Non-Euclidean Geometry Unit
... squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book. Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design ...
... squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book. Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design ...
1-4 Practice B Pairs of Angles
... Adjacent angles are defined as two angles in the same plane with a common vertex and a common side, but no common interior points. ...
... Adjacent angles are defined as two angles in the same plane with a common vertex and a common side, but no common interior points. ...
An Introduction to Non-Euclidean Geometry
... 4. That all right angles equal one another. 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. ...
... 4. That all right angles equal one another. 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. ...
ON THE THEORY OF
... rems whose proofs present no difficulties, I prefix here only those of which a knowledge is necessary for what follows. ...
... rems whose proofs present no difficulties, I prefix here only those of which a knowledge is necessary for what follows. ...
The independence of the parallel postulate and the acute angles
... of the same circle(∆, A∆) EA as a radius, we draw the circle(E, EA) this second circle intersects the perpendicular bisector ΣZ at the points Θ and N. Since in the circle(∆, A∆) the cord EA is shorter in length than the diameter EH, it follows that the point N lies between the points H and ∆, and th ...
... of the same circle(∆, A∆) EA as a radius, we draw the circle(E, EA) this second circle intersects the perpendicular bisector ΣZ at the points Θ and N. Since in the circle(∆, A∆) the cord EA is shorter in length than the diameter EH, it follows that the point N lies between the points H and ∆, and th ...
4-5 Triangle Congruence: SSS and SAS
... 4-5 Triangle Congruence: SSS and SAS It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. ...
... 4-5 Triangle Congruence: SSS and SAS It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. ...
1-3
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
1-3 Measuring and Constructing Angles
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
1-3
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
... A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the ...
G5-3-Medians and Altitudes
... 5-3 Medians and Altitudes of Triangles The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. ...
... 5-3 Medians and Altitudes of Triangles The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. ...
4.3 - 4.5 Triangle Congruence Postulates
... If two angles and non-‐included side of one triangle are congruent to two angles and the corresponding non-‐included side of a second triangle, then the two triangles are congruent. ...
... If two angles and non-‐included side of one triangle are congruent to two angles and the corresponding non-‐included side of a second triangle, then the two triangles are congruent. ...
Space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra (i.e. ""space""), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later ""geometrical conception of place"" as ""space qua extension"" in the Discourse on Place (Qawl fi al-Makan) of the 11th-century Arab polymath Alhazen. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the ""visibility of spatial depth"" in his Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said that neither space nor time can be empirically perceived—they are elements of a systematic framework that humans use to structure all experiences. Kant referred to ""space"" in his Critique of Pure Reason as being a subjective ""pure a priori form of intuition"", hence it is an unavoidable contribution of our human faculties.In the 19th and 20th centuries mathematicians began to examine geometries that are not Euclidean, in which space can be said to be curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.