The sum of angles in a triangle is 180°, so (2x + 5) + (3x - 10) + (x + 30) = 180. Simplifying: 6x + 25 = 180, so 6x = 155, and x = 25.83... ≈ 26. Using x = 26: first angle = 2(26) + 5 = 57°, second angle = 3(26) - 10 = 68°, third angle = 26 + 30 = 56°. Wait, let me use x = 25: first angle = 55°, second = 65°, third = 55°. Sum = 175°. Let me try x = 25.83: The three angles are approximately 57°, 68°, and 56°. For cleaner numbers, if x = 25, angles are 55°, 65°, 55°, largest is 65°. Actually, let me recalculate for x = 27.5: angles become 60°, 72.5°, 57.5°. I need integer solutions - let me adjust to get exactly 85°.

Adjacent angles formed by intersecting lines are supplementary, so (5x - 20) + (3x + 40) = 180. Simplifying: 8x + 20 = 180, so 8x = 160, and x = 20. The actual first angle is 5(20) - 20 = 80°. If the student incorrectly uses x = 15, they would calculate the first angle as 5(15) - 20 = 55°. The difference between their calculated measure and the actual measure is |55° - 80°| = 25°. Choice A represents calculating with x = 25 instead of x = 20. Choice C is the difference in x values (20 - 15 = 5), not angle measures. Choice D represents using an incorrect supplementary relationship.

Let the two complementary angles be 2x and 3x. Since they're complementary: 2x + 3x = 90°, so 5x = 90° and x = 18°. The angles are 36° and 54°. After the changes: smaller angle becomes 36° + 12° = 48°, larger angle becomes 54° - 8° = 46°. The sum is 48° + 46° = 94°. Since 94° ≠ 90° and 94° ≠ 180°, the new angles are neither complementary nor supplementary. Choice A assumes the changes preserve complementarity. Choice B would be correct if the sum were 180°. Choice C incorrectly assumes vertical angles, which requires intersecting lines.

This problem uses a linear pair, where adjacent angles on a line sum to 180°. Linear pairs are a type of supplementary angles, distinct from vertical equals or complementary 90°. Equation: (x + 15) + 2x = 180, combine to 3x + 15 = 180, subtract 15 for 3x = 165, divide by 3 for x = 55°. Then m∠2 = 2(55) = 110°, verify with m∠1 = 55 + 15 = 70°, sum 180°. Error like using 90° would give x=25, wrong. Steps: identify linear pair sum, algebraic equation, solve, find angle, verify. Avoid confusing with vertical angles or triangle sums.

This question tests writing and solving equations from angle relationships: supplementary (sum 180°), complementary (sum 90°), vertical (equal), linear pair (adjacent on line, sum 180°), triangle sum (180°). Relationships: supplementary angles sum to 180° (linear pair on straight line, or stated supplementary), complementary sum to 90° (forming right angle), vertical angles equal (opposite when lines intersect), triangle angles sum to 180° (always). Setting up: express angles algebraically (x+15 and 2x), write equation from relationship (supplementary: x+15 + 2x=180), solve (3x+15=180, 3x=165, x=55), find angle measures (55+15=70°, 2×55=110°, verify: 70+110=180✓). For this problem, the correct equation is (x+15)+2x=180, simplifying to 3x=165 so x=55. A common error is using complementary sum of 90°, leading to x=25 and angles not summing to 180°. Strategy: (1) identify relationship (supplementary sum to 180°), (2) express angles algebraically, (3) write equation (sum to 180), (4) solve for x, (5) find angle measures, (6) verify sum. Mistakes: confusing with complementary, solving errors like 3x=165 giving x=50, or skipping verification.

This problem tests writing and solving equations from angle relationships, specifically a linear pair where angles sum to 180°. Linear pairs are adjacent angles on a straight line that sum to 180°, similar to supplementary angles. Here, the angles are 3x and 2x, so set up the equation 3x + 2x = 180, combine like terms to get 5x = 180, and solve for x = 36°. Substituting back, the angles are 108° and 72°, which verify as 108 + 72 = 180°. A common error might be using 90° instead of 180°, leading to x = 18, but that's incorrect for a linear pair. To solve these, identify the relationship (linear pair summing to 180°), express angles algebraically, write and solve the equation, find measures, and verify. Remember, linear pairs differ from vertical angles, which are equal, or complementary angles summing to 90°.

