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Modern Geometry

Saccheri quadrilateral

Let A and B be not on the same side of CD. Let AB cuts CD at F.

contradiction

As ABCD is a Saccheri quadrilateral so BAD =90? and CDA=90?. Now one of the triangles DAF and BFC contains a right angle and an obtuse angle because one of the linear pair of angles of F must be an obtuse or a right angle. In this case, angles FAD=90? and FDA=90?. As a result of which, the sum of the three angles of a triangle will be more than 180?. Contradiction!!!! Hence A and B must lie on the same side of CD

Let C and D be not on the same side of AB. Let CD cuts AB at E. 

sum of three angles

As ABCD is a Saccheri quadrilateral so BAD =90? and CDA=90?. Now one of the triangles DAE and BEC contains a right angle and an obtuse angle because one of the linear pair of angles of E must be an obtuse or a right angle. In this case angles EAD=90? and EDA=90?. So sum of three angles in Triangle EAD is more than 180?. Contradiction!!!!
Hence C and D must lie on the same side of AB

Let A and D be not on the same side of BC. Let AD cuts BC at F. 

equation

As ABCD is a Saccheri quadrilateral so BAD =90? and CDA=90?. Now one of the triangles DCF and BAF contains a right angle and an obtuse angle because one of the linear pair of angles of F must be an obtuse or a right angle. In this case angles FDC=90? and CFD≥90?. So sum of three angles in Triangle CDF is more than 180?. Contradiction!!!! Hence A and D must lie on the same side of BC.

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My Assignment Help (2021) Proof That A And B Must Lie On The Same Side Of CD [Online]. Available from: https://myassignmenthelp.com/free-samples/math5376-modern-geometry/pair-of-angles.html
[Accessed 24 February 2024].

My Assignment Help. 'Proof That A And B Must Lie On The Same Side Of CD' (My Assignment Help, 2021) <https://myassignmenthelp.com/free-samples/math5376-modern-geometry/pair-of-angles.html> accessed 24 February 2024.

My Assignment Help. Proof That A And B Must Lie On The Same Side Of CD [Internet]. My Assignment Help. 2021 [cited 24 February 2024]. Available from: https://myassignmenthelp.com/free-samples/math5376-modern-geometry/pair-of-angles.html.

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