postulates, all right angles are equal to one another.
• In the Euclidean Postulates, the 5th postulate is also called the parallel postulate.
• In the Euclidean Postulates, if the transversal falls on two lines in such a way that the interior angles on the side of the transversal is less than the two right angles, then the lines meet on the side of which the angles are less than the two right angles.
• If the 3rd and 5th interior angles are less than the right angles and sum are less than 180, then the lines must intersect at some point on the same side of the interior angles.
• Posidonius said "Two lines are equidistant."
• Proclus said "If a line intersects one of the two parallel lines, then it intersects the other."
• Legendre said "The sum of the interior angles of a Euclidean triangle is 180."
• Playfair is the most popular axiom as it is equivalent to the 5th Postulate.
• In Playfair axiom, through a point P not on a line I, there exists exactly one line passing through Point P parallel to I.
• Euclidean Triangles will always have a total sum of their interior angles as 180.
• Euclidean Triangles have two properties which are the congruency and the similarity.
• Euclidean Triangles are only congruent if they have the same size and shape.
• Congruence can be proven by SSS (Side-Side-Side), ASA (Angle-Side-Angle), and SAS (Side-Angle-Side).
• SSS is where three corresponding sides are congruent.
• SAS is where there are two corresponding sides and the angle between them are congruent.
• ASA is where two corresponding angles and their side are congruent.