2.4.2 Angles and Angle Measure
Printout
The right angle from which to approach any
problem is the try angle.
Definitions. An angle is the union of two
noncollinear
rays
with a common endpoint. The common endpoint is called the vertex of the angle, and the rays
are called the sides
of the angle.
The Ruler Postulate and
Ruler
Placement Postulate were motivated by the "real-world" use of
rulers.
A similar set of postulates, SMSG Postulates 11
Postulate 11. (Angle Measurement Postulate) To every angle there corresponds a real number between 0 and 180.
Postulate 12.
(Angle Construction Postulate) Let
Postulate 13.
(Angle Addition Postulate) If D is a point in the interior of
Note that an angle has measure
between 0 and 180. No angle has measure greater than or equal to 180, or less
than or equal to 0.
Definitions.
Two angles are congruent if they have the same
measure, denoted
The interior of an angle
The
interior of a triangle ABC is the
intersection of the set of points on the same side of line BC as A,
on the same side of line AC as B, and on the same side of line
AB as C.
The bisector of an angle
A right angle is an angle that
measures exactly 90.
An acute angle is an angle that
measures between 0 and 90.
An obtuse angle is an angle that
measures between 90 and 180.
Two lines are perpendicular if they contain a right
angle.
The next theorem, stated here
without proof, will be used in later sections.
Theorem 2.7.
Exercise
2.32. Find the axioms
from a high school book that correspond to SMSG Postulates 11, 12, and 13.
Exercise
2.33. Find the measures of the three angles determined by the points A(1,
1), B(1, 2) and C(2, 1) where the points are in the (a)
Euclidean Plane; and (b)
Poincaré Half-plane.
Also, find the sum of the measures of the
angles of the triangles.
Exercise
2.34. Find the angle bisector of
Exercise
2.35. Given
Exercise
2.36. Prove or disprove that all right angles are congruent.
Exercise
2.37. Prove or disprove that an angle has a unique bisector.
Exercise
2.38. (a) Prove that given a line and a point on the line, there
is a line perpendicular to the given line and point on the line.
(b) Prove the existence of two lines perpendicular to each other.
Exercise
2.39. Prove Theorem 2.7.
Exercise 2.40. Prove congruence of angles is an equivalence relation on the set of all angles.
© Copyright 2005, 2006 - Timothy Peil |