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
Related questions
Question
Question 1
![(SAS).
Case 4.
Н* А * В.
The proof of this case is
This time
very much like the proof of Case 3.
A is in the interior of LHCB and ZHGB (see Fig-
ure 4.10). Thus µ(LACB) :
and u(LAGB)
tor Postulate, Part 4). But µ(LACH) = µ(LAGH)
and µ(LHCB) = µ(LHCB) (Isosceles Triangle The-
orem). Thus ZACB = LAGB (subtraction) and so
ΔΑΒC ΔΑΒG (SAS ) .
µ(LHCB)
u(LHGB) – e(LHGA) (Protrac-
- μ(LHCA).
%3D
H
Case 5. A * B * H. The proof of this case is similar to
the proof of Case 4.
In each of the five cases we have proved that
FIGURE 4.10: Case 4, H * A * B
AABC = AABG. But AABG = ADEF, so the proof is complete.
Section 4.3
Three inequalities for triangles 77
Side-Side-Side is Euclid's Proposition 8. Euclid's proof is somewhat clumsy, relying
on a rather convoluted lemma (Proposition 7). Heath attributes the more elegant proof,
above, to Philo and Proclus (see [22], page 263). SSS can also be proved as a corollary of
the Hinge Theorem (see Exercise 4.3.5), but that proof is not considered to be as elegant
as the proof given here.
Before leaving the subject of triangle congruence conditions we should mention that
in neutral geometry the three angles of a triangle do not determine the triangle. This
may seem obvious to you since you remember from high school that similar triangles are
usually not congruent. However, that is a special situation in Euclidean geometry. One
of the surprising results we will prove in hyperbolic geometry is that (in that context)
Angle-Angle-Angle is a valid triangle congruence condition!
EXERCISES 4.2
1. Prove the Converse to the Isosceles Triangle Theorem (Theorem 4.2.2).
2. Prove the Angle-Angle-Side Triangle Congruence Condition (Theorem 4.2.3).
3. Suppose AABC and ADEF are two triangles such that ZBAC = LEDF, AC = DF,
and CB = FE (the hypotheses of ASS). Prove that either ZABC and ZDEF are
congruent or they are supplements.
4. Prove the Hypotenuse-Leg Theorem (Theorem 4.2.5).
5. Prove that it is possible to construct a congruent copy of a triangle on a given base
(Theorem 4.2.6).
4.3 THREE INEQUALITIES FOR TRIANGLES
The Exterior Angle Theorem gives one inequality that is always satisfied by the measures
of the angles of a triangle. In this section we prove three additional inequalities that will be
useful in our study of triangles. The first of these theorems, the Scalene Inequality, extends
the Isosceles Triangle Theorem and its converse. It combines Euclid's Propositions 18
and 19. The word scalene means "unequal" or “uneven." A scalene triangle is a triangle
that has sides of three different lengths.
Theorem 4.3.1 (Scalene Inequality). In any triangle, the greater side lies opposite the
greater angle and the greater angle lies opposite the greater side.
Restatement. Let AABC be a triangle. Then AB > BC if and only if µ(LACB) >
u(LBAC).
Proof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume
the hypothesis AB > BC and prove that µ(LACB) > µ(LBAC). Since AB > BC, there
exists a point D between A and B such that BD = BC (Ruler Postulate).
A
B
FIGURE 4.11: The greater side lies opposite greater angle
Now µ(LACB) > µ(LDCB) (Protractor Postulate, Part 4, and Theorem 3.3.10)
and ZDCB = ZCDB (Isosceles Triangle Theorem). But LCDB is an exterior angle for
78
Chapter 4
Neutral Geometry
AADC (see Figure 4.11), so µ(LCDB) > µ(LCAB) (Exterior Angle Theorem). The
conclusion follows from those inequalities.
The proof of the converse is left as an exercise (Exercise 1).
The second inequality is the familiar Triangle Inequality. It is Euclid's Proposition 20.
