Final answer:
To find the measures of the angles, we first solve for x by setting up the equation based on the angle addition and find x = 14.5. Substituting x back into the angle measures, we get m∠UHI (assumed m∠JHI) to be 36 degrees and m∠GHI to be 114 degrees.
Step-by-step explanation:
The question asks us to solve for the measures of angles m∠UHI and m∠GHI given the equations of m∠JHI and m∠GHI, and the measure of m∠JHG.
We have that m∠JHI = 2x + 7, m∠GHI = 8x - 2, and m∠JHG = 65 degrees. First, we need to find the value of x by using the fact that m∠JHI + m∠JHG = m∠GHI, which becomes 2x + 7 + 65 = 8x - 2.
Simplifying, we find x = 14.5. Therefore, m∠GHI = 8(14.5) - 2 = 116 - 2 = 114 degrees. Since the question asks for m∠UHI as well, we can assume it may have meant m∠JHI, calculating it to be m∠JHI = 2(14.5) + 7 = 29 + 7 = 36 degrees.