4.2.1 Holt Geometry
Chapter 2 contains an extensive development of inductive and deductive
reasoning, including formal rules of logic. Section 2.1 introduces inductive
reasoning and conjecturing in mathematics, science, and life outside science.
Next come Venn diagrams and Section 2.2 on conditional (if-then) statements.
Section 2.3 addresses deductive reasoning as a way to verify conjectures.
Section 2.4 is devoted to bi-conditional statements and definitions. Section 2.5
addresses algebraic proof, and Sections 2.6 and 2.7 begin geometric proof -
two-column and then flow chart and paragraph proofs. All sections include a
generous selection of examples and problems from geometry, other areas of
mathematics, and daily life. Various strategies and re presentations are
presented to support understanding and applications of these ideas. These rules
of logic and proof are used to develop geometry topics in the rest of the book.
Chapter 3 focuses on parallel and perpendicular lines. Section 3.1 provides
definitions of parallel and perpendicular lines, as well as skew lines and
parallel planes. This is followed by an informal introduction to examples of
parallel lines (e.g., the edges of a box). Terminology is developed here for the
four pairs of angles formed by two lines and a transversal line. Section 3.2
begins with a postulate (Postulate 3-2-1) that states the equality of
corresponding angles in a figure formed of two parallel lines and a transversal.
Then the consequences are stated and proved as examples or problems. Section 3.3
includes a new postulate (Postulate 3-3-1) that is the converse of Postulate 3-2-1; that is, sufficient conditions that two lines be parallel. This postulate
is used to prove theorems establishing that certain lines are parallel,
including the case of two lines perpendicular to the same
line. Section 3.3 ends with a Geometry Lab with constructions for parallel lines
by compass and straightedge and by paper folding.
Section 3.4 is devoted to perpendicular lines, including some theorems about
perpendicular transversals and compass and straightedge construction of the
perpendicular bisector of a segment. There it is also a statement that the
shortest segment from a point to a line is the perpendicular segment (the proof
will come later). The Geometry Lab introduces constructions of perpendicular
lines. Sections 3.5 and 3.6 deal with lines in the coordinate plane.
Intersections of lines are found by solving linear equations; the concept of
slope is developed and it is asserted as a theorem that parallel
lines have the same slope and that perpendicular lines have slopes whose product
is -1. The relationships between slope and parallelism are neither proved nor
justified informally.
This chapter does a thorough job of stating and proving the basic angle theorems
about parallel lines and transversals and also theorems about perpendicular
lines. The inclusion of some properties of distance in the section on
perpendiculars seems natural, though it does require assuming a theorem whose
proof must be deferred. It is puzzling that there is no attempt to explain the
slope relations for parallel and perpendicular lines, either by solving
simultaneous algebraic equations or drawing simple figures with slope. This is a
missed opportunity to help students make sense of the mathematics.
Chapter 6 (Parallelograms and Polygons) begins by introducing some basic
definitions and theorems about polygons in general and developing the theory of
parallelograms. A later part of the chapter moves on to special parallelograms
and other special quadrilaterals such as isosceles trapezoids and kites. Section
6.1 defines basic terminology such as vertex, interior angle, exterior angle,
and then states and proves theorems for general convex n-gons about the sum of
the interior angles and the sum of the exterior angles (an important theorem
that is not always given the prominence that is its due). Section 6.2 develops
the standard properties of parallelograms. The properties are proved as theorems
and also are studied by construction and drawing, and there are examples in the
coordinate plane. Section 6.3 proves and applies conditions for parallelograms,
that is, the converses of some of the theorems of 6.2. Examples and problems in
the coordinate plane apply some of these theorems. Section 6.4 is about
properties special parallelograms. These include parallelograms with adjacent
angles equal (rectangles) and those with adjacent sides equal (rhombi). It is
pointed out that squares are parallelograms with both properties. Section 6.5
proves and applies conditions for special parallelograms, including examples in
the coordinate plane. The remaining sections of this chapter are devoted to
other special quadrilaterals such as isosceles trapezoids and kites.
Sections 6.2 and 6.3 and Sections 6.4 and 6.5 follow a pattern of paired
sections found often in this text. Certain proofs are given in the first section
of the pair and then converses are developed in second section. Throughout the
chapter, there are mathematically illuminating applications of parallelograms
and special quadrilaterals, from carpentry to mechanical devices (e.g., car
jacks).
