Final answer:
By using geometric principles and algebra, we can determine the measure of angles gde and edh to be 55 degrees and 27 degrees, respectively.
Step-by-step explanation:
The question is asking to find the measures of angles in a geometric scenario where a straight angle cde is bisected by de, creating two angles, gde and edh, with given algebraic expressions for their measures. Additionally, there is another angle cdf with a given numerical measure. To find the measures, one would combine relevant geometric properties such as the fact that a straight angle measures 180 degrees, the Angle Bisector Theorem indicating that de bisects gdh into two equal angles, and algebraic methods to solve for the variable x and subsequently the angle measures.
Firstly, by understanding that cde is a straight angle and therefore measures 180 degrees, and given that m cdf= 43 degrees, one can deduce that the measure of gdh is 180 - 43 = 137 degrees. Since de bisects gdh, it follows that gde and edh are equal. Setting the two expressions for gde and edh equal to each other, we find the value of x:
- 8x - 1 = 6x - 15
- 2x = 14
- x = 7
Using the value of x, we can now find the measures of the angles:
- m gde = 8x - 1 = 8(7) - 1 = 55 degrees
- m edh = 6x - 15 = 6(7) - 15 = 27 degrees