https://file.aiquickdraw.com/mathbot/file/1728584791286vt7wbewo.jpg

Part A (1 point):  What type of angle pair are angles ILM and KML?
Part B (1 point):  Are they congruent or supplementary?
Part C (3 points):  Set up an equation and solve for x.
Part D (2 points):  What are the measures of angles ILM and KML?

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?

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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 \(180^\circ\).

Part A: What type of angle pair are angles ILM and KML?
Part B: Are they congruent or supplementary?
Part C: Set up an equation and solve for x.
Part D: What are the measures of angles ILM and KML?

This geometry problem involves solving for the measures of angles ILM and KML, which form a linear pair. Learn how to set up an algebraic equation based on the given angles and solve for x. Suitable for students in Grades 8-10.

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