Introduction to shapes
... Quick Facts About Triangles • A Triangle has 3 sides • A right triangle has 90 degree angle in it • An acute triangle all of its sides are no greater than 90 degrees • Obtuse triangles have one angle that is greater than 90 degrees • A scalene triangle none of its sides are equal • In an Isosceles ...
... Quick Facts About Triangles • A Triangle has 3 sides • A right triangle has 90 degree angle in it • An acute triangle all of its sides are no greater than 90 degrees • Obtuse triangles have one angle that is greater than 90 degrees • A scalene triangle none of its sides are equal • In an Isosceles ...
Triangles and Angles
... An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. ...
... An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. ...
Reading and creating angle measures
... The angle at each vertex of a regular octagon is 135o. What type of angle is at each vertex of a regular octagon? ...
... The angle at each vertex of a regular octagon is 135o. What type of angle is at each vertex of a regular octagon? ...
Chapter 3 Terms
... Steps: 1. Put the vertex in the middle of the line on the bottom of the protractor. 2. Decide whether it is an acute or obtuse angle. 3. Choose the larger number for obtuse angles and the smaller number for acute angles. 4. Name the angle with the vertex in the middle of three letters. 5. Write = th ...
... Steps: 1. Put the vertex in the middle of the line on the bottom of the protractor. 2. Decide whether it is an acute or obtuse angle. 3. Choose the larger number for obtuse angles and the smaller number for acute angles. 4. Name the angle with the vertex in the middle of three letters. 5. Write = th ...
Parallels and Euclidean Geometry Lines l and m which are coplanar
... Theorem If two coplanar lines are cut by a transversal so that some pair of alternate interior angles are congruent, then the two lines are parallel. // Proposition i.29 is the converse of this theorem, but it is not a theorem of absolute geometry: to prove it, Euclid made use of a new axiom, his l ...
... Theorem If two coplanar lines are cut by a transversal so that some pair of alternate interior angles are congruent, then the two lines are parallel. // Proposition i.29 is the converse of this theorem, but it is not a theorem of absolute geometry: to prove it, Euclid made use of a new axiom, his l ...
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.