Let's go through each part of the problem, analyzing the image and providing solutions:

Part A: Angle Pair Type

The angles $\angle ILM$ and $\angle KML$ are positioned on a straight line with their vertex points touching line $FMG$. Since they share a common side and are on opposite sides of the transversal line $FMG$, they are linear pair angles.

Part B: Congruent or Supplementary?

Linear pair angles are always supplementary because they are adjacent angles that form a straight line. Therefore, their measures add up to $180^\circ$.

Part C: Equation Setup and Solution

We have been given that:

• $\angle KML = 115^\circ$
• $\angle ILM = (8x + 17)^\circ$

Since $\angle ILM$ and $\angle KML$ are supplementary, their sum is $180^\circ$: $(8x + 17) + 115 = 180$

Solving for $x$:

1. Combine like terms: $8x + 132 = 180$
2. Subtract $132$ from both sides: $8x = 48$
3. Divide by $8$: $x = 6$

Part D: Measure of Angles $ILM$ and $KML$

Now, we substitute $x = 6$ back into the expression for $\angle ILM$: $\angle ILM = 8x + 17 = 8(6) + 17 = 48 + 17 = 65^\circ$

Since $\angle KML = 115^\circ$, we can verify the sum: $\angle ILM + \angle KML = 65 + 115 = 180^\circ$ The calculations confirm that $\angle ILM$ and $\angle KML$ are supplementary.

Summary of Answers:

• Part A: Linear pair
• Part B: Supplementary
• Part C: $x = 6$
• Part D: $\angle ILM = 65^\circ$ and $\angle KML = 115^\circ$

Would you like further details or have any questions about this solution?

Here are five related questions for further exploration:

1. What are vertical angles, and how do they relate to linear pairs?
2. If $\angle ILM$ is decreased by $10^\circ$, what would $x$ be?
3. How would the equation change if $\angle KML$ was instead given as $(9x + 2)^\circ$?
4. Can two angles be supplementary if they are not adjacent?
5. What is the difference between complementary and supplementary angles?

Tip: For any two adjacent angles that form a straight line, remember that they are always supplementary and add up to $180^\circ$.