342 Define a Lambert quadrilateral to be a quadrilateral ABC

34.2 Define a Lambert quadrilateral to be a quadrilateral ABCD with right angles at A, B, C. Show that the fourth angle D is acute, right, or obtuse according as the geometry is semihyperbolic, semi-Euclidean, or semielliptic

Solution

Obtuse angle : An angle which is larger than 90 degrees.

Acute angle : An angle which is smaller than 90 degrees.

Right angle : An angle with measure 90 degrees.

We know that sum of angles of quadrilateral equals 360 degrees.

Proof : If we have the fourth angle of the quadrilateral as obtuse angle then our quadrilateral would have an angle sum greater than 360 degrees which is not possible.

If we have fourth angle as right angle then rectangle would exist and all triangles would have to have defect 0. Since there is a triangle with angle sum less than 180 degrees, we have a traingle with positive defect. Thus fourth angle cannot be a right angle as well.

Hence the fourth angle of Lambert quadrilater is acute.

Hence the proof.