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Topic: Angle of parallelism

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Hyperbolic geometry

	m324
		The summit angles of a Saccheri quadrilateral are acute.
		The summit angles of a Saccheri quadrilateral are congruent.
		The measure of an exterior angle of a triangle is equal to the sum of the measures of the corresponding interior angles.
	ac.marywood.edu /johnsonc/www/m324.htm (1053 words)

	Angle of parallelism - Wikipedia, the free encyclopedia
		In hyperbolic geometry, the angle of parallelism Φ is the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides.
		The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism Φ.
		The angle at the center of Q subtended by the radius to (0, y) is also Φ because the two angles have sides that are perpendicular, left side to left side, and right side to right side.
	en.wikipedia.org /wiki/Angle_of_parallelism (329 words)

	Caster angle - Wikicars
		Caster (or castor) angle is the angular displacement from the vertical axis of the suspension of a steered wheel in a car or other vehicle, measured in the longitudinal direction.
		It is the angle between the pivot line (an imaginary line that runs through the center of the upper ball joint to the center of the lower ball joint) and vertical.
		The pivot points of the steering are angled such that a line drawn through them intersects the road surface slightly ahead of the contact point of the wheel.
	wikicars.org /es/Caster_angle (484 words)

	Method of manufacturing magnetic head - Patent 6881124 (Site not responding. Last check: 2007-11-02)
		Also, the angle adjustment mechanism, which consists of a pair of angle adjustment portions driven in an arc form in the direction perpendicular to each other, enables three-dimensional adjustment of the relative position of the surface of the workpiece to be lapped and the surface of the lapping plate.
		On the bridge 5, a reciprocating motion drive unit 7 consisting of a linear actuator is installed, and an angle adjustment mechanism 4 comprising two gonio-stages 4a and 4b is mounted on a slider table 15 of the reciprocating motion drive unit 7.
		After the above-described angle adjustment (parallelism) of the surface of workpiece to be lapped and the surface of lapping plate is made by using the completely fixed abrasive lapping plate prepared as described above, the workpiece is brought into contact with the lapping plate 8 and is lapped.
	www.freepatentsonline.com /6881124.html (6425 words)

	Ethan Frome (Site not responding. Last check: 2007-11-02)
		The students will be able to measure and find the sum of the angles of a triangle, construct tiling patterns from several polygons (squares, rectangles, parallelograms, right triangles, non-right triangles), review various geometric concepts (parallel lines, congruent angles, shape identification), and also discover new patterns and properties, in particular "saws" and " ladders".
		Supporting Materials Needed: straws or d-stix, protractor, acetate angle measure overlays, unmarked clock face, overlay divided onto fifteen degree angle increments, overlay that is divided into 360 degrees, Pattern Blocks, square grid, pictures of tiling patterns, rectangular grid, triangular grid, overlay of saws and ladders, colored pencils.
		Finally, a triangle is shown with only two angle measurements identified and students are asked to predict the measurement of the unmarked angle.They are then asked to explain their reasoning.
	www.ship.edu /~eisen/Eisen99/Intermediate/angle_measure.html (1268 words)

	Homework Problems 24
		If the sum of the angles at A and B is greater than or equal to \pi, then C would have to be smaller than or equal to the angle at B. This contradicts Thm.
		However, thes e are the angles of parallelism for the lines B\Omega and B'\Omega at the points A and A' respectively.
		9.4, this angle is greater than the angle of parallelism at C to l, thus, the angle of paralle lism decreases as the distance from l increases.
	www-math.cudenver.edu /~wcherowi/courses/m3210/hghw24.old (802 words)

	Homework Problems 24
		The angle of parallelism at A to l, is an exterior angle of the omega triangle AC\Omega.
		9.4, this angle is greater than the angle of parallelism at C to l, thus, the angle of parallelism decreases as the distance from l increases.
		Since the angles at A and B are congruent and the sides AM and MB are congruent, the omega triangles AM\Omega and BM\Omega are congruent by Thm.
	www-math.cudenver.edu /~wcherowi/courses/m3210/hghw24.html (753 words)

	Project 4 Solutions
		By Theorem 4.5.11, angle C equals 1/2 the measure of arc DEB, while angle E equals 1/2 the measure of arc DCB.
		Theorem: Suppose angle ABC is the angle of parallelism for point B and segment AB perpendicular to line l and that angle DEF is the angle of parallelism for point E and segment DE perpendicular to line m.
		Thus the alternate interior angles BDM and ACM are congruent.
	www.unc.edu /depts/cmse/science/proj4solns.htm (703 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism), which we will here designate by
		Upon the other side of the perpendicular EE' will in like manner the prolongations AH' and AK' of the parallels AH and AK likewise be parallel to BC; the remaining lines pertain, if in the angle K'AH', to the intersecting, but if in the angles K'AE, H'AE' to the non-intersecting.
		For both assumptions it serves as the mark of parallelism that the line becomes intersecting for the smallest deviation toward the side where lies the parallel, so that if AH is parallel to DC, every line AF cuts DC, how small soever the angle HAF may be.
	userwww.service.emory.edu /~gharri2/kaneon/loba/16.txt (428 words)

