Final answer:
To determine the measures of angles LDP and ULS, we use the given angle measures and expressions, along with the sum of angles around a point. By solving for x and calculating the measures, we find the measure of angle LDP to be 77°. However, there is a discrepancy with angle ULS, which should be re-examined with correct or additional information.
Step-by-step explanation:
To find the measure of angle LDP and angle ULS, we must understand the relationships between the angles. If DLS is 72°, and DUP is an expression in terms of x, and the same goes for SUP, then by using the fact that the sum of angles around a point is 360°, we can set up an equation to solve for x.
Since DLS is given as 72°, the other angles around the same point will add up to 360° - 72° = 288°. If DUP is 2x + 15° and SUP is 7x - 6°, we add these expressions to get:
- DUP + SUP = 2x + 15 + 7x - 6
- DUP + SUP = 9x + 9
Now we set this sum equal to the remaining degree measure:
- 9x + 9 = 288
- 9x = 279
- x = 31
With the value of x, we can find DUP and SUP:
- DUP = 2(31) + 15 = 77°
- SUP = 7(31) - 6 = 211°
Therefore, the measure of angle LDP is equal to the measure of angle DUP, which is 77°, because they are the same angle or vertical angles. The measure of angle ULS is equivalent to 360° - the sum of angles DLS, LDP, and SUP, which gives:
- ULS = 360 - (72 + 77 + 211)
- ULS = 360 - 360
- ULS = 0°
It appears there is an issue with the provided information as the measure of angle ULS should not be 0°. However, if the angles mentioned are not all around the same point or if there is a different configuration, additional information or a diagram would be required for proper calculation.