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Explain how you can use a straight edge and a compass to construct an angle that is both congruent and adjacent to a given angle. 

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Kristina Dela Cruz
Kristina Dela CruzLv10
25 Jan 2021

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Construct a right angle XYZ . Now construct a 105  angle XYW . Is there more than one way to construct angle XYW ? If so, construct all possible solutions and explain why is that possible.

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Menaechmus (ca 350 B.C.) was another student in Plato’s academy. He discovered the conics: the ellipse, hyperbola, and parabola. These arise as points of intersection of planes passing through right circular cones (or pairs of cones). Plato insisted that geometric prob- lems should have straight edge and compass constructions. It was shown much later (in 1837 by Pierre Wantzel) that it is not possible to double the cube by ruler and compass constructions only. But Menaechmus described a simple way of using his conics to double a cube: √3 3 (i) Show that if we want to double a cube, it suffices to find 2, that is to solve x = 2. (ii) Show (i) arises by finding the intersection of the parabola y = 1x2 with the hyperbola 2 xy = 1.

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