To solve the triangle $DQRS$ with sides $q = 1.7$ m, $r = 4.3$ m, and $s = 5.6$ m, we need to find the measures of the angles $D$, $Q$, and $R$ to the nearest degree. We will use the Law of Cosines to find one angle and then use the Law of Sines to find the remaining angles.

Step-by-Step Solution

1. Find Angle $\angle D$ Using the Law of Cosines

The Law of Cosines states: $s^2 = q^2 + r^2 - 2qr \cos(D)$

Rearranging to solve for $\cos(D)$: $\cos(D) = \frac{q^2 + r^2 - s^2}{2qr}$

Substitute the given values: $\cos(D) = \frac{1.7^2 + 4.3^2 - 5.6^2}{2 \cdot 1.7 \cdot 4.3}$

Calculate each term: $1.7^2 = 2.89$ $4.3^2 = 18.49$ $5.6^2 = 31.36$

So: $\cos(D) = \frac{2.89 + 18.49 - 31.36}{2 \cdot 1.7 \cdot 4.3} = \frac{-10.08}{14.62} \approx -0.69$

Find $D$ using the inverse cosine function: $D = \cos^{-1}(-0.69) \approx 134.6^\circ \approx 135^\circ$

2. Find Angle $\angle Q$ Using the Law of Sines

Now, we can use the Law of Sines to find another angle. Let's find angle $Q$:

The Law of Sines states: $\frac{\sin(Q)}{q} = \frac{\sin(D)}{s}$

Rearranging to solve for $\sin(Q)$: $\sin(Q) = \frac{q \cdot \sin(D)}{s}$

Using the previously found value of $\sin(D)$: $\sin(135^\circ) = \sin(180^\circ - 135^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707$

So: $\sin(Q) = \frac{1.7 \cdot 0.707}{5.6} \approx \frac{1.202}{5.6} \approx 0.215$

Find $Q$ using the inverse sine function: $Q = \sin^{-1}(0.215) \approx 12.4^\circ \approx 12^\circ$

3. Find Angle $\angle R$

Finally, the sum of the angles in a triangle is $180^\circ$:

$R = 180^\circ - D - Q = 180^\circ - 135^\circ - 12^\circ = 33^\circ$

Summary of the Angles

$\angle D \approx 135^\circ$ $\angle Q \approx 12^\circ$ $\angle R \approx 33^\circ$

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

1. How do we use the Law of Sines and the Law of Cosines together?
2. What are the steps to verify the correctness of our angle calculations?
3. Can the Law of Cosines be used to solve for any side of a triangle?
4. How do the properties of angles in a triangle help in solving such problems?
5. What are some practical applications of solving triangles in real-life scenarios?

Tip: Always double-check your angle measures to ensure they add up to $180^\circ$ in a triangle.