What is measurrment: kih, jih, kji, and hki?

The figure as drawn is impossible. Taking the side lengths to be correct, triangle GIJ can be solved using the Law of Cosines. That solution can be used to solve triangle GJK using the angle at G and the Law of Sines.

The angle GJK, marked as 28°, is actually about 29.7767°.

The size of the angle at H (84°) ensures that the quadrilateral is not cyclic, so the solution of triangle HIK involves three simultaneous quadratic equations in the side lengths HI, GH, and HK (using the Law of Cosines).

I used a machine solver for that, with the results shown above.

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The figure shows the same result using a free geometry drawing program, GeoGebra. The tricky part is making sure the angle at H is 84°. That is done by making IK the chord of a circle through points H, I, K, such that the chord subtends an arc of 168°. The angle values shown on the figure are those measured by the drawing program.

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step-by-step explanation: