Lesson 2: Circles, Chords, Diameters, and Their
... Students are asked to construct the perpendicular bisector of a line segment and draw conclusions about points on that bisector and the endpoints of the segment. They relate the construction to the theorem stating that any perpendicular bisector of a chord must pass through the center of the circle. ...
... Students are asked to construct the perpendicular bisector of a line segment and draw conclusions about points on that bisector and the endpoints of the segment. They relate the construction to the theorem stating that any perpendicular bisector of a chord must pass through the center of the circle. ...
Plane Separation, Interior of Angles, Crossbar Theorem
... two like terms, we have 2BC = 0. Divide both sides by 2 and note that this means that B = C by D2. This is not true because B is between A and C by hypothesis and is, thus, distinct from C by the definition of betweeness. Our assumption is incorrect and C is not between A and B. A similar proof show ...
... two like terms, we have 2BC = 0. Divide both sides by 2 and note that this means that B = C by D2. This is not true because B is between A and C by hypothesis and is, thus, distinct from C by the definition of betweeness. Our assumption is incorrect and C is not between A and B. A similar proof show ...
Triangles
... hanger by its angle measures. The triangle has one right angle, so it is a right triangle. ...
... hanger by its angle measures. The triangle has one right angle, so it is a right triangle. ...
Sam Otten - Michigan State University
... exists. If a rectangle exists, then every triangle has angle sum equal to 180º. Though the parallel postulate is never used as the basis for a proof in neutral geometry, there is still work that occurs surrounding it. For instance, it can be proven in neutral geometry that certain statements are log ...
... exists. If a rectangle exists, then every triangle has angle sum equal to 180º. Though the parallel postulate is never used as the basis for a proof in neutral geometry, there is still work that occurs surrounding it. For instance, it can be proven in neutral geometry that certain statements are log ...
Ex - CMS
... A flowchart proof is a visual way of organizing a proof in which statements are written in boxes with the reasons for them written underneath; arrows then connect the boxes to show how facts lead to conclusions. A column proof is a proof written in a two-column table in which the statements are list ...
... A flowchart proof is a visual way of organizing a proof in which statements are written in boxes with the reasons for them written underneath; arrows then connect the boxes to show how facts lead to conclusions. A column proof is a proof written in a two-column table in which the statements are list ...
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.