Question Video: Identifying the Relation between Two Sides in a Triangle given Their Angles’ Measures Mathematics • 11th Grade

From the figure, how do 𝐴𝐵 and 𝐵𝐶 compare?

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Video Transcript

From the figure, how do 𝐴𝐵 and 𝐵𝐶 compare?

In this question, we are given a figure containing a triangle 𝐴𝐵𝐶, with two of its sides extended, and the measure of two external angles at these sides. We want to use this information to compare the lengths of two of the sides in the triangle.

To do this, we can start by highlighting the two line segments whose lengths we want to compare. We can see that these are sides of a triangle. So we can compare the lengths of these sides using the side comparison theorem in triangles. We can recall that this tells us that in a triangle, if one side of the triangle is opposite an angle of larger measure than another side of the triangle, then it must be the longer side. Therefore, we can determine which side is longer by comparing the measures of the angles opposite each side in triangle 𝐴𝐵𝐶.

We can find the measure of the internal angle at 𝐵 by noting that it combines with the angle of measure 128 degrees to make a straight angle. So its measure is 180 degrees minus 128 degrees, which we can calculate is 52 degrees. We can follow the same process for the other angle. We see that its measure is given by 180 degrees minus 108 degrees, which we can calculate is 72 degrees.

We can find the measure of angle 𝐴 by recalling that the sum of the measures of the internal angles in a triangle is 180 degrees. So we have 52 degrees plus 72 degrees plus the measure of angle 𝐴 equals 180 degrees. We can then subtract 72 degrees and 52 degrees from both sides of the equation to obtain that the measure of angle 𝐴 is 180 degrees minus 72 degrees minus 52 degrees, which we can calculate is 56 degrees.

Finally, we can see that the measure of the angle opposite 𝐴𝐵 has larger measure than the angle opposite 𝐵𝐶 in triangle 𝐴𝐵𝐶. So by the side comparison theorem, side 𝐴𝐵 is longer than side 𝐵𝐶. Hence, 𝐴𝐵 is greater than 𝐵𝐶.

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