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IM Geo.1.5 Lesson: Construction Techniques 3: Perpendicular Lines and Angle Bisectors

Tekijä:GeoGebra Classroom Activities, IM Algebra 1, Geometry, Algebra 2

Points A and B are each at the centers of circles of radius AB.

Compare the distance  to the distance . Be prepared to explain your reasoning.

Compare the distance to the distance . Be prepared to explain your reasoning.

Draw line EF.

Write a conjecture about its relationship with segment .

Here is a line l with a point labeled C. Use straightedge and compass tools to construct a line perpendicular to l that goes through C.

Here is an angle:

Estimate the location of a point so that angle  is approximately congruent to angle .

Use compass and straightedge tools to create a ray that divides angle  into 2 congruent angles. How close is the ray to going through your point ?

Take turns with your partner, drawing and bisecting other angles. For each angle that you draw, explain to your partner how each straightedge and compass move helps you to bisect it. For each angle that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

For thousands of years since the ancient Greeks started playing with straightedge and compass constructions, people strived to find a construction to trisect an arbitrary angle into three equal angles. Many claimed to have found such a construction, but there was always some flaw in their reasoning. Finally, in 1837, Pierre Wantzel used a new field of mathematics to prove it was impossible—which still did not stop some from claiming to have found a construction. If we allow other tools besides just a straightedge and compass, though, it is possible. For example, try this method of using origami (paper folding) to trisect an angle. 
Video Trisecting an Angle with Origami available at https://player.vimeo.com/video/298418799.

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