Transcript
GeometryGeometry
•the branch of mathematics that is concerned with the properties of configurations of geometric objects – points, (straight) lines, and circles.
GeometryGeometry
•The word ‘geometry’ comes from the Greek words geo, meaning earth, and metria, meaning measure.
A Greek mathematician who lived around the year 300 BC is often referred to as the Father of Geometry for his amazing geometry works that included his influential book, ‘Elements’.
Types of GeometryTypes of Geometry• Plane geometryPlane geometry deals with
objects that are flat, such as triangles and lines, that can be drawn on a flat piece of paper in two dimensions.
• Solid geometrySolid geometry deals with objects in that space, having width, depth and height, such as cubes and spheres.
Tools of GeometryTools of Geometry• The compass and
straight edge were powerful tools in the advancement of geometry, allowing the construction of various lengths, angles and geometric shapes.
Geometry in ActionGeometry in Action• Design and
manufacturing– Architecture– Assembly planning– CAD/CAM– Robotics– Modeling– Textile layout– VLSI design
• Graphics and visualization– Computer
graphics– Graph drawing– Virtual reality– Video game
programming
Geometry in ActionGeometry in Action• Medicine and
biology– Medical imaging– Biochemical
modeling
• Information systems– Cartography and
geographic information systems
– Data mining and multidimensional analysis
• Physical sciences– Astronomy– Molecular modeling
CAD/CAM
Architecture
VLSI design
Textile layout Biomedical
imaging
Cartography
Molecular modeling
Lines, Points and Lines, Points and PlanesPlanes
PointsPoints
• A point has no dimension. • It is usually represented by a
small dot.A
LinesLines
• A line extends in one dimension.
• It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions.
Line l or AB
Bl
A
LinesLines
• Collinear points are points that lie on the same line.
• Points A, B, and C are collinear.
C Line l
A B
LinesLines
• The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B.
CLine
l
A
B
A B
Segment AB
LinesLines
• The ray AB (symbolized by BC) consists of the initial point B and all points on Line l that lie on the same side of B as point C.
CLine l
A B
CB
Ray BC
PlanesPlanes
• A plane extends in two dimensions.
• The plane extends without end even though the drawing of a plane appears to have edges.
A
BC
M
PlanesPlanes
• A plane extends in two dimensions.
• The plane extends without end even though the drawing of a plane appears to have edges.
A
BC
M
• Coplanar points are points that lie on the same plane.
PlanesPlanes
• D, E, F, and G lie on the same plane, so they are coplanar.
• D, E, F, and H are also coplanar; although, the plane containing them is not drawn.
G
DE F
H
PlanesPlanes
A line intersects a plane
in one point.
PlanesPlanes
Two planes intersect
plane in a line.
Parallel Lines, Parallel Lines, Transversals and Transversals and
AnglesAngles
Parallel Lines, Parallel Lines, Transversals and Transversals and
AnglesAngles
Parallel LinesParallel Lines
Two lines are parallel lines if they are coplanar and do not intersect.
Parallel PostulateParallel Postulate
• If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
ll
PP
TransversalTransversal
A line that intersects two or more coplanar lines at different points.
P1
P2
Transversal and AnglesTransversal and Angles• Angle 1 and 5 are
corresponding angles
• Angles 1 and 8 are alternate exterior angles
• Angles 3 and 5 are alternate interior angles.
1 2
5 67 8
3 4
• Angles 3 and 5 are consecutive interior angles.
Corresponding Angles Corresponding Angles PostulatePostulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
2
1 l
m
If l // m, then 1 2.
Alternate Interior Angles Alternate Interior Angles TheoremTheorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior s are .
21
l
m
If l // m, then 1 2.
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