Video Transcript
Is the measure of angle ๐ด๐ท๐ less than, equal to, or greater than the measure of angle ๐ด๐ถ๐ท?
Letโs begin by identifying the two angles that are referred to in the question. Angle ๐ด๐ท๐ is the angle formed by travelling from ๐ด to ๐ท to ๐. So itโs the obtuse angle marked in orange. Angle ๐ด๐ถ๐ท is the angle formed by travelling from ๐ด to ๐ถ to ๐ท. So itโs the obtuse angle marked in green. Now, letโs consider how to answer this question.
We havenโt been given the length of any sides in the diagram or any of the angles. And therefore, we canโt answer this question by calculating the two angles. Instead, we need to think about the relationship between the measures of these two angles based on their positions. Letโs consider part of the diagram: triangle ๐ด๐ถ๐ท.
With respect to this triangle, we see that angle ๐ด๐ถ๐ท is an interior angle and angle ๐ด๐ท๐ is an exterior angle. We can therefore answer this question using the exterior angle inequality, which tells us about the relationship between the measure of an exterior angle of a triangle and the measures of the two nonadjacent interior angles.
The exterior angle inequality tells us that in a triangle the measure of an exterior angle is greater than each of the two nonadjacent interior angles. By nonadjacent interior angles, we mean the two interior angles of the triangle that donโt lie on the straight line with the given exterior angle โ the two angles that Iโve marked with stars. One of these is of course the angle weโre interested in โ angle ๐ด๐ถ๐ท.
Therefore, we can conclude that by the exterior angle inequality, the measure of angle ๐ด๐ท๐ is greater than the measure of angle ๐ด๐ถ๐ท.