In Summary, Holt Geometry includes a full treatment of what is required by the
Standards and a bit more. The mathematics is developed rigorously, with proofs
of theorems based on postulates. Many of the examples and exercises are either
proofs of these theorems or applications of them to geometry problems. In
addition there are examples of applications and some geometry lab experiments
with constructions.
4.2.2 McDougal-Littell Geometry
Chapter 2 (Reasoning and Proof) begins with an extensive Section 2.1 explicitly
on inductive reasoning. This features numerical and geometrical patterns and
examples about data. Section 2.2 addresses conditional statements, including
if-then statements and their converses, contra positives and inverses , and the
relationship between definitions and biconditional statements. Some examples
address perpendicular lines and vertical angles. This section is rather short in
exposition, but there are several pages of exercises. Section 2.3 is about
applications of deductive reasoning, including statements of the Law of
Detachment and the Law of Syllogism. Examples involve mathematics and the real
world , but not much about geometry is proved in this section. An extension
addresses symbolic notation, including the standard arrow notation and truth
tables. Section 2.4 includes a list of postulates about the incidence relations
among points, lines, and planes along with some interesting comments about how
to interpret geometrical diagrams and what can be assumed in diagrams. Solution
of algebraic equations is reviewed in Section 2.5. Section 2.6 (Prove Statements
about Segments and Angles) includes proofs of minor results about lengths of
segments and measure of angles. An example of how to write a two-column proof is
provided in one example. Section 2.7 establishes standard angle pair
relationships, including the congruence of right angles and the vertical angle
theorem. Overall, this chapter presents the rules of logic and proof. However,
the examples and illust rations seem not to go very far in addressing the
difficulties inherent in understanding these concepts. The examples of proofs
are technical and minor, with little geometric interest.
Section 3.1 (Identify Pairs of Lines and Angles) begins with postulates that
state for a given line and a point, there is exactly one line through the point
parallel to the line and one perpendicular to the line. The usual terminology is
defined for pairs of angles formed by two lines and a transversal, but no
theorems are proved in this section. In Section 3.2, a Corresponding Angle
Postulate is stated (even though this is a really a theorem that follows from
the parallel postulate in 3.1). Then three additional congruence theorems (one
example and two exercises) are proved about pairs of angles defined by two
parallels and a transversal. In Section 3.3 the converses of the theorems from
3.2 are proved (sufficient conditions for lines to be parallel). These theorems
are used to prove the important fact that the parallel relation is transitive.
Most of the exercises are immediate applications of the theorems. Sections 3.4
and 3.5 are about equations of lines. In 3.4, slope is defined and there are
postulates that state if-and-only-if conditions on the slope for lines to be
parallel or perpendicular. There is no indication that these properties can in
fact be proved and do not need to be assumed as postulates. Section 3.6 is
devoted to proving theorems about perpendicular lines. There is a proof that a
linear pair of congruent angles is a pair of right angles and relates this to
the real-world consequence of folding paper. Special cases of parallels and
transversals when the transversal is perpendicular are spelled out. One strong
feature of Chapter 3 is the explicit attention to the transitive property of
parallelism. One weakness is the redundancy of assuming a parallel postulate and
then assuming an equivalent statement as a postulate in the next section rather
than proving it as a theorem (or at least noting that it can be done). Another
weakness is the absence of any explanation or proof for the slope properties of
parallels and perpendiculars, or even noting that these properties are really
theorems, not postulates.
The topic of parallelograms appears rather late (Chapter 8), after a chapter on
right angle trigonometry . Section 8.1 states the interior and exterior angle sum
theorems for convex polygons (proofs are exercises). This is a short section
with a few examples and exercises. The problem of finding the angle sum of a
convex polygon is presented as a challenge but the figures supplied as hints and
the answer key are incomplete in that they assume the polygon can be dissected
into triangles, all of which have the same shared vertex. This teacher notes do
not alert the teacher to the underlying mathematical difficulty, so the
opportunity for a more challenging discussion is not
supported. In Section 8.2 the usual properties of a parallelogram are stated and
proved in exercises. In some problems in the coordinate plane, students are
simply told that quadrilaterals are parallelograms, when students could (and
should) verify this fact. Section 8.3 states the four standard necessary
criteria for a quadrilateral to be a parallelogram; the opposite sides congruent
theorem is proved as an example and the others are left to exercises. Here,
there is a demonstration that a quadrilateral in the coordinate plane is a
parallelogram by showing that one pair of sides is congruent and parallel.
