In a quadrilateral ABCD, the diagonals BD and AC are equal, AB = CD, angle ABD = 85 °, angle ADB = 58 °. Find the ACD angle.

univerkov | April 11, 2021 | education

1) Consider several triangles, and find equal among them. Triangles ABD and ACD on the basis of equality of three sides: AD – common side; AB = CD, and diagonals AC = BD by condition.

2) Corollary: opposite to equal sides there are equal angles in such triangles, hence, <CAD = <BDA = 58 °; <BAD = <CDA = 180 ° – 58 ° – 85 ° = 37 °; <ABD = <ACD = 85 °.

3) We check the ACD triangle: <ACD = 180 ° – <CAD – <CDA = 180 ° – 58 ° – 37 ° = 85 °.

That is, solving in two ways, we got one answer <ACD = 85 °.

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