Complementary and supplementary angles

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Question from Tracey, a teacher:

I have looked at your answers for students asking if groups of more than two angles can be considered either complementary or supplementary. Your answer is basically "no" because of historic definition. However, I present to you the following case to consider:

Segments AB, CD and EF intersect at point G creating 6 angles numbered 1-6 in a clockwise manner. If Angle 1=25 degrees, and angle 2 = 106 degrees, would the only way to calculate the measures of angles 3 and 6 not be to consider the definition of supplementary angles? And, if one was to be doing a proof of this, would not the reason be "definition of supplementary angles"?

This, then, creates a group of 3 angles that are supplementary.

Help me correct my logic if it is flawed.

Thanks

Hi Tracey,

There is nothing wrong with your logic, I only have two comments on your argument. The first is the word only

"would the only way to calculate the measures of angles 3 and 6 not be to consider the definition of supplementary angles?"

I would calculate the measure of angles 3 and 6 using two facts. First the sum of the measures of all six angles is 360 degrees and the second is that when two lines intersect the opposite angles are congruent. Hence angles 1 and 4 each have measure 25 degrees and angles 2 and 5 each have measure 106 degrees. The remaining two angles (3 and 6) are congruent and the sum of their measures is 360 - 2(25 + 106) degrees. Thus angles 3 and 6 each have measure 98/2 = 49 degrees.

My second comment is just about the use of language in mathematics. It is extremely important that we all use the same definitions if we are to be able to communicate meaningfully. Your argument does make sense that the concept contained in the word supplementary might apply to more than two angles but the mathematical definition applies to two angles only.

I hope this helps,
Penny

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