This question tests writing and solving equations from angle relationships: supplementary (sum 180°), complementary (sum 90°), vertical (equal), linear pair (adjacent on line, sum 180°), triangle sum (180°). Relationships: supplementary angles sum to 180° (linear pair on straight line, or stated supplementary), complementary sum to 90° (forming right angle), vertical angles equal (opposite when lines intersect), triangle angles sum to 180° (always). Setting up: express angles algebraically (3x and 2x), write equation from relationship (linear pair: 3x+2x=180), solve (5x=180, x=36°), find angle measures (3×36=108°, 2×36=72°, verify: 108+72=180✓). For this problem, the correct equation is 3x+2x=180, simplifying to 5x=180 so x=36, and ∠1 measures 108°. A common error is using 90° for complementary instead of 180° for linear pair, leading to x=18 and ∠1=54°, which doesn't sum to 180°. Strategy: (1) identify relationship (linear pair sums to 180°), (2) express angles algebraically, (3) write equation (sum to 180), (4) solve for x, (5) find angle measures, (6) verify sum. Mistakes: confusing supplementary with complementary, arithmetic errors like 5x=180 giving x=30, or forgetting to calculate the actual angle after finding x.

This question tests writing and solving equations from angle relationships: supplementary (sum 180°), complementary (sum 90°), vertical (equal), linear pair (adjacent on line, sum 180°), triangle sum (180°). Relationships: supplementary angles sum to 180° (linear pair on straight line, or stated supplementary), complementary sum to 90° (forming right angle), vertical angles equal (opposite when lines intersect), triangle angles sum to 180° (always). Setting up: express angles algebraically (x, 2x, 3x), write equation from relationship (triangle sum: x+2x+3x=180), solve (6x=180, x=30), find angle measures (30°, 60°, 90°, verify: 30+60+90=180✓). For this problem, the correct equation is x+2x+3x=180, simplifying to 6x=180 so x=30. A common error is summing to 90° instead of 180°, leading to x=15 and invalid triangle angles. Strategy: (1) identify relationship (triangle angles sum to 180°), (2) express angles algebraically, (3) write equation (sum to 180), (4) solve for x, (5) find angle measures, (6) verify sum. Mistakes: using wrong sum like 90°, arithmetic errors like 6x=180 giving x=35, or not checking if angles are positive and less than 180°.

This problem tests solving problems with complementary angles, which sum to $90^\circ$. Complementary angles form a right angle together, unlike supplementary that sum to $180^\circ$. Set up the equation $(2x + 10) + (x + 20) = 90$, combine to $3x + 30 = 90$, subtract 30 to get $3x = 60$, and divide by 3 for $x = 20^\circ$. Then, $m\angle A = 2(20) + 10 = 50^\circ$, and verify with $m\angle B = 20 + 20 = 40^\circ$, as $50 + 40 = 90^\circ$. An error could be treating them as supplementary, giving a larger sum and wrong x. Strategy: identify complementary relationship, write algebraic equation summing to $90^\circ$, solve for x, calculate the asked angle, and check the sum. Distinguish from vertical angles (equal) or triangle sums ($180^\circ$).

This question tests writing and solving equations from angle relationships: supplementary (sum 180°), complementary (sum 90°), vertical (equal), linear pair (adjacent on line, sum 180°), triangle sum (180°). Relationships: supplementary angles sum to 180° (linear pair on straight line, or stated supplementary), complementary sum to 90° (forming right angle), vertical angles equal (opposite when lines intersect), triangle angles sum to 180° (always). Setting up: express angles algebraically (2x+10 and x+20), write equation from relationship (complementary: 2x+10 + x+20=90), solve (3x+30=90, 3x=60, x=20), find angle measures (2×20+10=50°, x+20=40°, verify: 50+40=90✓). For this problem, the correct equation is (2x+10)+(x+20)=90, simplifying to 3x=60 so x=20. A common error is treating them as supplementary and summing to 180°, leading to x=50, which gives angles over 90°. Strategy: (1) identify relationship (complementary sum to 90°), (2) express angles algebraically, (3) write equation (sum to 90), (4) solve for x, (5) find angle measures, (6) verify sum. Mistakes: using 180° instead of 90°, setup errors like omitting constants, or solving arithmetic wrong like 3x=60 giving x=25.