Let A, B and C not be on the exact same side of CD.

Let AB trim CD at F.

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MATH 5376 Modern Geometry

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ABCD being a Saccheri quadrilateral means that BAD =90

CDA=90

Because one of the angles DAF and BFC contain a right-angle and an obtuse, one of F’s linear pairs must be either an right or an obtuse.

In this example, angles FAD=90

FDA=90 and FAD=90 respectively.

This means that the sum of the triangle’s angles will exceed 180?

Therefore, A and B must be on the same CD side.

Let C, and D not be on opposite sides of AB.

Let CD be cut at E.

ABCD being a Saccheri quadrilateral means that BAD =90

CDA=90

Because one of DAE and BEC contain a right-angle and an obtuse triangle, one of the pair of angles in E must be an either an obtuse oder right angle.

In this example, angles EAD=90

EDA=90

Triangle EAD sums to more than 180.

C and D have to be on the exact same side of AB.

Let A, D and BC not be on the exact same side.

Let AD cut BC at F.

ABCD being a Saccheri quadrilateral means that BAD =90

CDA=90

Because one of the angles DCF and BAF contain a right angle or an obtuse angle respectively, one of the pairs of linear angles F must either be an obtuse oder a right angle.

For this example angles FDC=90

CFD>=90

What is the sum of all three angles in Triangle CDF?

Therefore, A and D must be on the same side as BC.

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