GEOMETRY
... 2. Two intersecting lines that intersected in point B 3. A line parallel to the first one 4. A perpendicular line from C to the BC 4. Four different lines passing through point C 5. A ray which vertex is in point D 6. A line segment which one of its endpoints is E 7. An acute angle 8. An ob ...
... 2. Two intersecting lines that intersected in point B 3. A line parallel to the first one 4. A perpendicular line from C to the BC 4. Four different lines passing through point C 5. A ray which vertex is in point D 6. A line segment which one of its endpoints is E 7. An acute angle 8. An ob ...
geometry institute - day 5
... Algebra, measurement, coordinate geometry OTHER LINKS Architecture, surveying LESSON OVERVIEW “The shortest distance between two points is a straight line.” Young children instinctively know that it is shorter to cut across the lawn than to go around it. This is axiomatic for the student of geometry ...
... Algebra, measurement, coordinate geometry OTHER LINKS Architecture, surveying LESSON OVERVIEW “The shortest distance between two points is a straight line.” Young children instinctively know that it is shorter to cut across the lawn than to go around it. This is axiomatic for the student of geometry ...
4-6
... 4-6 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
... 4-6 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
Ag_mod05_les03 congruent parts of congruent triangles
... Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
... Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
No Slide Title
... 4-7 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
... 4-7 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
7-2 - cloudfront.net
... Use the angles formed by a transversal to prove two lines are parallel. ...
... Use the angles formed by a transversal to prove two lines are parallel. ...
4-7
... 4-7 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
... 4-7 Triangle Congruence: CPCTC Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. ...
Geometry_CH-04_Lesson-5 _Using Indirect Reasoning _ Geometric
... If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Therefore, at least one of the items costs more than $25. This means that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the g ...
... If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Therefore, at least one of the items costs more than $25. This means that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the g ...
The Coarse Baum-Connes Conjecuture for Relatively Hyperbolic
... on closed manifold can be computed by topological data. t − index(DM ) = a − index(DM ) The Baum-Connes conjecture says that a purely “TOPOLOGICAL” object coincides with a purely ...
... on closed manifold can be computed by topological data. t − index(DM ) = a − index(DM ) The Baum-Connes conjecture says that a purely “TOPOLOGICAL” object coincides with a purely ...
Non Euclidean Geometry
... very strange ways. This is a 90-90-90 equilateral triangle. Such a triangle ONLY exists on the sphere! There is no way you could draw a 90-90-90 triangle on a piece of paper. Remember how the sum of the angles of a EUCLIDEAN triangle always has to be 180? Well, here you see an example of a spherica ...
... very strange ways. This is a 90-90-90 equilateral triangle. Such a triangle ONLY exists on the sphere! There is no way you could draw a 90-90-90 triangle on a piece of paper. Remember how the sum of the angles of a EUCLIDEAN triangle always has to be 180? Well, here you see an example of a spherica ...
The Rise of Projective Geometry
... would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the ...
... would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the ...
Giovanni Girolamo Saccheri
... states: “If two straight lines in a plane are cut by a transversal making the sum of the measures of two interior angles on the same side of the transversal less than 180˚, then the two straight lines will meet on that side of the transversal.” This assertion is linguistically much more complex than ...
... states: “If two straight lines in a plane are cut by a transversal making the sum of the measures of two interior angles on the same side of the transversal less than 180˚, then the two straight lines will meet on that side of the transversal.” This assertion is linguistically much more complex than ...
20_Testbank
... 1) Explain how we estimate that there are about 50-100 billion galaxies in the observable universe. Answer: Obviously it's impossible to count so many galaxies one by one, but by observing a small part in detail, we can extrapolate to get the total number. As an example, the Hubble deep field shows ...
... 1) Explain how we estimate that there are about 50-100 billion galaxies in the observable universe. Answer: Obviously it's impossible to count so many galaxies one by one, but by observing a small part in detail, we can extrapolate to get the total number. As an example, the Hubble deep field shows ...
