r t (5y-9)° (5x+4)° 3x° Note: Figure not drawn to scale In the figure above, r || t. What is the value of x+y? A) 37 C) 43 B) 40 D) 46

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In this problem, we have two parallel lines r and t intersected by a transversal. We're given three angles: 5y minus 9 degrees, 5x plus 4 degrees, and 3x degrees. We need to find the value of x plus y. Let's analyze the angle relationships created by these parallel lines. To solve this problem, we need to use the properties of parallel lines cut by a transversal. First, let's identify that the angle measuring 5x plus 4 degrees has a supplementary angle of 180 degrees minus 5x plus 4 degrees. Since lines r and t are parallel, this supplementary angle corresponds to the angle measuring 5y minus 9 degrees. These corresponding angles are equal. So we can write: 180 degrees minus 5x minus 4 degrees equals 5y minus 9 degrees. Simplifying, we get 176 degrees minus 5x equals 5y minus 9 degrees. Adding 5x to both sides and adding 9 degrees, we get 185 degrees equals 5x plus 5y, which means 185 degrees equals 5 times the quantity x plus y. Therefore, x plus y equals 185 divided by 5, which is 37. The answer is A. Let's review the key angle relationships in parallel lines cut by a transversal. First, corresponding angles are equal - these are angles in the same position relative to the parallel lines and the transversal. Second, supplementary angles sum to 180 degrees - this is what we used when we identified that 180 degrees minus 5x plus 4 degrees is supplementary to the angle 5x plus 4 degrees. Third, alternate interior angles are equal - these are angles between the parallel lines on opposite sides of the transversal. Fourth, alternate exterior angles are equal - these are angles outside the parallel lines on opposite sides of the transversal. In our problem, we used the corresponding angles property to set up our equation: 180 degrees minus 5x plus 4 degrees equals 5y minus 9 degrees, which led us to our answer: x plus y equals 37. Let's verify our solution that x plus y equals 37. First, recall our equation: 180 degrees minus 5x plus 4 degrees equals 5y minus 9 degrees. Let's try specific values that satisfy x plus y equals 37, such as x equals 17 and y equals 20. When we substitute x equals 17, we get 5x plus 4 equals 5 times 17 plus 4, which equals 89 degrees. When we substitute y equals 20, we get 5y minus 9 equals 5 times 20 minus 9, which equals 91 degrees. Now let's check if our equation is satisfied. The left side is 180 degrees minus 89 degrees, which equals 91 degrees. The right side is 91 degrees. Since 91 degrees equals 91 degrees, our equation is satisfied. We can also check the third angle in our figure, which is 3x degrees. With x equals 17, this angle is 3 times 17, or 51 degrees. The sum of angles in a triangle should be 180 degrees. We have 89 degrees plus 51 degrees plus the third angle equals 180 degrees, which means the third angle is 40 degrees. This confirms that our answer, x plus y equals 37, is correct. Let's summarize what we've learned. First, when parallel lines are cut by a transversal, corresponding angles are equal. This is a fundamental property we used to solve our problem. Second, supplementary angles sum to 180 degrees, which was crucial for setting up our equation. Third, for this specific problem, we used the equation: 180 degrees minus 5x plus 4 degrees equals 5y minus 9 degrees. Fourth, solving this equation gave us x plus y equals 37, which corresponds to answer choice A. Finally, verification is important: we confirmed our answer by checking all angle relationships and ensuring they were consistent. These angle relationships in parallel lines are fundamental concepts in geometry that appear frequently in many problems.