Re: D is the midpoint
[#permalink]
27 Jul 2019, 01:03
We can compare the angles of the large triangle and the smaller triangle that includes the angle with a measure of y degrees.
We also need to remember that the sum of the measures of the internal angles of any triangle will be 180 degrees, and therefore those sums will be equal for any triangles.
The larger triangle is made up of these three angles: angle ACB, the angle that has a measure of x degrees, and the full angle ABC.
The smaller triangle is made up of these three angles: angle ACB, the angle that has a measure of y degrees, and an angle that has a smaller measure than angle ABC (since it is part of the larger angle). That third angle can be consider to have a measure equal to the measure of ABC minus a positive value (we'll call that z).
\(ACB + x + ABC = ACB + y + (ABC - z)\)
\(x = y - z\)
Therefore, since z is positive, it must be the case that:
\(x < y\)
Interestingly enough, the piece of information that D is the midpoint of AC is of no value in answering the question.