	Hypercycles and Horocycles (Site not responding. Last check: 2007-11-02)
		Thus the bisectors of the angles of the trilateral
		Theorem 18.4: If three points P, Q, and R lie on three parallels in the same direction so that P and Q are corresponding points on their parallels and Q and R are corresponding points on theirs, then P, Q, and R are noncollinear.
		Theorem 18.5: If three points P, Q, and R lie on three parallels in the same direction so that P and Q are corresponding points on their parallels and Q and R are corresponding points on theirs, then P and R are corresponding points on their parallels.
	www.math.uncc.edu /~droyster/math3181/notes/hyprgeom/node68.html (1393 words)

	Interactive Geometry with Polyhedra
		In the former case, the student undertakes the teacher's role and thus is free to choose a subject by himself and interrupt the resolution whenever he or she wants.
		is parallel to BG and passes through a point that corresponds to J in which CE intersects DF in Figure 2a.
		Draw on the upper base of this truncated pyramid (Figure 4a) a line parallel to AB so that the distance between it and AB is equal to the length of edge [AD].
	geocentral.net /polyhedron/dyngeom (2743 words)

	Euclidean: (Site not responding. Last check: 2007-11-02)
		angle of parallelism (for the distance CF) = the angles between the perpendicular (CF) and one of the two parallel lines (left and right parallels).
		The two angles of parallelism for the same distance are congruent and acute.
		Reflect the smaller, angle FCE, to the side of angle FCD with respect to the perpendicular CF.
	pegasus.cc.ucf.edu /~xli/chap9.htm (468 words)

	FRANZ ADOLPH TAURINUS (1794–1874)
		and this angle sum increased as the area of the triangle decreased which was noticed by Lambert.
		In his second book, “Geometriae Prima Elementa” in 1826, he had to admit that the “third system of geometry”, where the angle sum of a triangle is less than two right angles, is not contradicted and he called this the “logarithmic-spherical geometry”.
		The angle at B is called the angle of parallelism for line through BC From this Taurinus calculates Schweikart’s constant in terms of the sphere’s radius,
	www.southernct.edu /~pinciuv/mat530pr8.html (753 words)

	Szydlik: Example - The Angle of Parallelism
		Either of the congruent angles XPQ and X'PQ is called the angle of parallelism determined by P and Q:
		In this way the phrase, "the angle of parallelism determined by P and Q," is well-defined.
		Construct and measure the angle of parallelism determined by P and Q.
	www.joma.org /images/upload_library/4/vol1/hypertoolbox/parang.htm (328 words)

	Hyperbolic geometry - Wikipedia, the free encyclopedia
		A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging parallel lines.
		The parallel postulate in Euclidean geometry states (for two dimensions) that given a line l and a point P not on l, there is exactly one line through P that does not intersect l.
		Both Poincaré models preserve hyperbolic angles, and are thereby conformal.
	en.wikipedia.org /wiki/Hyperbolic_geometry (1357 words)

	IMAGES \| Two- and Three-Dimensional Geometry
		Suggest a general topic or theme for their story or relate it to what you are teaching in language arts, science, or social studies.
		Use dynamic geometric software to measure the interior angles and the to Explore Triangles lengths of the sides of a triangle.
		Have students create different types of triangles, discover the triangle inequality property, and determine that the sum of interior angles is always 180 degrees.
	images.rbs.org /content_strands/two_and_three_d.shtml (1368 words)

	Meaning of say ``two parallel lines'' (Site not responding. Last check: 2007-11-02)
		For a Omega triangle AB, the exterior angle at A by extending AB is larger than its opposite interior angle.
		Case II: angle CA = angle CB Let D be the midpoint of segemnt AB.
		But angle DFA is also the angle of parallelism and so it cannot be a right angle.
	pegasus.cc.ucf.edu /~xli/omega.htm (264 words)

	NAEP - Mathematics Content Areas 2005
		Students may be asked to select appropriate units and tools for measuring, to measure length with a ruler at all three grades, to measure angles with a protractor at grades 8 and 12, and to solve application problems related to units of measurement.
		As students move to middle school and beyond, increased understanding should deepen of two- and three-dimensional figures, especially parallelism, perpendicularity, angle relations in polygons, congruence, similarity, and the Pythagorean theorem.
		Students at all grades are expected to show knowledge of symmetry and transformations of shapes and to identify images resulting from flips, rotations, or turns.
	nces.ed.gov /nationsreportcard/mathematics/contentareas2005.asp (690 words)

	Angle Plates
		These slotted angle plates have convenient clamping slot for mounting and widely used for machining, layout and inspection
		Giant angle plates are best suited for work mounting and setup on such large machine tools.
		These useful stepped angle plates with tapped holes for clamping jobs at any position are made of alloy steel which is then hardened and ground.
	www.tfttools.com /AnglePlates.htm (96 words)

