Quadrilateral ABCD is inscribed in a circle. ABC angle is 138 ° CAD angle is 83 °. Find angle ABD.

The angles of the quadrilateral ABCD are determined by the half of the arc on which these corners rest. Hence, <DBC = <DAC = 83 °, since both angles rest on the same DC arc, and are equal to half the value of this arc.

Now consider <ABC: <ABC = <ABD + <DBC.

Whence we find <ABD = <ABC – <DBC = 138 ° – 83 ° = 55 °.

Answer: <ABD = 55 °.

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