Triangle 𝐴𝐵𝐶 has been drawn between two parallel lines. The triangle’s sides have been extended as shown. By finding the values of 𝑥 and 𝑦, show that triangle 𝐴𝐵𝐶 is isosceles.
First, we need to make sure that we know that this diagram is not drawn to scale. So we’ll use what we know about intersecting lines to help us identify these angle measures. Angle 𝐶𝐵𝐴 can be found because we know that angles on a straight line must sum to 180 degrees. If this measure is 80 degrees, then 𝐶𝐵𝐴 must measure 100 degrees. We’ll say angle 𝐶𝐵𝐴 measures 100 degrees because angles on a straight line sum to 180 degrees.
What about this angle, angle 𝐵𝐶𝐴? Angle 𝐵𝐶𝐴 has an alternate exterior angle. And alternate exterior angles are equal. Angle 𝐵𝐶𝐴 must be 40 degrees. 𝑦 is angle 𝐵𝐶𝐴. It equals 40 degrees because alternate angles are equal.
The last value we’re missing is the value for 𝑥. But we know that, inside a triangle, all three angles must sum to 180 degrees. So we take 180 degrees. And we subtract 140 degrees, which is the other two angles combined. 180 degrees minus 140 degrees equals 40 degrees. 𝑥 equals angle 𝐵𝐴𝐶. And that equals 40 degrees because angles in a triangle sum to 180 degrees. We’re trying to prove that triangle 𝐴𝐵𝐶 is isosceles. An isosceles triangle has two equal angles. For us, 𝑥 and 𝑦 are both 40 degrees.
Since 𝑦 equals 𝑥, we can say that the triangle 𝐴𝐵𝐶 is isosceles.