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.