plane geometry the building blocks of geometry. geometry plays an important part of many types of...
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Plane GeometryPlane GeometryPlane GeometryPlane Geometry
The building blocks of geometryThe building blocks of geometry
Geometry plays an important part of many types of careers from engineers to carpenters. Here is the Cooper River Bridge in Charleston SC. This bridge
would not be possible without geometry.
Points, lines, and planes: Here are some definitions you will need to
remember.
• Point – names an exact location on a plane.• Line – a collection of points forming a straight path
that extends infinitely in opposite directions.• Plane – a perfectly flat surface that extends forever
in all directions.• Segment – part of a line between two endpoints.• Ray – part of a line that starts at one endpoint and
extends forever in one direction.• Angle – formed by 2 rays with a common endpoint
called a vertex. Pleural of vertex is vertices.
Congruent - figures that have the same size and shape.
Segments that have the same length are congruent.
Angles that have the same measure are congruent.
The symbol for congruence is , which is read “ congruent to”.
The definitions up until now apply to angles when we look at one angle alone, but there
are also some special relationships between pairs of angles
Adjacent angles – 2 angles which share a vertex, share a side but do not overlap.
Angle 1 and angle 2 are adjacent angles.
Angle 1 and angle ABC are not adjacent
Vertical angles – 2 angles formed by
intersecting lines. They can not be adjacent, and they are always equal in
measure. They are across from one another.Angle 1 and angle 3 are
vertical angles.Angle 2 and angle 4 are
vertical angles.Angle 1 and angle 2 are
not vertical.
Complementary angles – 2 angles
whose measures add up to 90°.
Complementary angles can be placed so that
they form perpendicular lines.Angle 1 and angle 2 are complementary.
Angle XYZ and angle 1 are not
complementary.Line segment XY is
perpendicular to line segment YZ
Supplementary angles – 2 angles whose measures add up to 180°. Supplementary angles can be placed so that they form a
straight line.Angle 1 and angle 2 are supplementary.
The line passing through points A, B, and C is a straight line.
Parallel lines and angles
Angles formed by parallel lines and transversals (lines intersecting parallel lines), have a very interesting relationship.
The most important angles needed for most math applications are called alternate interior angles, alternate exterior angles and corresponding angles.
Transversal – a line that intersects 2 or more lines.Corresponding angles – angles formed by a transversal that are in the same relative position.Alternate interior angles – a pair of angles on the inner sides of two lines cut by a transversal and are on opposite sides of the transversal.Alternate exterior angles – a pair of angles on the outer sides of two lines cut by a transversal and are on opposite sides of the transversal.Adjacent angles – angles that share a common vertex and a side.
Certain angle “names” describe “where” the angles are located.
Alternate interior angles are between the parallel lines.
Alternate interior angles are congruent (equal)!
Alternate exterior angles can be easily found because their “name” describes “where” they
are.Alternate exterior angles are outside the
parallel lines.Alternate exterior angles are congruent
(equal)!
Corresponding angles are on the same side of the transversal, one is interior and the
other is exterior and they are not adjacent (they don’t touch).
Corresponding angles are congruent (equal)!
Adjacent angles create a straight angle or line.
Since a straight angle is 180°, adjacent angles add up to 180°. (Adjacent angles share a vertex, share a side, and do not
overlap.)