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SES 4.3 1. Complete the proof of the Scalene Inequality (Theorem 4.3.1). Provo the Trionoelo Inoguolity (The om 43 2)

SES 4.3 1. Complete the proof of the Scalene Inequality (Theorem 4.3.1). Provo the Trionoelo Inoguolity (The om 43 2)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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11:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB required. В D, D. D, A FIGURE 4.17: Two possible locations for Dy in the proof of continuity SES 4.3 1. Complete the proof of the Scalene Inequality (Theorem 4.3.1). 2. Prove the Triangle Inequality (Theorem 4.3.2). 3. Prove the following result: A, B, and C are three points such that AB + BC = A, B, and C are collinear. (It follows that the assumption in the definition of between that A, B, and C are collinear is redundant.) АС, then 4. Prove: If A, B, and C are any three points (collinear or not), then AB + BC > AC. 5. Use the Hinge Theorem to prove SSS. 6. Prove that the hypotenuse is always the longest side of a right triangle. 7. Prove that the shortest distance from a point to a line is measured along the perpendic- ular (Theorem 4.3.4). 8. Prove the Pointwise Characterization of Angle Bisectors (Theorem 4.3.6). 9. Prove the Pointwise Characterization of Perpendicular Bisectors (Theorem 4.3.7). 82 Chapter 4 Neutral Geometry HE ALTERNATE INTERIOR ANGLES THEOREM We come now to our first theorem about parallel lines. The Alternate Interior An- gles Theorem is one of the major theorems of neutral geometry. It is also Euclid's Proposition 27. Before we can state the theorem we need some definitions and notation. Definition 4.4.1. Let l and l' be two distinct lines. A third line t is called a transversal for l and l' if t intersects l in one point B and t intersects l' in one point B' with B' + B. Notice that there are four angles with vertex B that are formed by rays on t and l and there are four angles with vertex B' that are formed by rays on t and l'. We will give names to some of these eight angles. In order to do so, choose points A and C on l and A' and C' on l' so that A * B * C, A' * B' * C', and A and A' are on the same side of t. (Such points exist by the Ruler and Plane Separation Postulates.) The four angles LABB', ZA' B' B, LC BB', and LC' B' B are called interior angles for l and l' with transversal t. The two pairs {LABB', LBB'C'} and {LA'B' B, LB' BC} are called alternate interior angles. See Figure 4.18. A В
Transcribed Image Text:11:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB required. В D, D. D, A FIGURE 4.17: Two possible locations for Dy in the proof of continuity SES 4.3 1. Complete the proof of the Scalene Inequality (Theorem 4.3.1). 2. Prove the Triangle Inequality (Theorem 4.3.2). 3. Prove the following result: A, B, and C are three points such that AB + BC = A, B, and C are collinear. (It follows that the assumption in the definition of between that A, B, and C are collinear is redundant.) АС, then 4. Prove: If A, B, and C are any three points (collinear or not), then AB + BC > AC. 5. Use the Hinge Theorem to prove SSS. 6. Prove that the hypotenuse is always the longest side of a right triangle. 7. Prove that the shortest distance from a point to a line is measured along the perpendic- ular (Theorem 4.3.4). 8. Prove the Pointwise Characterization of Angle Bisectors (Theorem 4.3.6). 9. Prove the Pointwise Characterization of Perpendicular Bisectors (Theorem 4.3.7). 82 Chapter 4 Neutral Geometry HE ALTERNATE INTERIOR ANGLES THEOREM We come now to our first theorem about parallel lines. The Alternate Interior An- gles Theorem is one of the major theorems of neutral geometry. It is also Euclid's Proposition 27. Before we can state the theorem we need some definitions and notation. Definition 4.4.1. Let l and l' be two distinct lines. A third line t is called a transversal for l and l' if t intersects l in one point B and t intersects l' in one point B' with B' + B. Notice that there are four angles with vertex B that are formed by rays on t and l and there are four angles with vertex B' that are formed by rays on t and l'. We will give names to some of these eight angles. In order to do so, choose points A and C on l and A' and C' on l' so that A * B * C, A' * B' * C', and A and A' are on the same side of t. (Such points exist by the Ruler and Plane Separation Postulates.) The four angles LABB', ZA' B' B, LC BB', and LC' B' B are called interior angles for l and l' with transversal t. The two pairs {LABB', LBB'C'} and {LA'B' B, LB' BC} are called alternate interior angles. See Figure 4.18. A В
11:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB uuurU, tu I HI v unu IUVIUD (-15 pugo 20J DU vUI UIJU U pruruu uv u VUIVHUI y un 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! ERCISES 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 LABC 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). 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 u(LACB) > pe(LBAC). Proof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume the hypothesis AB > BC and prove that u(LACB) > µ(LBAC). Since AB > BC, there exists a point D between A and B such that BD = BC (Ruler Postulate). A D В FIGURE 4.11: The greater side lies opposite greater angle Now u(LACB) > µ(LDCB) (Protractor Postulate, Part 4, and Theorem 3.3.10) and ZDCB = LCDB (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 The.. he conclusion follows from those inequalities. The proof of the converse is left as an exercise (Exercise 1).
Transcribed Image Text:11:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB uuurU, tu I HI v unu IUVIUD (-15 pugo 20J DU vUI UIJU U pruruu uv u VUIVHUI y un 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! ERCISES 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 LABC 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). 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 u(LACB) > pe(LBAC). Proof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume the hypothesis AB > BC and prove that u(LACB) > µ(LBAC). Since AB > BC, there exists a point D between A and B such that BD = BC (Ruler Postulate). A D В FIGURE 4.11: The greater side lies opposite greater angle Now u(LACB) > µ(LDCB) (Protractor Postulate, Part 4, and Theorem 3.3.10) and ZDCB = LCDB (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 The.. he conclusion follows from those inequalities. The proof of the converse is left as an exercise (Exercise 1).
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