The vertices of the quadrilateral ABCD lie on a circle and divide it into four arcs, the degree measures of which

| January 22, 2021 | education

The vertices of the quadrilateral ABCD lie on a circle and divide it into four arcs, the degree measures of which are 56.74.97 and 133 degrees. Find the degree measure of the smaller angle of the quadrilateral.

Let the circle be described around the AВСD, the values obtained when separating the arcs: arc AB = 56 °, arc BC = 74 °, arc DC = 97 ° and arc AC = 133 °, the sum of these angles is 56 ° + 74 ° + 97 ° + 133 ° = 360 °.

In the quadrilateral AВСD, all angles <A, <B, <C, <D are inscribed angles, the value of which is equal to half of the arcs on which they rest. That is, <A = 1/2 (arc ВСD) = 1/2 (arc BC + СD).

The smaller angle is equal to the sum of two smaller adjacent arcs, these arcs: ABC = 56 ° + 74 ° = 130 °; ВСD = 74 ° + 97 ° = 171 °; ADС = 97 ° + 133 ° = 230 °; DAВ = 133 ° + 56 ° = 189 °. Smaller angle <ADС = 1/2 * 130 ° = 65 °.



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