Constructing an Angle of 90°

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Suppose that you have a line L and some point A on L:

Line

How can you construct the perpendicular line to L through A? The steps of the construction are outlined below:

Step 1: Taking A as center and any radius, draw two circular arcs which intersect L on both sides of A (at B and C):

Circular arcs intersecting

Step 2: Taking B and C as centers and a radius equal to more than half of BC, draw two arcs on the same side of L, which intersect each other at D.

Perpendicular intersecting

Step 3: Draw a line through A and D. This is the required perpendicular to L.

Perpendicular intersection

Proof: Compare \(\Delta ABD\) and \(\Delta ACD\):

1. AB = AC (arcs of equal radii)

2. BD = CD (arcs of equal radii)

3. AD = AD (common)

By the SSS criterion, the two triangles are congruent, which means that

\(\angle BAD\) = \(\angle CAD\) = ½ (1800) = 900

Thus, AD is perpendicular to L.

Note that you can construct an angle of 450 by bisecting an angle of 900, and you can further construct an angle of 22.50 by bisecting an angle of 450.