Students are asked to use other methods to verify that the quadrilateral is a
parallelogram. An appendix to Section 8.3 is a Problem Solving Workshop that
demonstrates two methods for determining whether or not a figure in the
coordinate plane is a parallelogram. This is a valuable addition to the section.
Section 8.4 contains if-and-only-if conditions for quadrilaterals to be
rhombuses, rectangles, and squares. A Venn diagram shows how the set of squares
is the intersection of the set of rhombuses and the set of rectangles. A
definition of a square is given here, but rectangles and squares have been used
regularly in earlier chapters (e.g., in the proofs of the Pythagorean theorem).
There is no acknowledgement of the earlier appearance of squares when squares
are defined in this chapter. This undercuts the presentation of geometry as an
axiomatic and logical system .
The McDougal-Littell text covers the Washington Standards items checked in this
review, but the impression of the mathematics in this text is mixed. The
reasoning section seems rather shallow, though there is good discussion about
how to reason from figures. The exercises routinely have examples of incorrect
proofs in which students are asked to find the error. There is more attention
than usual devoted to the transitive property of parallelism, and there is an
extra section with explicit examples of multiple solutions of a problem. On the
other hand, most of the exercises are routine or else do not really exploit the
mathematical possibilities of potentially rich problems. Whether or not it is a
good choice to postpone parallelograms and rectangles to the second half of the
text is something that should be considered. Rectangles and squares appear
informally in many earlier places in the text without any explicit efforts to
reconcile the delay of rigorous development. Teachers will have to deal with
possible confusion coming from this departure from logical development.
4.2.3 Glencoe McGraw-Hill Geometry
Chapter 2 addressed reasoning and proof. Section 2.1 presents inductive
reasoning as using examples to form a conclusion that may – as a conjecture –
lead to a prediction. Several contexts are presented, including number
sequences, geometrical figures, and data. Section 2.2 introduces some aspects of
formal logic including truth tables, conjunctions, and disjunctions. (The book
uses this technical terminology for logical “and” and “or.”) Venn diagrams are
also introduced. Section 2.3 is about conditional (ifthen) statements;
mathematical and real world examples are included. The converse, inverse, and
contrapositive are defined, and there is a proof using truth tables showing
which statements are equivalent. There is an extension about bi-conditional
statements. Section 2.4 introduces deductive reasoning, including the Law of
Detachment and the Law of Syllogism. An extensive set of examples is given, some
of which are quite illuminating about the uses of if-then statements and
possible pitfalls in understanding them. A data analysis example used to provide
a contrasting example with inductive reasoning. Section 2.5 is about postulates
and paragraph proofs. Some postulates about the relations among points, lines
and planes are presented and then some proofs are based on these postulates.
This is all correct, but the modest toolkit of postulates at this point limits
the interest and challenge of what can be proved. The chapter concludes with
Sections 2.6 (algebraic proof), 2.7 (proving segment relationships), and 2.8
(proving angle relationships). These sections focus on short proofs of technical
and rather trivial propositions. This writing in this chapter is not a clear
development of the mathematical ideas. Some helpful examples are included,
but others range so far afield that they are a distraction from what is
important for proof in geometry. The chapter may unintentionally communicate
that the goal of proof is to find the right terminology rather than to find
reasons for important mathematical statements. This seems to divert attention
away from the study of geometry. In writing mathematics logically, more
technical detail is not necessarily better. Focus on, and clarity about, the
mathematics content being studied is essential.
Chapter 3 is devoted to parallel and perpendicular lines. Section 3.1 defines
parallel and skew lines, as well as parallel planes, with exercises to find such
lines in a wedge of cheese or a cubical box. Terminology about angle pairs
defined by a transversal is introduced, along with practice using this
terminology. Section 3.2 is about angles and parallel lines. Based on a
postulate about corresponding angles, the congruence of other angle pairs is
proved. The special case of a perpendicular transversal is a theorem, and there
are examples and exercises about angle measures in geometry figures and in
realworld examples. Section 3.3 includes postulates about the slope
relationships for parallel and perpendicular lines; there are no explanations
for why these are true. In Section 3.4, most of the work is finding the
equations of lines through two points, but there is also an example of a line
through a point that is parallel to a given line. In the Geometry Lab at the end
there is a more substantial example developed, which is to find the equation of
the perpendicular bisector of a segment in the coordinate plane.