Theorem 4.3.2 (Triangle Inequality). If A, B, and C are three noncollinear points, then
АС < АВ + BС.
Proof. Exercise 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F9658e1cb-3627-4986-a3a8-287a76a97672%2F11wco7j_processed.png&w=3840&q=75)
Transcribed Image Text:(SAS).
Case 4.
Н* А * В.
The proof of this case is
This time
very much like the proof of Case 3.
A is in the interior of LHCB and ZHGB (see Fig-
ure 4.10). Thus µ(LACB) :
and u(LAGB)
tor Postulate, Part 4). But µ(LACH) = µ(LAGH)
and µ(LHCB) = µ(LHCB) (Isosceles Triangle The-
orem). Thus ZACB = LAGB (subtraction) and so
ΔΑΒC ΔΑΒG (SAS ) .
µ(LHCB)
u(LHGB) – e(LHGA) (Protrac-
- μ(LHCA).
%3D
H
Case 5. A * B * H. The proof of this case is similar to
the proof of Case 4.
In each of the five cases we have proved that
FIGURE 4.10: Case 4, H * A * B
AABC = AABG. But AABG = ADEF, so the proof is complete.
Section 4.3
Three inequalities for triangles 77
Side-Side-Side is Euclid's Proposition 8. Euclid's proof is somewhat clumsy, relying
on a rather convoluted lemma (Proposition 7). Heath attributes the more elegant proof,
above, to Philo and Proclus (see [22], page 263). SSS can also be proved as a corollary of
the Hinge Theorem (see Exercise 4.3.5), but that proof is not considered to be as elegant
as the proof given here.
Before leaving the subject of triangle congruence conditions we should mention that
in neutral geometry the three angles of a triangle do not determine the triangle. This
may seem obvious to you since you remember from high school that similar triangles are
usually not congruent. However, that is a special situation in Euclidean geometry. One
of the surprising results we will prove in hyperbolic geometry is that (in that context)
Angle-Angle-Angle is a valid triangle congruence condition!
EXERCISES 4.2
1. Prove the Converse to the Isosceles Triangle Theorem (Theorem 4.2.2).
2. Prove the Angle-Angle-Side Triangle Congruence Condition (Theorem 4.2.3).
3. Suppose AABC and ADEF are two triangles such that ZBAC = LEDF, AC = DF,
and CB = FE (the hypotheses of ASS). Prove that either ZABC and ZDEF are
congruent or they are supplements.
4. Prove the Hypotenuse-Leg Theorem (Theorem 4.2.5).
5. Prove that it is possible to construct a congruent copy of a triangle on a given base
(Theorem 4.2.6).
4.3 THREE INEQUALITIES FOR TRIANGLES
The Exterior Angle Theorem gives one inequality that is always satisfied by the measures
of the angles of a triangle. In this section we prove three additional inequalities that will be
useful in our study of triangles. The first of these theorems, the Scalene Inequality, extends
the Isosceles Triangle Theorem and its converse. It combines Euclid's Propositions 18
and 19. The word scalene means "unequal" or “uneven." A scalene triangle is a triangle
that has sides of three different lengths.
Theorem 4.3.1 (Scalene Inequality). In any triangle, the greater side lies opposite the
greater angle and the greater angle lies opposite the greater side.
Restatement. Let AABC be a triangle. Then AB > BC if and only if µ(LACB) >
u(LBAC).
Proof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume
the hypothesis AB > BC and prove that µ(LACB) > µ(LBAC). Since AB > BC, there
exists a point D between A and B such that BD = BC (Ruler Postulate).
A
B
FIGURE 4.11: The greater side lies opposite greater angle
Now µ(LACB) > µ(LDCB) (Protractor Postulate, Part 4, and Theorem 3.3.10)
and ZDCB = ZCDB (Isosceles Triangle Theorem). But LCDB is an exterior angle for
78
Chapter 4
Neutral Geometry
AADC (see Figure 4.11), so µ(LCDB) > µ(LCAB) (Exterior Angle Theorem). The
conclusion follows from those inequalities.
