Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem18

Video Transcription

Question number 18.  In the triangle ABC, the points M and N on the side AB are such that the length of the segment AN is equal to the length of the side AC, and the length of the side BC is equal  to the length of the segment BM.  We have to find the measure of the vertex angle C, so that's the angle ACB. If we know that the measure of the angle MCN is 43 degrees  as indicated, we are looking at two isosceles triangles.  The side AC has the same length as the segment AN, so that's in blue one isosceles triangle,  and in red we have the side BC is the same length as the segment BM, and that's another isosceles triangle.  Now the vertex angle C, I'll just suppress the angle symbol.  This is the measure of the angle ACB, and that's equal to 43 degrees plus alpha plus beta,  where alpha will be the angle here, and beta will be the angle here.  Now let's write some equations.  We look at the blue isosceles triangle, and we look at the supplement of angle A, which is twice the measures of angle A alpha plus 43 degrees, and similarly the relationship  with the red isosceles triangle gives us 2 times beta plus 43  is the supplement of angle B, and adding these equations together,  we have 360 minus A minus B is equal to 2 alpha plus 2 beta  plus 4 times 43, and hidden in here is the measure of angle C,  because 180 plus 180 minus A minus B, that quantity here is exactly the measure of angle C,  and here let me factor out A2 for a moment,  and I'll leave the 4 times 43 as before.  So now I'll substitute for the measure of angle C from our initial observation.  I have 180 plus 43 plus alpha plus beta is equal to 2 times alpha plus beta plus 4 times 43,  and I'll solve for alpha plus beta. We have 180 equal to now alpha plus beta plus 3 times 43, and subtracting 3 times 43, which is 129 gives us that 51 is the measure  of angle alpha plus beta.  So now going back to our calculation, the vertex angle C measures 43 plus 51 or 94 degrees,  and the answer is E.

Video Summary

In triangle ABC, points M and N on side AB create two isosceles triangles: one where AN = AC and another where BM = BC. The task is to find angle ACB, knowing angle MCN is 43 degrees. The solution involves setting up equations for the angles in these isosceles triangles and finding the supplementary relationships. By solving these equations, the individual angle sum alpha + beta is determined to be 51 degrees. Adding this to the given 43 degrees, the vertex angle ACB is found to be 94 degrees. The final answer is E.

Keywords

triangle geometry

isosceles triangles

angle calculation

supplementary angles

vertex angle