What is the sum of the outer angles of a triangle, a convex quadrilateral, and a pentagon?

| March 22, 2021 | education

The outer corners of polygons are the adjacent corners of their inner corners, that is, the sum of the inner and outer angles is 180 °.

Find the sum of the outer angles of the triangle.
Since the sum of the interior angles of a triangle is 180 °, then:

∠a + ∠b + ∠c = 180 °.

Let ∠a1, ∠b1 and ∠c1 be the outer corners of the triangle, then:

∠a + ∠a1 = 180 ° → ∠a1 = 180 ° – ∠a;

∠b + ∠b1 = 180 ° → ∠b1 = 180 ° – ∠b;

∠c + ∠c1 = 180 ° → ∠c1 = 180 ° – ∠c.

Find the sum of the outer angles of the triangle:

∠a1 + ∠b1 + ∠c1 = 180 ° – ∠a + 180 ° – ∠b + 180 ° – ∠c = 540 ° – (∠a + ∠b + ∠c) = 540 ° – 180 ° = 360 °.

Using the same principle, we find the sum of the outer angles of a convex quadrilateral:
∠a + ∠b + ∠c + ∠d = 360 °;

∠a1 + ∠b1 + ∠c1 + ∠d1 = 180 ° – ∠a + 180 ° – ∠b + 180 ° – ∠c + 180 ° – ∠d = 720 ° – (∠a + ∠b + ∠c + ∠ d) = 720 ° – 360 ° = 360 °.

Find the sum of the outer angles of a convex pentagon:
∠a + ∠b + ∠c + ∠d + ∠e = 540 °.

∠a1 + ∠b1 + ∠c1 + ∠d1 + ∠e1 = 180 ° – ∠a + 180 ° – ∠b + 180 ° – ∠c + 180 ° – ∠d + 180 ° – ∠e = 900 ° – (∠ a + ∠b + ∠c + ∠d + ∠e) = 900 ° – 540 ° = 360 °.



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