The proof of the converse is left as an exercise (Exercise 1).
The second inequality is the familiar Triangle Inequality. It is Euclid's Proposition 20.
Theorem 4.3.2 (Triangle Inequality). If A, B, and C are three noncollinear points, then
АС < АВ + BС.
Proof. Exercise 2.
Transcribed Image Text:74
Chapter 4
Neutral Geometry
2. Let AABC be the spherical triangle shown in Figure 4.3. Perform the construction in
the proof of Theorem 4.1.2 on the sphere, starting with this triangle. Convince yourself
that the point F constructed is in the interior of LACD if and only if u(LBAC) < 90°.
Draw diagrams illustrating both possibilities.
4.2 TRIANGLE CONGRUENCE CONDITIONS
The statement that two triangles are congruent means that the three interior angles of
the first triangle are congruent to the corresponding angles in the second and that the
three sides of the first triangle are congruent to the corresponding sides of the second.
The Side-Angle-Side Postulate indicates that it is sometimes possible to conclude all six
of these congruences from only three of them. SAS is just the first of several similar
results, which are known as triangle congruence conditions. In this section we build on the
Side-Angle-Side Postulate to prove the other familiar triangle congruence conditions. We
begin with the Angle-Side-Angle triangle congruence condition. It is one half of Euclid's
Proposition 26; the other half (Angle-Angle-Side) will be left as an exercise.
Theorem 4.2.1 (ASA). If two angles and the included side of one triangle are congruent to
the corresponding parts of a second triangle, then the two triangles are congruent.
Restatement. IfAABC and ADEF are two triangles such that ZCAB = LFDE, AB = DE,
and LABC = LDEF, then AABC = ADEF.
Proof. Let AABC and ADEF be two triangles such that ZCAB = LFDE, AB = DE,
and LABC = LDEF (hypothesis). We must show that AABC = ADEF.
A
В D
E
FIGURE 4.6: One possible location for C' in proof of ASA
There exists a point C' on AČ such that AC' = DF (Point Construction Postulate).
Now AABC' =ADEF (SAS) and so LABC' = LDEF (definition of congruent triangles).
Since ZABC = LDEF (hypothesis), we can conclude that ZABC = LABC'. Hence
B = BC' (Protractor Postulate, Part 3). But BC can only intersect AC in at most one
point (Theorem 3.1.7), so C = C' and the proof is complete.
Angle-Side-Angle can be used to prove the converse to the Isosceles Triangle
Theorem, which is Euclid’s Proposition 6.
Theorem 4.2.2 (Converse to the Isosceles Triangle Theorem). If AABC is a triangle such
that LA BC 2 LACB, then АВ — АС.
Proof. Exercise 1.
Theorem 4.2.3 (AAS). If AABC and ADEF are two triangles such that LABC = LDEF,
ZBCA = LEFD, and AC = DF, then AABC = ADEF.
Section 4.2
Triangle congruence conditions 75
Proof. Exercise 2.
The two conditions ASA and AAS show that any two angles plus a side determine
the entire triangle. Two sides plus one angle may or may not determine the triangle.
If the angle is included between the two sides, then SAS implies that the triangle is
determined. If the angle is not included between the sides, then the triangle is not
completely determined and therefore ASS is not a valid congruence condition. The
failure of Angle-Side-Side is illustrated in Figure 4.7. Even though the figure makes it
clear that ASS is not a valid triangle congruence condition, two triangles that satisfy the
hypotheses of Angle-Side-Side must be closely related-the relationship between them is
specified in Exercise 3.
C
A
В
A
В
FIGURE 4.7: There is no Angle-Side-Side Theorem
There is one significant special case in which Angle-Side-Side does hold; that is the
case in which the given angle is a right angle. The theorem is known as the Hypotenuse-Leg
Theorem.
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