https://file.aiquickdraw.com/mathbot/file/1728247806867et3dgheo.jpg

Let's solve each of the two problems step by step.

### Problem 11: 
Equation: 
\[
m \angle F + m \angle G = 180^\circ
\]
You are given that:
\[
m \angle F = 4x \quad \text{and} \quad m \angle G = 8x
\]

Since the sum of the two angles is \(180^\circ\), these angles form a **linear pair**. The equation becomes:
\[
4x + 8x = 180
\]
Simplifying:
\[
12x = 180
\]
Solving for \(x\):
\[
x = \frac{180}{12} = 15
\]

So, the type of angle pair is **linear pair**, and \(x = 15\).

---

### Problem 12:
Equation: 
\[
m \angle H + m \angle J = 90^\circ
\]
You are given:
\[
m \angle H = x + 24 \quad \text{and} \quad m \angle J = x
\]

Since the sum of the two angles is \(90^\circ\), they form a **complementary pair**. The equation becomes:
\[
(x + 24) + x = 90
\]
Simplifying:
\[
2x + 24 = 90
\]
Subtracting 24 from both sides:
\[
2x = 66
\]
Solving for \(x\):
\[
x = \frac{66}{2} = 33
\]

So, the type of angle pair is **complementary pair**, and \(x = 33\).

---

### Summary:
- For problem 11: Type of pair = **Linear Pair**, \(x = 15\)
- For problem 12: Type of pair = **Complementary Pair**, \(x = 33\)

Would you like more details on these solutions or have any further questions?

Here are 5 related questions:
1. What are supplementary angles, and how are they different from complementary angles?
2. How do you identify a linear pair of angles geometrically?
3. Why do complementary angles add up to 90°?
4. Can two obtuse angles be complementary? Why or why not?
5. How can you use angle properties to solve real-life geometry problems?

**Tip:** Always check the sum of angles in a pair (90° for complementary and 180° for supplementary) to verify your solution.

Identify the type of angle pair and solve for 'x' in each of the following problems:

11. m∠F + m∠G = 180° with m∠F = (4x)° and m∠G = (8x)°.
12. m∠H + m∠J = 90° with m∠H = (x + 24)° and m∠J = (x)°.

This lesson covers solving for 'x' in linear and complementary angles, including problems on a linear pair with a 180° sum and complementary angles with a 90° sum. Step-by-step solutions apply basic algebra and geometry principles, ideal for students in Grades 7-9.

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