To solve this problem, let's break it down step by step.

Given Information:

1. $m \angle BAD = 145^\circ$
2. $m \angle BAC = (4x + 31)^\circ$
3. $m \angle CAD = (5x + 24)^\circ$

Since $\angle BAD$ is a straight angle that includes both $\angle BAC$ and $\angle CAD$, we have: $m \angle BAD = m \angle BAC + m \angle CAD$ Thus: $145 = (4x + 31) + (5x + 24)$

Step 1: Solve for $x$

Combine terms: $145 = 4x + 31 + 5x + 24$ $145 = 9x + 55$

Subtract 55 from both sides: $90 = 9x$

Divide by 9: $x = 10$

Step 2: Find $m \angle BAC$

Substitute $x = 10$ into $m \angle BAC = (4x + 31)$: $m \angle BAC = 4(10) + 31 = 40 + 31 = 71^\circ$

Step 3: Find $m \angle CAD$

Substitute $x = 10$ into $m \angle CAD = (5x + 24)$: $m \angle CAD = 5(10) + 24 = 50 + 24 = 74^\circ$

Final Answers:

• $x = 10$
• $m \angle BAC = 71^\circ$
• $m \angle CAD = 74^\circ$

If you need further explanation, feel free to ask!

Here are five questions that expand on this problem:

1. How would the solution change if $m \angle BAD$ were a different angle?
2. What is the relationship between supplementary angles?
3. Can you generalize the steps taken here to solve for any unknowns in similar angle problems?
4. How does solving for $x$ help in determining specific angle measures?
5. What if the values of $m \angle BAC$ and $m \angle CAD$ were expressions in terms of different variables?

Tip: When working with linear pairs or supplementary angles, remember that their measures always add up to $180^\circ$.