In the figure, point 𝐷 is on the line 𝐴𝐶, 𝐴𝐵 equals 𝐵𝐷 which equals 𝐶𝐷, and 𝐴𝐷 equals 18. What is the measure, in degrees, of angle 𝐵𝐶𝐷?
Let’s start by labeling this image 𝐴𝐵 equals 𝐵𝐷 which equals 𝐶𝐷. 𝐴𝐵 is the same length as 𝐵𝐷. And it’s the same length as 𝐶𝐷. What can we say then about triangle 𝐴𝐵𝐷? We can say that triangle 𝐴𝐵𝐷 is isosceles. And in an isosceles triangle, the angles opposite the same length sides have the same measure. And so we can say that angle 𝐴𝐷𝐵 measures 30 degrees, just like angle 𝐵𝐴𝐷. What other information were given is that 𝐴𝐷 measures 18 and point 𝐷 is on the line 𝐴𝐶. Because point 𝐷 is on the line 𝐴𝐶, we know something about the angle 𝐶𝐷𝐵. Angle 𝐶𝐷𝐵 and angle 𝐵𝐷𝐴 must measure 180 degrees because they make a straight line.
We know that the measure of 𝐶𝐷𝐴 Is 30 degrees. This means the measure of angle 𝐶𝐷𝐵 is equal to 180 degrees minus 30 degrees. Angle 𝐶𝐷𝐵 measures 150 degrees. If we look closely at the other triangle, triangle 𝐵𝐷𝐴, it is also an isosceles triangle. It has two sides that are the same length. This tells us that the angle 𝐷𝐵𝐶 and the angle 𝐷𝐶𝐵 must be equal to one another. We could label them as 𝑥 degrees.
And because we know that inside of a triangle all the angles add up to 180 degrees, we can say 150 degrees plus 𝑥 plus 𝑥 must equal 180 degrees. 𝑥 plus 𝑥 equals two 𝑥 and then we subtract 150 degrees from both sides of the equation. Two 𝑥 equals 30 degrees. And then we divide both sides by two, and we see that 𝑥 equals 15 degrees. The two other angles in this isosceles triangle must measure 15 degrees each. We’re interested in angle 𝐵𝐶𝐷, which is here. Angle 𝐵𝐶𝐷 must measure 15 degrees.