A D E H rectangle is divided into 3 congruent square pieces. Calculate α + β .
The
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sin ( α + β ) = sin α cos β + cos α sin β sin ( α + β ) = a 1 0 a ⋅ a 5 2 a + a 1 0 3 a ⋅ a 5 a sin ( α + β ) = 5 0 2 + 5 0 3 sin ( α + β ) = 5 0 5 sin ( α + β ) = 5 2 5 sin ( α + β ) = 2 1 sin ( α + β ) = 2 1 2 ∴ α + β = 4 5 ∘
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We see isosceles right triangle B F M with an angle of 4 5 ∘ at the base. And ∠ α = ∠ M B N , ∠ β = ∠ F B N and ∠ F B N + ∠ M B N = 4 5 ∘ .