Find the measures of the angles of
Let’s begin by considering the
figure that we are given. We can see that there are two
parallel lines marked on the diagram, the ray 𝐴𝐷 and the line segment 𝐵𝐶. And we also have two congruent line
segments, which are 𝐴𝐶 and 𝐵𝐶. Let’s consider the angle 𝐷𝐴𝐶,
which is created by transversal 𝐴𝐶 between the two parallel lines. The angle 𝐴𝐶𝐵 is alternate to
the angle 𝐷𝐴𝐶, and so these will both have a measure of 39.5 degrees.
Next, we can return to the two
congruent line segments. These are part of the triangle
𝐴𝐵𝐶, and so, by definition, this is an isosceles triangle. That’s because we can recall that
an isosceles triangle has two congruent sides. And by the isosceles triangle
theorem, we know that the angles opposite the congruent sides in an isosceles
triangle are congruent.
In triangle 𝐴𝐵𝐶, these congruent
base angles will be angle 𝐵𝐴𝐶 and angle 𝐴𝐵𝐶. As these will both have the same
measure, we can define them with the same letter, for example, the letter 𝑦. Since we know that the internal
angle measures in a triangle sum to 180 degrees, we can write that the three angles
of 𝑦, 𝑦, and 39.5 degrees sum to give 180 degrees. And simplifying the 𝑦 plus 𝑦
gives us two 𝑦. We can then subtract 39.5 degrees
from both sides to give us that two 𝑦 equals 140.5 degrees. Finally, dividing both sides by
two, we have that 𝑦 equals 70.25 degrees.
We have therefore found all three
of the required angle measures in triangle 𝐴𝐵𝐶. The measure of angle 𝐴𝐵𝐶 is
70.25 degrees. The measure of angle 𝐵𝐴𝐶 is also
70.25 degrees. And the first angle that we
determined, the measure of angle 𝐴𝐶𝐵, is 39.5 degrees.