4. We work in absolute geometry. (a) Suppose A, B and P are non-collinear and drop the per- pendicular from P to Q Є AB. If P lies between the perpendiculars l, m to AB through A and B, prove that Q is interior to AB. (Hint: show that the other cases are impossible) (b) Suppose there exists a triangle with angle sum 180°. Show A that there exists a right-triangle with angle sum 180° and P Q B therefore a rectangle. (Since rectangles are impossible in hyperbolic geometry, this proves part 2 of Theorem 4.19)
4. We work in absolute geometry. (a) Suppose A, B and P are non-collinear and drop the per- pendicular from P to Q Є AB. If P lies between the perpendiculars l, m to AB through A and B, prove that Q is interior to AB. (Hint: show that the other cases are impossible) (b) Suppose there exists a triangle with angle sum 180°. Show A that there exists a right-triangle with angle sum 180° and P Q B therefore a rectangle. (Since rectangles are impossible in hyperbolic geometry, this proves part 2 of Theorem 4.19)
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
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Transcribed Image Text:4. We work in absolute geometry.
(a) Suppose A, B and P are non-collinear and drop the per-
pendicular from P to Q Є AB.
If P lies between the perpendiculars l, m to AB through A
and B, prove that Q is interior to AB.
(Hint: show that the other cases are impossible)
(b) Suppose there exists a triangle with angle sum 180°. Show A
that there exists a right-triangle with angle sum 180° and
P
Q
B
therefore a rectangle.
(Since rectangles are impossible in hyperbolic geometry, this proves part 2 of Theorem 4.19)
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