Given that 𝐴𝐵𝐶𝐷 is similar to 𝑄𝑆𝑅𝑃, find the values of 𝑥 and 𝑦.
Since these two quadrilaterals are similar, they’ll have the properties of similar polygons. In similar polygons, corresponding angles are congruent and corresponding sides are proportional. We’ll only need the first part of this property: corresponding angles are congruent.
To use this property, we need to identify which angles are corresponding. To do that, we use the names of each quadrilateral. 𝐴𝐵𝐶𝐷 is similar to 𝑄𝑆𝑅𝑃. This means the angle at vertex 𝐴 corresponds to the angle at vertex 𝑄. These two angles are congruent, which makes the angle at vertex 𝑄 81 degrees. We can also see that the angle at vertex 𝐵 is congruent to the angle at vertex 𝑆. And that means the angle at vertex 𝐵 must be 97 degrees.
Next, we have the angle at vertex 𝐶 corresponding to the angle at vertex 𝑅. This means we can set up the equation 84 is equal to 𝑦 plus 35, because we know that the angle at vertex 𝑅 must be equal to 84 degrees. If we subtract 35 from both sides of this equation, we’ll see that 𝑦 must be equal to 49. Our final pair of corresponding angles is the angle at vertex 𝐷 and the angle at vertex 𝑃. Therefore, three 𝑥 plus 65 must equal 98 degrees. If we subtract 65 from both sides of this equation, we find three 𝑥 equals 33. Dividing both sides by three gives us 𝑥 equal to 11. Using the property that corresponding angles in similar polygons are congruent, we found that 𝑥 must be 11 and 𝑦 equals 49.