** 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 1113,
which are motivated by the "real-world" protractor, do the same for
angles. Hence, these axioms are sometimes referred to as the
*protractor
postulates*.

*Postulate 11*.
(*Angle Measurement Postulate*) To every angle there corresponds a real
number between 0 and 180.

*Postulate 12*.
(*Angle Construction Postulate*) Let be a ray on the edge of the half-plane `H`.
For every `r` between 0 and 180, there is exactly one ray
with `P` in `H` such that .

*Postulate 13*.
(*Angle Addition Postulate*) If `D` is a point in the interior of
,
then .

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* is the intersection of set of all points on the same
side of line *BC* as *A* and the set of all points on the same
side of line *AB* as *C*, denoted .
*(Note that this definition uses the
Plane
Separation Postulate.)*

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* is a ray *BD*
where *D* is in the interior of and .

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. and D is on the same side of line as C if and only if .*

** **

*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 ,
if *A*(0, 5), *B*(0, 3), and where the points are in the (a) Euclidean
Plane; and (b) Poincaré Half-plane.

*Exercise
2.35.* Given ,
,
and .
Prove or disprove .

*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.