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9.4 Provide the missing betweenness argu- ments to complete Euclid's proof of (1.7) in the case he considers. Namely, as- suming that the ray AD is in the inte- rior of the angle LCAB, and assuming that D is outside the triangle ABC, prove that CB is in the interior of the angle LACD and DA is in the interior of the angle LCDB. A D B

9.4 Provide the missing betweenness argu- ments to complete Euclid's proof of (1.7) in the case he considers. Namely, as- suming that the ray AD is in the inte- rior of the angle LCAB, and assuming that D is outside the triangle ABC, prove that CB is in the interior of the angle LACD and DA is in the interior of the angle LCDB. A D B

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Geometry] How do you solve 9.4?

9.1 (Difference of angles). Suppose we are given congruent angles LBAC = LB'A'C'. Suppose also that we are given a ray AD in the interior of LBAC. Then there exists a ray A'D' in the interior of LB'A'C' such that LDAC LD'A'C' and LBADLB'A'D'. This statement corresponds to Euclid's Common Notion 3: "Equals subtracted from equals are equal," where "equal" in this case means congruence of angles. 9.2 Suppose the ray AD is in the interior of the angle LBAC, and the ray AE is in the interior of the angle LDAC. Show that AE is also in the interior of LBAC. مراه 9.4 Provide the missing betweenness argu- ments to complete Euclid's proof of (1.7) in the case he considers. Namely, as- suming that the ray AD is in the inte- rior of the angle LCAB, and assuming that D is outside the triangle ABC, prove that CB is in the interior of the angle LACD and DA is in the interior of the angle LCDB. A A B с A B D 9.3 Consider the real Cartesian plane where congruence of line segments is given by the absolute value distance function (Exercise 8.7). Using the usual congruence of angles that you know from analytic geometry (Section 16), show that (C4) and (C5) hold in this model, but that (C6) fails. (Give a counterexample.) E D B B
Transcribed Image Text:9.1 (Difference of angles). Suppose we are given congruent angles LBAC = LB'A'C'. Suppose also that we are given a ray AD in the interior of LBAC. Then there exists a ray A'D' in the interior of LB'A'C' such that LDAC LD'A'C' and LBADLB'A'D'. This statement corresponds to Euclid's Common Notion 3: "Equals subtracted from equals are equal," where "equal" in this case means congruence of angles. 9.2 Suppose the ray AD is in the interior of the angle LBAC, and the ray AE is in the interior of the angle LDAC. Show that AE is also in the interior of LBAC. مراه 9.4 Provide the missing betweenness argu- ments to complete Euclid's proof of (1.7) in the case he considers. Namely, as- suming that the ray AD is in the inte- rior of the angle LCAB, and assuming that D is outside the triangle ABC, prove that CB is in the interior of the angle LACD and DA is in the interior of the angle LCDB. A A B с A B D 9.3 Consider the real Cartesian plane where congruence of line segments is given by the absolute value distance function (Exercise 8.7). Using the usual congruence of angles that you know from analytic geometry (Section 16), show that (C4) and (C5) hold in this model, but that (C6) fails. (Give a counterexample.) E D B B
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