A Brief History of the Fifth Euclidean Postulate and Two New Results
... the diagram of a square produces an area of double the size. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and ...
... the diagram of a square produces an area of double the size. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and ...
4-3 to 4-5 Notes - Blair Community Schools
... If two angles and a non-included side of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. ...
... If two angles and a non-included side of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. ...
G5-3-Medians and Altitudes
... 5-3 Medians and Altitudes of Triangles The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. ...
... 5-3 Medians and Altitudes of Triangles The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. ...
3-2
... 1. m1 = 120°, m2 = (60x)° Alt. Ext. s Thm.; m2 = 120° 2. m2 = (75x – 30)°, m3 = (30x + 60)° Corr. s Post.; m2 = 120°, m3 = 120° 3. m3 = (50x + 20)°, m4= (100x – 80)° Alt. Int. s Thm.; m3 = 120°, m4 =120° 4. m3 = (45x + 30)°, m5 = (25x + 10)° Same-Side Int. s Thm.; m3 = 120°, m5 =6 ...
... 1. m1 = 120°, m2 = (60x)° Alt. Ext. s Thm.; m2 = 120° 2. m2 = (75x – 30)°, m3 = (30x + 60)° Corr. s Post.; m2 = 120°, m3 = 120° 3. m3 = (50x + 20)°, m4= (100x – 80)° Alt. Int. s Thm.; m3 = 120°, m4 =120° 4. m3 = (45x + 30)°, m5 = (25x + 10)° Same-Side Int. s Thm.; m3 = 120°, m5 =6 ...
7-2 Ratios in Similar Polygons 7-2 Ratios in Similar Polygons
... Warm Up 1. If ∆QRS ≅ ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. ∠Q ≅ ∠Z; ∠R ≅ ∠Y; ∠S ≅ ∠X; QR ≅ ZY; RS ≅ YX; QS ≅ ZX Solve each proportion. ...
... Warm Up 1. If ∆QRS ≅ ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. ∠Q ≅ ∠Z; ∠R ≅ ∠Y; ∠S ≅ ∠X; QR ≅ ZY; RS ≅ YX; QS ≅ ZX Solve each proportion. ...
7-2 - Plainfield Public Schools
... Warm Up 1. If ∆QRS ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. Q Z; R Y; S X; QR ZY; RS YX; QS ZX Solve each proportion. ...
... Warm Up 1. If ∆QRS ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. Q Z; R Y; S X; QR ZY; RS YX; QS ZX Solve each proportion. ...
Shape of the universe
The shape of the universe is the local and global geometry of the Universe, in terms of both curvature and topology (though, strictly speaking, the concept goes beyond both). The shape of the universe is related to general relativity which describes how spacetime is curved and bent by mass and energy.There is a distinction between the observable universe and the global universe. The observable universe consists of the part of the universe that can, in principle, be observed due to the finite speed of light and the age of the universe. The observable universe is understood as a sphere around the Earth extending 93 billion light years (8.8 *1026 meters) and would be similar at any observing point (assuming the universe is indeed isotropic, as it appears to be from our vantage point).According to the book Our Mathematical Universe, the shape of the global universe can be explained with three categories: Finite or infinite Flat (no curvature), open (negative curvature) or closed (positive curvature) Connectivity, how the universe is put together, i.e., simply connected space or multiply connected.There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite. Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one.The exact shape is still a matter of debate in physical cosmology, but experimental data from various, independent sources (WMAP, BOOMERanG and Planck for example) confirm that the observable universe is flat with only a 0.4% margin of error. Theorists have been trying to construct a formal mathematical model of the shape of the universe. In formal terms, this is a 3-manifold model corresponding to the spatial section (in comoving coordinates) of the 4-dimensional space-time of the universe. The model most theorists currently use is the so-called Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat, but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space and the Picard horn.