In this problem, we are dealing with a parallelogram, where the opposite angles are congruent and adjacent angles are supplementary (they sum to $180^\circ$).

Given:

• Lines $RS \parallel VT$ and $RV \parallel ST$
• $\angle S = 3a + 24^\circ$
• $\angle V = 6a - 12^\circ$
• $\angle T = 10b + 5^\circ$

From the properties of parallelograms:

• Opposite angles are equal: $\angle S = \angle T$ and $\angle V = \angle R$
• Adjacent angles are supplementary: $\angle S + \angle V = 180^\circ$

Step 1: Solve for $a$ using the supplementary angle relationship

Since $\angle S$ and $\angle V$ are adjacent angles, we have:

$\angle S + \angle V = 180^\circ$

Substitute the expressions for $\angle S$ and $\angle V$:

$(3a + 24) + (6a - 12) = 180$

Simplify:

$3a + 24 + 6a - 12 = 180$ $9a + 12 = 180$

Subtract 12 from both sides:

$9a = 168$

Solve for $a$:

$a = \frac{168}{9} = 18.67$

Step 2: Solve for $b$ using the opposite angle property

We know $\angle S = \angle T$, so:

$\angle S = \angle T$

Substitute the expressions for $\angle S$ and $\angle T$:

$3a + 24 = 10b + 5$

Substitute $a = 18.67$:

$3(18.67) + 24 = 10b + 5$

Simplify:

$56.01 + 24 = 10b + 5$ $80.01 = 10b + 5$

Subtract 5 from both sides:

$75.01 = 10b$

Solve for $b$:

$b = \frac{75.01}{10} = 7.5$

Final Answers:

• $a = 18.67$
• $b = 7.5$

Would you like further explanation or clarification on any part of the solution?

Here are some related questions you might find interesting:

1. How do opposite angles in a parallelogram relate to each other?
2. How do you determine the value of an unknown angle using the supplementary property?
3. What are some other properties of parallelograms?
4. How can you prove that opposite angles in a parallelogram are congruent?
5. What happens if two angles in a parallelogram are equal?

Tip: Always use the property that adjacent angles in a parallelogram add up to $180^\circ$ when trying to solve for unknown angles.