	Lens-back parallelism-Angle-finder OK? - Large Format Photography Forum
		It is also difficult to w ork on film - lens parallelism, especially in low light conditions.
		The Horseman angle finder II I've seen is an expensive, not particularly precise and fairly worthless tool.
		I also work with ultra wide angle lenses, and can appreciate the challenges they present.
	www.largeformatphotography.info /forum/showthread.php?t=1399 (329 words)

	Teaching Ideas (Site not responding. Last check: 2007-11-02)
		For example, in the Klein model, constructing segments and lines is trivial, dropping and raising perpendiculars is fairly straightforward, measuring angles can provide a nice application of hyperbolic trigonometry, and drawing circles is hard.
		In hyperbolic geometry, if m and m' are parallel lines for which there exists a pair of points A and B on m equidistant from m', then m and m' have a common perpendicular that is also the shortest segment between m and m'.
		A second, slightly more advanced exercise is, given two points P and Q in a model, to have students construct and measure the angle of parallelism corresponding to segment PQ.
	www.uwosh.edu /faculty_staff/szydliks/noneuclid/teaching.htm (526 words)

	ADE: Arizona Standards
		Identify the properties of geometric figures using appropriate terminology and vocabulary (e.g., parallelism, perpendicularity and congruency)
		Classify triangles by their angles and sides (e.g., equilateral, acute, isosceles.
		Verify characteristics of a given geometric figure using coordinate formulas such as distance, mid-point, and slope to confirm parallelism, perpendicularity, and congruency
	seamonkey.ed.asu.edu /emc300/azstandards/mstandard4.htm (816 words)

	5th Grade Geometry
		Use the triangles and angles worksheet to review the different types of angles and triangles.
		Using the worksheets, walk around the school and list various things that are viewed that demonstrate the concepts of perpendicular and parallel.
		Using the worksheet, walk around the school and list various things that are viewed that demonstrate the concept of congruent.
	edweb.tusd.k12.az.us /ekowalcz/geometry (590 words)

	TS-Aligner Jr. vs MasterGage
		Optional angle blocks and can be used to further increase accuracy and ease of use.
		However, no instructions were provided for such a use and the vertical motion of the Level Arm is not stable enough for accurate results.
		Such warps are extremely common around the throat plate area, where the thin cross section of cast iron allows room for operation of the blade tilt mechanism.
	www.ts-aligner.com /tsjrvsmg.htm (3151 words)

	ROV15_6.html
		We calculate angles of a triangle using the procedure angle.
		Calculate the sum of its angles, and the area using the procedure
		Calculate the angle between them and the angle of parallelism.
	math.haifa.ac.il /ROVENSKI/rovenski/rov15_6.html (466 words)

	Math 390 -- Reveiw topics for final exam (Site not responding. Last check: 2007-11-02)
		(Specifically, you may be asked to compute an angle of parallelism in the H-model as we did in Section 6F.)
		I MAY ask you to reason with the usual models for Euclidean, hyperbolic, and spherical geometries.
		In the material since the last exam, you should know the definitions of angle of parallelism, asymptotically parallel, and ideal point.
	www.uidaho.edu /math/nielsen/390video/revfe.html (186 words)

	Szydlik: Teaching Examples
		In the Klein model, for example, constructing lines and segments are trivial, calculating lengths, dropping and raising perpendiculars is relatively straightforward, while calculating angle measure, constructing midpoints, angle bisectors, and circles is quite challenging.
		On the other hand, calculating angle measure in the Poincaré models is relatively straightforward since these models are conformal, while in the Klein model, this action requires the use of hyperbolic trigonometry.
		In my classroom, I have had students develop the tools for constructing lines and segments in the three models as an introduction to the models and to writing scripts in Geometer's Sketchpad after reading the appropriate section of Greenberg for the Poincaré disk construction.
	www.joma.org /images/upload_library/4/vol1/hypertoolbox/szydtch.htm (540 words)

	Mathematics Standards (Site not responding. Last check: 2007-11-02)
		Visualize and draw two- and three-dimensional geometric figures with special attention to analyzing and reasoning informally about their properties (e.g., parallelism, perpendicularity and congruence).
		Apply geometric properties and relationships such as congruence, similarity, angle measure, parallelism and perpendicularity to real-world situations.
		Select and use appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation.
	coe.west.asu.edu /students/desertsky/standards.html (996 words)

	klein view page
		orthogonal to C, so that The angle BAC is zero.
		the angles of a trebly asymptotic triangle is zero, and its area is π.
		The sum of the angles of a hyperbolic triangle is less than π.
	www.maths.gla.ac.uk /~wws/cabripages/hyperbolic/asymptotic.html (568 words)

	[No title]
		Show how to use Euclidean geometry to construct lines between 2 points, lines perpendicular to given lines, etc., in the Poincaré disk and/or in the Beltrami-Klein models of hyperbolic geometry.
		Focus on one or more features such as SAS, the angle of parallelism, Saccheri quadrilaterals, equivalence and area, etc., and discuss them using a particular model such as the Poincaré disk.
		Show how to use Euclidean geometry to construct lines between 2 points, lines perpendicular to given lines, etc., in the Klein and in the spherical models of elliptic geometry.
	oregonstate.edu /instruct/mth338/garity/Topics.htm (504 words)

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