Geometry Curriculum Map (including Honors) 2014
... G.CO.3.9. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicula ...
... G.CO.3.9. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicula ...
Geometry curriculum guide
... the center of dilation through each vertex of the original figure by the scale factor to find each new vertex, Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the s ...
... the center of dilation through each vertex of the original figure by the scale factor to find each new vertex, Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the s ...
TOPIC 1
... A chord is a line segment which has its endpoints on the circle. A secant is a line which intersects a circle two times. An arc is a part of the circle, defined by two points on the circle. A tangent line is a line (or line segment) which hits a circle one time. A central angle is an angle whose ver ...
... A chord is a line segment which has its endpoints on the circle. A secant is a line which intersects a circle two times. An arc is a part of the circle, defined by two points on the circle. A tangent line is a line (or line segment) which hits a circle one time. A central angle is an angle whose ver ...
ISOMETRIES BETWEEN OPEN SETS OF CARNOT GROUPS AND
... homogeneous spaces that also admit dilations. These spaces, called Carnot groups, are particular nilpotent groups equipped with general left-invariant geodesic distances. Our method of proof also shows that, as in Riemannian geometry, global isometries of homogeneous spaces are uniquely determined b ...
... homogeneous spaces that also admit dilations. These spaces, called Carnot groups, are particular nilpotent groups equipped with general left-invariant geodesic distances. Our method of proof also shows that, as in Riemannian geometry, global isometries of homogeneous spaces are uniquely determined b ...
February Invitational Team Questions February
... Identify the following statements as true or false. Find the number of false statements plus 10 times the number of true statements. Two planes always intersect in a line. Two coplanar lines that do not intersect are parallel. The slope of perpendicular lines are reciprocals. All triangles have exac ...
... Identify the following statements as true or false. Find the number of false statements plus 10 times the number of true statements. Two planes always intersect in a line. Two coplanar lines that do not intersect are parallel. The slope of perpendicular lines are reciprocals. All triangles have exac ...
Unit 2
... How does the location of the vertex of an angle effect the formula for finding the angle measure? How many ways can a line intersect a circle? When two lines intersect they create an angle. How does the location of the vertex of that angle effect the formula that I will use to find the lengths of th ...
... How does the location of the vertex of an angle effect the formula for finding the angle measure? How many ways can a line intersect a circle? When two lines intersect they create an angle. How does the location of the vertex of that angle effect the formula that I will use to find the lengths of th ...
(Semester) Pacing Guide
... Recall definitions of angles, circles, perpendicular and parallel lines and line segments. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. Given a geometric figure and a rotation, reflection or ...
... Recall definitions of angles, circles, perpendicular and parallel lines and line segments. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. Given a geometric figure and a rotation, reflection or ...
10.2 Arcs and Chords
... • PQ and RS are in congruent circles and m PQ = m RS = 80°. So, PQ RS ...
... • PQ and RS are in congruent circles and m PQ = m RS = 80°. So, PQ RS ...
6. The sum of angles in a triangle is =180° 7. The sum of angles in a
... perpendicular line to the given line through the given point ...
... perpendicular line to the given line through the given point ...
Adjacent angles share a common vertex and side. What does
... When a pair of lines is drawn, two distinct regions are formed: The interior region lies between the two lines. The exterior region is the area remaining outside the lines. A line that intersects two or more lines in different points is called a transversal. ...
... When a pair of lines is drawn, two distinct regions are formed: The interior region lies between the two lines. The exterior region is the area remaining outside the lines. A line that intersects two or more lines in different points is called a transversal. ...
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous ""line-sphere correspondence"" between the space of lines and the space of spheres in 3-dimensional space.