Given that 𝐴𝐵𝐶 is similar to 𝑃𝑄𝑅, find the value of 𝑥.
𝐴𝐵𝐶 and 𝑃𝑄𝑅 are both triangles. Since these are similar triangles, the properties of similar polygons will apply. In similar polygons, corresponding angles are congruent, and corresponding sides are proportional. Here, we’ll be primarily concerned with the angles. And our first job will be identifying which of the angles in these triangles are corresponding.
Based on the naming convention, since triangle 𝐴𝐵𝐶 is similar to triangle 𝑃𝑄𝑅, the angle at vertex 𝐴 will be corresponding to the angle at vertex 𝑃. Since these two angles are corresponding, we can say that angle 𝑃 must be 54 degrees. We don’t need it to solve this question, but we can go ahead and identify that the angle at vertex 𝐵 corresponds to the angle at vertex 𝑄 and the angle at vertex 𝐶 corresponds to the angle at vertex 𝑅. This makes the angle 𝐴𝐶𝐵 85 degrees.
To solve for 𝑥, we remember that the three angles inside a triangle sum to 180 degrees. And therefore, 54 plus two 𝑥 plus two plus 85 equals 180. Combining like terms, we reduce this to two 𝑥 plus 141 equals 180. Subtracting 141 from both sides gives us two 𝑥 equals 39. And dividing through by two, we find that 𝑥 must be equal to 19.5.
If we plug in 19.5 into two 𝑥 plus two, we find that the angle at vertex 𝑄 is 41 degrees, which would also be the case for the angle at vertex 𝐵. It’s possible to check your working by adding 41, 54, and 85 to confirm that they equal 180 degrees, which they do. Therefore, we can say that 𝑥 equals 19.5.