Download Sections 10.1 and 10.2

Chapter 10: Geometry
Section 10.1: Visualization
Important Objects
Object
Visualization
Characteristics
Point
A tiny dot
Has no size or shape
Line
An infinitely long stretched
string with no beginning or
end
An infinite flat piece of paper
with no beginning or end
Has no thickness
Plane
Has no thickness
Line segment: part of a line lying between two points, called the endpoints of
that segment
Ray: part of a line lying on one side of a point
Where objects lie
• In the plane: on an infinite 2-dimensional surface
• In space: in an infinite 3-dimensional room
Section 10.2: Angles
• Def: (a) An angle is the amount of rotation about a fixed point
(b) An angle is the region between two rays with a common
endpoint
• Def: Two angles are congruent if they both represent the same
amount of rotation
Measuring Angles
• Def: Angles are measured in degrees, where a full circle rotation is
considered to be 360˚.
• Def’s: acute angle: < 90˚
right angle: = 90˚
obtuse angle: > 90˚
straight angle: = 180˚
Angles formed by two lines
• Theorem 1: When two lines meet in a plane, they form four angles,
which sum to 360˚.
• Def: If the angles formed by two intersecting lines are all 90˚, then the
two lines are perpendicular.
• Def: If two lines never meet, they are parallel.
Configurations of 3 lines in a plane
• Parallel Postulate (or Axiom): If parallel lines are cut by another line,
then the corresponding angles are equal, i.e.
𝑎 = 𝑎′ , 𝑏 = 𝑏 ′ , 𝑐 = 𝑐 ′ , and 𝑑 = 𝑑′ .
Configurations of 3 lines in a plane
• Converse of the Parallel Postulate: If the angles 𝑎 and 𝑎′ are
congruent when two lines 𝑙1 and 𝑙2 are crossed by a third line, then
the lines 𝑙1 and 𝑙2 are parallel, or 𝑙1 ǁ 𝑙2 .
Configurations of 3 lines in a plane
• Theorem 2: The three angles in a triangle sum to 180˚.