Sum it Up

By

Scottie Benford

Can you create a triangle with two angles that measure 30 and 50 degrees?

How about 90 and 95 degrees?

How about 10 and 120 degrees?

How about 105 and 85 degrees?

Based on this investigation, a triangle must have what angle sum?

Devise and explain several different ways to illustrate the sum of

the angles in a triangle.

I want to begin my investigation by using GSP to inscribe a triangle inside of a circle.  I will attempt to show that the inscribed angles of the triangle are equal to one half the measure of the arc that the angle touches.  GSP has a tool that will construct the circle then I will choose and label three points on the circle to inscribe a triangle.

Next, I will choose one of the angles of the triangle and show the correlation between its arc measure and the measure of the angle.

The above diagram shows that the inscribed angle BAC is half of the arc measure BDC.   When I divide the arc by the angle I get two.  This means that the arc is twice the measure of the angle and that the angle is one half the arc.  This should hold true for each set of corresponding angles and arc measures.

I chose another angle of the triangle BCA.  I again found that the angle measure is half of the arc measure.

My final angle ABC is also half of the measure of its related arc measure.  I can conclude from my observations that the angles of the inscribed triangle have a sum of 180 degrees.  If I follow the pattern between the angles and the arcs I would divide the degrees of a circle, 360, in half and get 180 degrees.  Using this conclusion I can answer my initial questions.   I can create a triangle with angle measures of 30 and 50.  The measure of the final angle would be 100 degrees.  I cannot create a triangle with angles of 90 and 95 degrees because the sum of the two angles is more than 180 degrees.  With angles of 120 degrees and 10 degrees I can add an angle with measure of 50 degrees to complete the triangle.  The last situation also cannot be created because angles of 105 and 85 have a sum equal to more than 180.