Chapter 10: Geometry

Section 10.1: Visualization

Important ObjectsObject Visualization Characteristics

Point A tiny dot Has no size or shape

Line An infinitely long stretched string with no beginning or

end

Has no thickness

Plane An infinite flat piece of paper with no beginning or end

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 sameamount 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. and

Configurations of 3 lines in a plane• Converse of the Parallel Postulate: If the angles and are congruent

when two lines and are crossed by a third line, then the lines and are parallel, or ǁ .

Configurations of 3 lines in a plane

• Theorem 2: The three angles in a triangle sum to 180˚.