Is the measure of angle 𝐴𝐷𝑌 less than, equal to, or greater than the measure of angle 𝐴𝐶𝐷?
Let’s begin by identifying the two angles that are referred to in the question. Angle 𝐴𝐷𝑌 is the angle formed by travelling from 𝐴 to 𝐷 to 𝑌. So it’s the obtuse angle marked in orange. Angle 𝐴𝐶𝐷 is the angle formed by travelling from 𝐴 to 𝐶 to 𝐷. So it’s the obtuse angle marked in green. Now, let’s consider how to answer this question.
We haven’t been given the length of any sides in the diagram or any of the angles. And therefore, we can’t 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. Let’s consider part of the diagram: triangle 𝐴𝐶𝐷.
With respect to this triangle, we see that angle 𝐴𝐶𝐷 is an interior angle and angle 𝐴𝐷𝑌 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 don’t lie on the straight line with the given exterior angle — the two angles that I’ve marked with stars. One of these is of course the angle we’re interested in — angle 𝐴𝐶𝐷.
Therefore, we can conclude that by the exterior angle inequality, the measure of angle 𝐴𝐷𝑌 is greater than the measure of angle 𝐴𝐶𝐷.