Section 3.5 is about proving lines are parallel in the plane. Postulate 3.4
asserts that if two lines are cut by a transversal so that all the corresponding
angles are congruent, then the lines are parallel. This is followed by a
description of the construction of a line through a point parallel to a given
line. Then comes Postulate 3.5, which is a version of the Euclidean parallel
postulate. Next are four theorems that state the congruence of a pair of angles
implies that two lines are parallel. The proofs are left to the exercises.
Several aspects of the mathematical development in this section are troubling.
First, Postulate 3.4 is unusual and awkward, since it is sufficient that only
one pair of the corresponding angles be congruent. In fact the statement that
one pair of corresponding angles is missing, though one theorem correctly
asserts that if one pair of congruent alternating interior angles implies the
lines are parallel. Second, there is the curious appearance of the Euclidean
Parallel Postulate. It is stated that the straightedge and compass construction
proves that there is at least one parallel line, but this Postulate is needed to
prove that there is only one. However, the two postulates about corresponding
angles already given are sufficient to prove the Euclidean Parallel Postulate,
so the insertion of this additional postulate is unnecessary and confusing.
Also, the historical note (i.e., Euclid needed only five postulates to prove the
theorems “in his day”) is very odd.
Section 3.6 on perpendiculars and distance begins by asserting without proof
that the distance from a point to a line is the length of the perpendicular
segment from the point to the line. The uniqueness of the perpendicular is
stated as a Postulate in the text, but the fact that the length is minimal is
not justified. At the end of Section 3.6, the concept of distance between two
parallel lines is introduced as the distance from any point on one line to the
other line. This is followed by a detailed example in which the distance between
two parallel lines in the coordinate plane is computed. This section has some
logical difficulties. Early on, an alternate definition of parallel lines is
given; namely, two lines are parallel if they are equidistant. Since the proof
of equidistance depends on rectangle properties that are not yet developed, the
definition can only be stated here without proof. If distance is going to enter
into this chapter, there should at least be a coherent explanation so that it is
clear that there are statements that must be proved later, so that students will
not be confused about the underlying mathematics. Worse, students are asked to
prove that if two lines are equidistant from a third, then the two lines are
parallel. Since the logical development is deficient here, no proof could be
correct.
The answer in the teacher’s edition is based on the coordinate plane, so there
is real confusion about whether a proof is supposed to be in the Euclidean plane
(no coordinates) or in the coordinate plane.
A strong point of this chapter is that after a rather lengthy review of the
various forms of the equation of a line, there are some substantial applications
of the algebra to constructing parallel lines and perpendicular bisectors,
finding distance from a point to a line, and other applications. On the other
hand, the development of angles defined by transversals introduces an unusually
large number of terms for the pairs of angles; the attention necessary for
mastering this terminology diverts the narrative from more important geometric
content. The chapter also provides rather weak support for understanding and
proving, as opposed to memorizing, these properties. It is unfortunate that the
slope properties of parallels and perpendiculars are presented as postulates
rather than as theorems than can be explained and proved (with algebra and at
least informally with geometry). There are some exercises that call for proof,
but there is little support for learning how to write proofs. And the logical
flaws in the development of the parallel postulate and in the treatment of
distance pointed out above detract significantly from the mathematical rigor and
clarity.
Chapter 6 deals with parallelograms and polygons. Section 6.1 presents the
interior and exterior angle sum formulas for a convex polygon. These formulas
are considered in a number of exercises about general polygons and also previews
of some special cases. In Section 6.2 the standard properties of parallelograms
are stated and proved (i.e., one example of a proof, the rest as exercises).
Some examples of parallelogram arms from the real world are shown. In Section
6.3 sufficient conditions for a quadrilateral to be a parallelogram are proved.
Section 6.4 is about rectangles, with a proof of equal diagonals being a
necessary and sufficient condition for a parallelogram to be a rectangle.
Section 6.5 is about rhombi and squares, including the definitions and
properties of the diagonals. This chapter develops the ideas clearly and
correctly, with several examples of proofs provided as models. The inclusion of
examples for the coordinate plane meets the requirements of Performance
Expectation G.4.C.
The Glencoe text covers the topics required by the Washington Standards. In many
places the treatment is clear and correct. But as noted in the section
summaries, there are several instances of logical flaws, a conflation of genuine
postulates and unproved theorems and some confusing mathematical statements that
detract from the text.
