**Header:** Teaching 13-year-olds multiple problem-solving approaches for geometry questions involving intersecting lines and angle calculations.

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Let's examine this geometry problem involving three intersecting lines. We have lines q, r, and s, with a 125-degree angle marked where lines q and r meet. We need to find the values of angles x and y, then calculate their sum. Our first method uses the vertical angles property. When lines q and r intersect, they form two pairs of vertical angles. Since vertical angles are equal, the angle opposite to the 125-degree angle is also 125 degrees. This gives us a key relationship to work with. The second method uses linear pairs. Since angles on a straight line sum to 180 degrees, the angle adjacent to 125 degrees must be 55 degrees. Using the properties of parallel lines and transversals, we can determine that both x and y equal 55 degrees. The third method treats lines q and r as parallel lines with s as a transversal. Using alternate interior angles and supplementary angle properties, we find that both x and y equal 55 degrees. Therefore, x plus y equals 110 degrees, which is option B. In conclusion, we explored three different methods to solve this geometry problem. Whether using vertical angles, linear pairs, or parallel line properties, we consistently find that x equals 55 degrees and y equals 55 degrees. Therefore, x plus y equals 110 degrees, making option B the correct answer.