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It is a case of Trigonometry/Geometry. The two triangles formed by the angles and sides of artillery aiming are "Similar" (not congruent) since two angles and a side (base) are Similar. Because the Range of the projectile is 2x the base (which is congruent) of the triangles, the range MUST be the same.

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Q: Why do complementary angles of launch result in the same projectile range?
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