2.4.2 Angles and Angle Measure
The right angle from which to approach any problem is the try 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.
(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.
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.
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.
© Copyright 2005, 2006 - Timothy Peil