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5. Suppose AABC is a triangle and l is a line such that none of the vertices A, B, or C lies on l. Prove that l cannot intersect all three sides of AABC. Is it possible for a line to intersect all three sides of a triangle?

5. Suppose AABC is a triangle and l is a line such that none of the vertices A, B, or C lies on l. Prove that l cannot intersect all three sides of AABC. Is it possible for a line to intersect all three sides of a triangle?

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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7:34 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB none of the points A, B, and C lies on l (hypothesis). Let H1 and H2 be the two half-planes determined by l (Axiom 3.3.2). The points A and B are in opposite half- planes (hypothesis and Proposition 3.3.4). Let us say that A E Hj and B E H2 (notation). It must be the case that either C is in H1 or C is in H2 (Axiom 3.3.2). If CE H2, then AC ne#Ø (Axiom 3.3.2, Part 2). If C Е H, then BC Nеt 0 (Axiom 3.3.2, Part 2). В FIGURE 3.17: Pasch's Axiom SES 3.3 1. Prove that the intersection of two convex sets is convex. Show by example that the union of two convex sets need not be convex. Is the empty set convex? 2. Let A and B be two distinct points. Prove that each of the sets {A}, AB, AB, and AB is a convex set. 3. Let l be a line and let H be one of the half-planes bounded by l. Prove that H U l is a convex set. 4. Betweenness on the sphere. In Exercise 3.2.6, betweenness and segments were defined for the sphere S². Define ray in the usual way (Definition 3.2). A great circle divides the sphere into two hemispheres, which we could think of as half-planes determined by the great circle. (a) Does the Plane Separation Postulate hold in this setting? Explain. (b) Let A and B be distinct, nonantipodal points on S?. Find all points C such that AB. A * B * C. Sketch the ray (c) Does Theorem 3.3.9 hold in this setting? Explain. 5. Suppose AABC is a triangle and l is a line such that none of the vertices A, B, or C lies on l. Prove that l cannot intersect all three sides of AABC. Is it possible for a line to intersect all three sides of a triangle? 6. Prove that the vertex of an angle is well defined (i.e., if ZBAC = LEDF, then A 7. Prove that the vertices and edges of a triangle are well defined (i.e., if AABC = ADEF, then {A, B, C} = {D, E, F}). D). %3D NGLE MEASURE AND THE PROTRACTOR POSTULATE The fifth axiom spells out the properties of angle measure (the last undefined term). Axiom 3.4.1 (The Protractor Postulate). For every angle LBAC there is a real number µ(LBAC), called the measure of LBAC, such that the following conditions are satisfied. 1. 0° < µ(LBAC) < 180° for every angle LBAC. 52 Chapter 3 Axioms for Plane Geometry AĆ. 2. µ(LBAC) = 0° if and only if AB 3. (Angle Construction Postulate) For each real number r, 0 <r < 180, and for each half-plane H bounded by AB there exists a unique ray AÉ such that E is in H and u(LBAE) = r°. 4. (Angle Addition Postulate) If the ray AD is between rays AB and AĆ, then u(LBAD) + µ(LDAC) = µ(LBAC). In Part 3 of the postulate, it is the ray that is unique, not the point E. Many dif points determine the same ray. See Figure 3.13 for an illustration of the angles in P We are familiar with the measurement of angles from previous mathematics co and it is intuitively clear that a measurement function like that described in the Protractor Postulate exists in the Cartesian plane. It is not easy to write down a precise formula, however and we make no attemnt to do so at this noint Most of this book is devoted to.
Transcribed Image Text:7:34 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB none of the points A, B, and C lies on l (hypothesis). Let H1 and H2 be the two half-planes determined by l (Axiom 3.3.2). The points A and B are in opposite half- planes (hypothesis and Proposition 3.3.4). Let us say that A E Hj and B E H2 (notation). It must be the case that either C is in H1 or C is in H2 (Axiom 3.3.2). If CE H2, then AC ne#Ø (Axiom 3.3.2, Part 2). If C Е H, then BC Nеt 0 (Axiom 3.3.2, Part 2). В FIGURE 3.17: Pasch's Axiom SES 3.3 1. Prove that the intersection of two convex sets is convex. Show by example that the union of two convex sets need not be convex. Is the empty set convex? 2. Let A and B be two distinct points. Prove that each of the sets {A}, AB, AB, and AB is a convex set. 3. Let l be a line and let H be one of the half-planes bounded by l. Prove that H U l is a convex set. 4. Betweenness on the sphere. In Exercise 3.2.6, betweenness and segments were defined for the sphere S². Define ray in the usual way (Definition 3.2). A great circle divides the sphere into two hemispheres, which we could think of as half-planes determined by the great circle. (a) Does the Plane Separation Postulate hold in this setting? Explain. (b) Let A and B be distinct, nonantipodal points on S?. Find all points C such that AB. A * B * C. Sketch the ray (c) Does Theorem 3.3.9 hold in this setting? Explain. 5. Suppose AABC is a triangle and l is a line such that none of the vertices A, B, or C lies on l. Prove that l cannot intersect all three sides of AABC. Is it possible for a line to intersect all three sides of a triangle? 6. Prove that the vertex of an angle is well defined (i.e., if ZBAC = LEDF, then A 7. Prove that the vertices and edges of a triangle are well defined (i.e., if AABC = ADEF, then {A, B, C} = {D, E, F}). D). %3D NGLE MEASURE AND THE PROTRACTOR POSTULATE The fifth axiom spells out the properties of angle measure (the last undefined term). Axiom 3.4.1 (The Protractor Postulate). For every angle LBAC there is a real number µ(LBAC), called the measure of LBAC, such that the following conditions are satisfied. 1. 0° < µ(LBAC) < 180° for every angle LBAC. 52 Chapter 3 Axioms for Plane Geometry AĆ. 2. µ(LBAC) = 0° if and only if AB 3. (Angle Construction Postulate) For each real number r, 0 <r < 180, and for each half-plane H bounded by AB there exists a unique ray AÉ such that E is in H and u(LBAE) = r°. 4. (Angle Addition Postulate) If the ray AD is between rays AB and AĆ, then u(LBAD) + µ(LDAC) = µ(LBAC). In Part 3 of the postulate, it is the ray that is unique, not the point E. Many dif points determine the same ray. See Figure 3.13 for an illustration of the angles in P We are familiar with the measurement of angles from previous mathematics co and it is intuitively clear that a measurement function like that described in the Protractor Postulate exists in the Cartesian plane. It is not easy to write down a precise formula, however and we make no attemnt to do so at this noint Most of this book is devoted to.
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