Let’s analyze the given problem.

Known values from the diagram:

1. $AD = 4.5 \, \text{km}$
2. $DC = 3 \, \text{km}$
3. $\angle B = 90^\circ$ (right angle at $B$)
4. $\angle QAD = 40^\circ$
5. $\angle ADC = 110^\circ$

Goal:

Find $\angle Q$.

Step 1: Relation between angles in triangle $ADC$

From triangle $ADC$, the sum of angles is always $180^\circ$: $\angle ADC + \angle DAC + \angle ACD = 180^\circ$ Substitute known values: $110^\circ + \angle DAC + \angle ACD = 180^\circ$ $\angle DAC + \angle ACD = 70^\circ$

Step 2: Use triangle $ABQ$

In triangle $ABQ$, we are given:

• $\angle QAD = 40^\circ$ (already marked),
• $\angle DAC$ (calculated in Step 1),
• And we know the sum of angles in triangle $ABQ$.

Let me finalize this mathematically.Let us solve step-by-step to find $\angle Q$:

Step 1: Decompose $\angle QAD$

From the diagram: $\angle QAD = 40^\circ$ $\angle QAD$ is part of triangle $ABQ$, which will help us later in solving for $\angle Q$.

Step 2: Triangle $ADC$ Properties

In triangle $ADC$: $\angle ADC = 110^\circ$ The other two angles in triangle $ADC$ must add up to: $\angle DAC + \angle ACD = 70^\circ$

We do not have enough specific information to calculate individual values of $\angle DAC$ and $\angle ACD$ directly, but we know their sum.

Step 3: Identify $\angle Q$

Since $\angle Q$ is an external angle, its relationship depends on adjacent angles. To fully solve for $\angle Q$, additional relationships in the diagram (e.g., lengths of sides or specific geometry properties for triangle $ABQ$) would be required.

Could you clarify if there are additional details (e.g., side lengths of $AB$, or any missing angles)? Let me know, and I'll refine the solution further!

Would you like me to break this into detailed trigonometric steps or assumptions? Let me know!

Follow-up Questions:

1. Can you confirm if any side lengths for $AB$ or $AQ$ are provided?
2. Is $Q$ the exterior angle between $AQ$ and $AB$?
3. Should I explore laws of sine/cosine to relate triangles $ADC$ and $ABQ$?
4. Are there alternative interpretations for the problem?
5. Is any approximation acceptable for the final answer?

Tip: Always label all angles and side lengths clearly in triangle problems to avoid ambiguity.