Mathematical Propositions
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Exercise 42: Formulate definition of supplementary angles ($\S 22$) and vertical angles ($\S 26$) using the notion of sides of an angle.
$\S 22$ does this for supplementary angles: angles which have a common side and their other sides are continuations of each other.
$\S 26$ does this for vertical angles: 2 angles whose sides are continuations of the other angle's.
Exercise 43: Find in the text the definitions of an angle, its vertex and sides, in terms of the notion of a ray drawn from a point.
Angle - a figure formed by two rays drawn from the same point.
Angle vertex - the common endpoint of the rays that form an angle.
Angle sides - the rays drawn from a point forming an angle.
Exercise 44: In Introduction, find the definitions of a ray and a straight segment in terms of the notion of a straight line and a point. Are there definitions of a point, line, surface, geometric solid? Why?
Straight line - defined by experiential examples. A thing thread stretched tight or a ray of light emitting through a small hole.
Ray - a straight line which terminates in one direction only.
Point, line, surface, and geometric solid are undefined. The base of this knowledge must start somewhere and those basic items cannot be defined in terms of the unknown. Thus knowledge of these items draws from human observation and experience.
Exercise 45: Is the following proposition from $\S 6$ a definition, axiom or theorem: "Two segments are congruent if they can be laid one onto the other so that their endpoints coincide"?
Theorem proven using $\S 1$ properties of geometric figures and the definition of Congruence.
Exercise 46: In the text, find the definitions of a geometric figure, and congruent geometric figures. Are there definitions of congruent segments, congruent arcs, congruent angles? Why?
Geometric figure - $\S 1$: a set of points, lines, surfaces, or solids positioned in a certain way in space.
Congruent geometric figures - $\S1$: two geometric figures where either one can be moved through space and superimposed onto the other such that they become identified in all their parts.
Congruence of specific figures (segments, arcs, angles) is not defined. This is covered under the general definition of Congruence.
Exercise 47: Define a circle.
The set of all points within a plane that are the same distance from a given point. Also references the curved line through these points.
Exercise 48: Formulate the proposition converse to the theorem: "If a number is divisible by 2 and by 3, then it is divisible by 6." Is the converse true? Why?
If a number is divisible by 6 then it is divisible by 2 and by 3.
True. 2 and 3 are factors of 6.
Exercise 49: In the proposition from $\S 10$: "Two arcs of the same circle are congruent if they can be aligned so that their endpoints coincide," separate the hypothesis from the conclusion, and state the converse proposition. Is the converse proposition true? Why?
Hypothesis: two arcs of the same circle are congruent
Conclusion: the arcs can be aligned so that their endpoints coincide
Converse: If two arcs of the same circle can be aligned so that their endpoints coincide, then these arcs are congruent.
From the $\S 9$ definition of a circle it follows that all points of the arcs in the same circle will be the same distance from the center. Combining this with $\S 1$ geometric figure congruence, moving an arc and superimposing it onto the other with the endpoints coinciding will cause all their points to coincide. Therefore the arcs are congruent and the converse is True.
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