Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
Question 4
![11:03
Gerard A. Venema Foundations of Geometry 2011
PDF - 3 MB
FIGURE 3.35: The Continuity AXiom
ВС. The
define a function that relates the distance and the angle measure. Let d
Ruler Placement Postulate (Theorem 3.2.16) gives a one-to-one correspondence from the
interval [0, d] to points on BC such that C corresponds to 0 and B corresponds to d. Let
Dx be the point that corresponds to the number x; i.e., Dx is the point on BC such that
CDx
= x. Define a function f : [0, d] → [0, µ(LCAB)] by f(x) = µ(LCAD,).
Theorem 3.5.15 (The Continuity Axiom). The function f described in the previous
paragraph is a continuous function, as is the inverse of f.
Proof. Let f be the function described above. By Theorems 3.3.10 and 3.4.5, f is a strictly
increasing function. By the Crossbar Theorem and Theorem 3.4.5, f is onto. Therefore,
f is continuous (Lemma 3.5.14). It is obvious that the inverse of f is increasing and onto,
so the inverse is also continuous.
SES 3.5
1. If l I m, then l and m contain rays that make four different right angles.
2. Prove existence and uniqueness of a perpendicular to a line at a point on the line
(Theorem 3.5.9).
3. Prove existence and uniqueness of perpendicular bisectors (Theorem 3.5.11).
4. Prove that supplements of congruent angles are congruent.
5. Restate the Vertical Angles Theorem (Theorem 3.5.13) in if-then form. Prove the
theorem.
6. Prove the following converse to the Vertical Angles Theorem: If A, B, C, D, and E are
points such that A * B * C, D and E are on opposite sides of AB, and LDBC = LABE,
then D, B, and E are collinear.
7. Use the Continuity Axiom and the Intermediate Value Theorem to prove the Crossbar
Theorem.
HE SIDE-ANGLE-SIDE POSTULATE
So far we have formulated one axiom for each of the undefined terms. It would be
reasonable to expect this to be enough axioms since we now know the basic properties of
each of the undefined terms. But there is still something missing: The postulates stated
so far do not tell us quite enough about how distance (or length of segments) and angle
measure interact with each other. In this section we will give an example that illustrates
the need for additional information and then state one final axiom to complete the picture.
The simplest objects that combine both segments and angles are triangles. We have
defined what it means for two segments to be congruent and what it means for two angles
to be congruent. We now extend that definition to triangles, where the two types of
congruence are combined.
Section 3.6
The Side-Angle-Side Postulate
63
Definition 3.6.1. Two triangles are congruent if there is a correspondence between the
vertices of the first triangle and the vertices of the second triangle such that corresponding
angles are congruent and corresponding sides are congruent.
The assertion that two triangles are congruent is really the assertion that t’
six congruences, three angle congruences and three segment congruences.
F](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F5fc42978-a4c7-40f8-a805-660ee41e0f65%2Fifz9qip_processed.png&w=3840&q=75)
Transcribed Image Text:11:03
Gerard A. Venema Foundations of Geometry 2011
PDF - 3 MB
FIGURE 3.35: The Continuity AXiom
ВС. The
define a function that relates the distance and the angle measure. Let d
Ruler Placement Postulate (Theorem 3.2.16) gives a one-to-one correspondence from the
interval [0, d] to points on BC such that C corresponds to 0 and B corresponds to d. Let
Dx be the point that corresponds to the number x; i.e., Dx is the point on BC such that
CDx
= x. Define a function f : [0, d] → [0, µ(LCAB)] by f(x) = µ(LCAD,).
Theorem 3.5.15 (The Continuity Axiom). The function f described in the previous
paragraph is a continuous function, as is the inverse of f.
Proof. Let f be the function described above. By Theorems 3.3.10 and 3.4.5, f is a strictly
increasing function. By the Crossbar Theorem and Theorem 3.4.5, f is onto. Therefore,
f is continuous (Lemma 3.5.14). It is obvious that the inverse of f is increasing and onto,
so the inverse is also continuous.
SES 3.5
1. If l I m, then l and m contain rays that make four different right angles.
2. Prove existence and uniqueness of a perpendicular to a line at a point on the line
(Theorem 3.5.9).
3. Prove existence and uniqueness of perpendicular bisectors (Theorem 3.5.11).
4. Prove that supplements of congruent angles are congruent.
5. Restate the Vertical Angles Theorem (Theorem 3.5.13) in if-then form. Prove the
theorem.
6. Prove the following converse to the Vertical Angles Theorem: If A, B, C, D, and E are
points such that A * B * C, D and E are on opposite sides of AB, and LDBC = LABE,
then D, B, and E are collinear.
7. Use the Continuity Axiom and the Intermediate Value Theorem to prove the Crossbar
Theorem.
HE SIDE-ANGLE-SIDE POSTULATE
So far we have formulated one axiom for each of the undefined terms. It would be
reasonable to expect this to be enough axioms since we now know the basic properties of
each of the undefined terms. But there is still something missing: The postulates stated
so far do not tell us quite enough about how distance (or length of segments) and angle
measure interact with each other. In this section we will give an example that illustrates
the need for additional information and then state one final axiom to complete the picture.
The simplest objects that combine both segments and angles are triangles. We have
defined what it means for two segments to be congruent and what it means for two angles
to be congruent. We now extend that definition to triangles, where the two types of
congruence are combined.
Section 3.6
The Side-Angle-Side Postulate
63
Definition 3.6.1. Two triangles are congruent if there is a correspondence between the
vertices of the first triangle and the vertices of the second triangle such that corresponding
angles are congruent and corresponding sides are congruent.
The assertion that two triangles are congruent is really the assertion that t’
six congruences, three angle congruences and three segment congruences.
F
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