Okay, here is a proof that the four angle measures in a quadrilateral add up to 360°:
Let ABCD be a quadrilateral.
Draw diagonal AC, dividing the quadrilateral into triangles ABC and ACD.
By the triangle sum theorem, the three angles in triangle ABC add up to 180°.
That is, m∠A + m∠B + m∠C = 180°
Similarly, the three angles in triangle ACD add up to 180°.
That is, m∠A + m∠C + m∠D = 180°
Adding these two equations:
m∠A + m∠B + m∠C + m∠A + m∠C + m∠D = 180° + 180°
m∠A + m∠B + m∠C + m∠D = 360°
Therefore, the four angle measures in quadrilateral ABCD add up to 360°.
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