Geometry sizes and shapes

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 Real Life

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.