Postulates-theorems
... 3.1: Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent 3.2: Slopes of Parallel Lines: Two nonvertical lines have the same slope if and only if they are parallel. All vertical lines are parallel 3.3: Slopes of Perpendic ...
... 3.1: Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent 3.2: Slopes of Parallel Lines: Two nonvertical lines have the same slope if and only if they are parallel. All vertical lines are parallel 3.3: Slopes of Perpendic ...
Lesson 28: Properties of Parallelograms
... How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms? To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we h ...
... How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms? To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we h ...
Lesson 28: Properties of Parallelograms
... How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms? To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we h ...
... How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now prove about the properties of parallelograms? To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we h ...
What is Congruent? - Miami Beach Senior High School
... • If you prove a fact for a category of objects, then you prove something for every object in that category. We will use the “Is A” chart (next slide) to help us know which quadrilaterals are also another kind of quadrilateral. • In this project, we will look at trapezoids and decide what we can pro ...
... • If you prove a fact for a category of objects, then you prove something for every object in that category. We will use the “Is A” chart (next slide) to help us know which quadrilaterals are also another kind of quadrilateral. • In this project, we will look at trapezoids and decide what we can pro ...
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.