Both the proposition and its converse (“If corresponding angles formed by a transversal are equal, then the lines are parallel”) are true, representing fundamental geometric properties defining parallel line relationships.
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Let square side be s. Two semicircles have total area 2 x (1/2) x π(s/2)² = (1/4)πs², identical to the quarter circle’s area (1/4)πs². Subtracting their common overlapping region proves Area(A) = Area(B).
Four top rectangles match five bottom ones in length, giving 4l = 5b, so l = 5b/4. Area 9lb = 72 yields lb = 8. Solving gives b = 2.53 cm, l = 3.16 cm, and perimeter ≈ 11.38 cm.
No, a larger stagger is needed. In a 400 m race, runners navigate two bends on a 400 m track but four bends on a 200 m track, requiring twice the total stagger distance.
Let OA = OB = r. Semicircle on diameter AB has radius r/√2 and area (1/4)πr². Subtracting circular segment AFB from both the semicircle and the quarter circle OAB leaves equal areas for both shaded regions.