Let's go through the problems in the image step by step.

Problem 28:

Given:

• $m\angle 6 = 7x - 24$
• $m\angle 7 = 5x + 14$

Since angles 6 and 7 are vertical angles (opposite each other when two lines intersect), they are equal: $7x - 24 = 5x + 14$ Solve for $x$: $7x - 5x = 14 + 24$ $2x = 38$ $x = 19$

Problem 29:

Given:

• $m\angle 4 = 2x - 5$
• $m\angle 5 = 4x - 13$

Since angles 4 and 5 are vertical angles, they are equal: $2x - 5 = 4x - 13$ Solve for $x$: $2x - 4x = -13 + 5$ $-2x = -8$ $x = 4$

Problem 30:

Given:

• $m\angle 1 = x + 10$
• $m\angle 2 = 3x + 18$

Since angles 1 and 2 are a linear pair (they form a straight line), their sum is 180 degrees: $(x + 10) + (3x + 18) = 180$ Solve for $x$: $4x + 28 = 180$ $4x = 152$ $x = 38$

Problem 31:

Given the diagram, the two angles on a straight line must sum up to 180 degrees. You have:

• One angle: $(2 + 3x)^\circ$
• Another angle: $62^\circ$

Thus, $(2 + 3x) + 62 = 180$ Solve for $x$: $3x + 64 = 180$ $3x = 116$ $x = \frac{116}{3} \approx 38.67$

Problem 32:

Given:

• One angle: $40^\circ$
• Another angle: $(6x + 2)^\circ$

Since these two angles form a linear pair, their sum is also 180 degrees: $(6x + 2) + 40 = 180$ Solve for $x$: $6x + 42 = 180$ $6x = 138$ $x = 23$

Would you like further details or explanations on any of these solutions? Here are some related questions to deepen your understanding:

1. What are vertical angles, and why are they always equal?
2. How do you determine when angles are supplementary or complementary?
3. How can you use algebra to solve for angles in more complex geometric figures?
4. What properties do linear pairs of angles always satisfy?
5. How can these types of problems help in understanding angles in polygons?

Tip: Always check if the angles in question form a special relationship like vertical, supplementary, or complementary. This simplifies solving for unknown variables!