L59 — Corresponding and Alternate Angles — Solutions

1. Corresponding Angles

   1. x = 55° ▶ View Solution
   2. x = 72° ▶ View Solution
   3. x = 108° ▶ View Solution
   4. x = 43° ▶ View Solution

2. Alternate Angles

   1. x = 48° ▶ View Solution
   2. x = 77° ▶ View Solution
   3. x = 103° ▶ View Solution
   4. x = 61° ▶ View Solution

3. Mixed — Which Rule?

   1. Same corner: Corresponding; x = 80° ▶ View Solution
   2. Opposite sides between lines: Alternate; x = 54° ▶ View Solution
   3. F-shape: Corresponding; x = 67° ▶ View Solution
   4. Z-shape: Alternate; x = 111° ▶ View Solution

4. Identify the Angle Pair

   1. Both in bottom-right corner: Corresponding angles ▶ View Solution
   2. One above upper line, one below lower line (opposite sides): Corresponding angles ▶ View Solution
   3. F-shape explanation: Parallel lines form horizontal bars of the F; transversal forms the vertical; matching corners at each crossing ▶ View Solution
   4. Z-shape explanation: Parallel lines form top and bottom of Z; transversal forms the diagonal; alternate angles at each end on opposite sides ▶ View Solution

5. Setting Up Equations

   1. x = 20 ▶ View Solution
   2. x = 30 ▶ View Solution
   3. x = 24; angle A = 96° ▶ View Solution
   4. Streets at 62°: 62° at corresponding position; 62° at alternate position ▶ View Solution

6. Equations with Angle Rules

   1. x = 14 (corresponding angles equal) ▶ View Solution
   2. x = 10; both angles = 47° (alternate angles equal) ▶ View Solution
   3. x = 20; angle = 75° (corresponding angles equal) ▶ View Solution
   4. x = 7; both angles = 45° (alternate angles equal) ▶ View Solution

7. Multi-step Problems

   1. Top-left upper to bottom-right lower (74°): 74° ▶ View Solution
   2. x = 36 ▶ View Solution
   3. 53° at top-left, upper — all 8 angles: Upper: 53°, 127°, 53°, 127°  |  Lower: same — 53°, 127°, 53°, 127° ▶ View Solution

8. Are the Lines Parallel?

   1. Corresponding angles both 65°: Yes — corresponding angles equal, so lines are parallel ▶ View Solution
   2. Alternate angles 78° and 82°: No — alternate angles must be equal; 78 ≠ 82 ▶ View Solution
   3. x = 15; angle = 40°; lines parallel when x = 15 ▶ View Solution
   4. Alternate angles both 112°: Yes — alternate angles equal, so lines are parallel ▶ View Solution

9. Triangles and Parallel Lines

   1. Show apex angle = 75°: Alternate angles to base angles (40°, 65°) appear at apex; 40 + apex + 65 = 180° on straight line → apex = 75° ✓ ▶ View Solution
   2. Identify corresponding and alternate pairs: Alternate pairs: (40° base, 40° apex-left) and (65° base, 65° apex-right) ▶ View Solution
   3. Why p + q + r = 180°: Alternate angles p and q appear at apex; p + r + q = 180° on straight line — true for any triangle ▶ View Solution

10. Design and Justify

    1. Any valid angle (e.g. 50°): Corresponding pairs equal; alternate pairs equal; adjacent non-paired angles supplementary (e.g. 50° + 130° = 180°) ▶ View Solution
    2. Beam exits at 38°: 38° — corresponding angles rule ▶ View Solution
    3. Z-shape vs C-shape: Incorrect — Z-shape shows ALTERNATE angles (equal); C-shape shows CO-INTERIOR angles (sum to 180°) ▶ View Solution