Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem21

Video Transcription

Question number 21. A rectangular piece of paper was folded three times as shown. What is the measure of angle beta if alpha is 70 degrees? So here we have our  rectangular piece of paper with the angle alpha and then we fold it over once in the following manner. The corner here is being folded, pulled down and  folded over like that and ends up over here. Now the angle alpha remains and it is in fact this angle over here. So we have alpha and alpha and then we can  compute the remaining angle in this triangle over here which would be equal to 180 degrees minus 2 alpha so that is exactly 40 degrees and then the 40  degree angle remains over here like that. Now we also notice that we have here in blue and I'll trace it out in orange a parallelogram after folding the  sheet of paper like that and the angle sum here is 360 degrees. So we have angle alpha over here. The opposite corner has to have the same angle measure  so this is angle alpha over here too and then what we have is, let me write that in green here, an angle that I will call not alpha but gamma for example and then  the opposite corner from that has measure over here in blue alpha plus gamma. So our angle sum of 360 degrees breaks down as 3, 4 times alpha plus 2  times gamma. The reason I'm doing this is so I can compute gamma so this is 4 times 70 degrees plus 2 times gamma and solving here for gamma we have 2 gamma  is equal to 80 degrees so gamma is 40 degrees. Now where is the angle gamma so I have it drawn here in green and that angle after folding over is this angle  here in green and by alternate interior angles it is that angle here so these are both gamma and the one in orange is 40 degrees. Now knowing gamma and the  orange angle are in fact both 40 degrees what we can compute is the measure of the angle here in purple and that would be equal to 100 degrees and that is the  same so this one is 100 degrees by symmetry we have another 100 degree angle and in the center here a 40 degree angle and then we can compute beta so  beta is equal to 360 degrees minus 2 of the purple angles so minus 200 degrees minus the central angle of 40 degrees and that comes out to 120 degrees so  after all of this we have our answer and that comes out to 120 degrees so answer C.

Video Summary

The problem involves a rectangular piece of paper being folded three times, requiring the measurement of angle beta given angle alpha as 70 degrees. By analyzing the folding process, the 180-degree angle sum rule is applied to calculate intermediate angles, determining a vital angle gamma (40 degrees). Using the properties of folded shapes, particularly a parallelogram formed by the fold, and considering symmetry and angle sums, the solution found that angle beta equates to 120 degrees. Therefore, the correct answer is 120 degrees.

Keywords

paper folding

angle calculation

geometry

parallelogram

symmetry