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Theorem, which is Euclid's Proposition 6. Theorem 4.2.2 (Converse to the Isosceles Triangle Theorem). If AABC is a triangle such that LABC = LACB, then AB = AC. Proof. Exercise 1.

Theorem, which is Euclid's Proposition 6. Theorem 4.2.2 (Converse to the Isosceles Triangle Theorem). If AABC is a triangle such that LABC = LACB, then AB = AC. Proof. Exercise 1.

Elementary Geometry For College Students, 7e
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
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4.2 question 1

u(LHGB). Thus ZACB = LAGB (addition) and so AABC = AABG LHCB) AS). Case 4. Н* А * В. The proof of this case is very much like the proof of Case 3. A is in the interior of LHCB and LHGB (see Fig- ure 4.10). Thus µ(LACB) = µ(LHCB) and µ(LAGB) tor Postulate, Part 4). But u(LACH) and µ(ZHCB) orem). Thus ZACB = LAGB (subtraction) and so ΔΑΒC ΔΑΒG (SAS). 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 This time u(LHCA) u(LHGA) (Protrac- u(LAGH) u(LHCB) (Isosceles Triangle The- A e(LHGB) H B FIGURE 4.10: Case 4, H * A * B ABC = AABG. But AABG = ADEF, so the proof is complete. Section 4.3 Three inequalities for triangles 7 Side-Side-Side is Euclid's Proposition 8. Euclid's proof is somewhat clumsy, relying a rather convoluted lemma (Proposition 7). Heath attributes the more elegant proof, ove, to Philo and Proclus (see [22], page 263). SSS can also be proved as a corollary of e Hinge Theorem (see Exercise 4.3.5), but that proof is not considered to be as elegant the proof given here. Before leaving the subject of triangle congruence conditions we should mention that neutral geometry the three angles of a triangle do not determine the triangle. This ay seem obvious to you since you remember from high school that similar triangles are ually not congruent. However, that is a special situation in Euclidean geometry. One the surprising results we will prove in hyperbolic geometry is that (in that context) ngle-Angle-Angle is a valid triangle congruence condition! 5 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 = and CB DF, 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). EE INEQUALITIES FOR TRIANGLES e Exterior Angle Theorem gives one inequality that is always satisfied by the measures the angles of a triangle. In this section we prove three additional inequalities that will be eful in our study of triangles. The first of these theorems, the Scalene Inequality, extends e Isosceles Triangle Theorem and its converse. It combines Euclid's Propositions 18 d 19. The word scalen at has sides of three different lengths. means "unequal" or "uneven." A scalene triangle is a triangle eorem 4.3.1 (Scalene Inequality). In any triangle, the greater side lies opposite the eater angle and the greater angle lies opposite the greater side. statement. Let AABC be a triangle. Then AB > BC if and only if µ(LACB) LBAC). oof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume e hypothesis AB > BC and prove that u(LACB) > µ(LBAC). Since AB > BC, there
Transcribed Image Text:u(LHGB). Thus ZACB = LAGB (addition) and so AABC = AABG LHCB) AS). Case 4. Н* А * В. The proof of this case is very much like the proof of Case 3. A is in the interior of LHCB and LHGB (see Fig- ure 4.10). Thus µ(LACB) = µ(LHCB) and µ(LAGB) tor Postulate, Part 4). But u(LACH) and µ(ZHCB) orem). Thus ZACB = LAGB (subtraction) and so ΔΑΒC ΔΑΒG (SAS). 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 This time u(LHCA) u(LHGA) (Protrac- u(LAGH) u(LHCB) (Isosceles Triangle The- A e(LHGB) H B FIGURE 4.10: Case 4, H * A * B ABC = AABG. But AABG = ADEF, so the proof is complete. Section 4.3 Three inequalities for triangles 7 Side-Side-Side is Euclid's Proposition 8. Euclid's proof is somewhat clumsy, relying a rather convoluted lemma (Proposition 7). Heath attributes the more elegant proof, ove, to Philo and Proclus (see [22], page 263). SSS can also be proved as a corollary of e Hinge Theorem (see Exercise 4.3.5), but that proof is not considered to be as elegant the proof given here. Before leaving the subject of triangle congruence conditions we should mention that neutral geometry the three angles of a triangle do not determine the triangle. This ay seem obvious to you since you remember from high school that similar triangles are ually not congruent. However, that is a special situation in Euclidean geometry. One the surprising results we will prove in hyperbolic geometry is that (in that context) ngle-Angle-Angle is a valid triangle congruence condition! 5 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 = and CB DF, 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). EE INEQUALITIES FOR TRIANGLES e Exterior Angle Theorem gives one inequality that is always satisfied by the measures the angles of a triangle. In this section we prove three additional inequalities that will be eful in our study of triangles. The first of these theorems, the Scalene Inequality, extends e Isosceles Triangle Theorem and its converse. It combines Euclid's Propositions 18 d 19. The word scalen at has sides of three different lengths. means "unequal" or "uneven." A scalene triangle is a triangle eorem 4.3.1 (Scalene Inequality). In any triangle, the greater side lies opposite the eater angle and the greater angle lies opposite the greater side. statement. Let AABC be a triangle. Then AB > BC if and only if µ(LACB) LBAC). oof. Let A, B, and C be three noncollinear points (hypothesis). We will first assume e hypothesis AB > BC and prove that u(LACB) > µ(LBAC). Since AB > BC, there
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 andADEF are two triangles such that LC AB = 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. F A В D E FIGURE 4.6: One possible location for C' in proof of ASA There exists a point C' on AC such that AC' = DF (Point Construction Postulate). Now AABC' = ADEF (SAS) and so LABC' = LDEF (definition of congruent triangles). Since LABC = LDEF (hypothesis), we can conclude that ZABC = LABC'. Hence = BC' (Protractor Postulate, Part 3). But BĆ can only intersect AC in at most one point (Theorem 3.1.7), so C = C' and the proof is complete. ВС 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 LABC = LACB, then AB = AC. Proof. Exercise 1. Theorem 4.2.3 (AAS). If AABC and ADEF are two triangles such that LABC = LDEF, ZBC A = 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. 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. Definition 4.2.4. A triangle is a right triangle if one of the interior angles is a right angle. The side opposite the right angle is called the hypotenuse and the two sides adjacent to the right angle are called legs.
Transcribed Image Text: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 andADEF are two triangles such that LC AB = 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. F A В D E FIGURE 4.6: One possible location for C' in proof of ASA There exists a point C' on AC such that AC' = DF (Point Construction Postulate). Now AABC' = ADEF (SAS) and so LABC' = LDEF (definition of congruent triangles). Since LABC = LDEF (hypothesis), we can conclude that ZABC = LABC'. Hence = BC' (Protractor Postulate, Part 3). But BĆ can only intersect AC in at most one point (Theorem 3.1.7), so C = C' and the proof is complete. ВС 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 LABC = LACB, then AB = AC. Proof. Exercise 1. Theorem 4.2.3 (AAS). If AABC and ADEF are two triangles such that LABC = LDEF, ZBC A = 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. 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. Definition 4.2.4. A triangle is a right triangle if one of the interior angles is a right angle. The side opposite the right angle is called the hypotenuse and the two sides adjacent to the right angle are called legs.
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