Let's solve for $\theta$ and the area of the given shape step by step.

Step 1: Solve for $\theta$

From the given image, the shape consists of two triangles. The known angles are:

• $60^\circ$ (upper triangle)
• $99^\circ$ (lower triangle)

Since the interior angles of a triangle sum to $180^\circ$, we can determine $\theta$ using:

$\theta = 180^\circ - 99^\circ - 60^\circ$

$\theta = 21^\circ$

Thus, $\theta = 21^\circ$.

---

Step 2: Find the Area of the Shape

The shape consists of two triangles. We will find their areas separately and sum them.

Upper Triangle (with sides 10 and 13, and angle $60^\circ$)

Using the formula for the area of a triangle:

$A = \frac{1}{2}ab \sin C$

$A_1 = \frac{1}{2} (10)(13) \sin 60^\circ$

Using $\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$:

$A_1 = \frac{1}{2} (10)(13)(0.866)$

$A_1 \approx 56.29$

---

Lower Triangle (with sides 13 and 25, and angle $99^\circ$)

Using the same formula:

$A_2 = \frac{1}{2} (13)(25) \sin 99^\circ$

Using $\sin 99^\circ \approx 0.999$:

$A_2 = \frac{1}{2} (13)(25)(0.999)$

$A_2 \approx 162.37$

---

Total Area:

$A_{\text{total}} = A_1 + A_2$

$A_{\text{total}} \approx 56.29 + 162.37$

$A_{\text{total}} \approx 218.66$

Thus, the total area of the shape is approximately 218.66 square units.

---

Final Answers:

1. $\theta = 21^\circ$
2. Area $\approx 218.66$ square units

Would you like a step-by-step breakdown for any part?

Related Questions:

1. How can we verify the angle sum property for irregular quadrilaterals?
2. What other methods could be used to calculate the area of the shape?
3. How would the solution change if one of the sides were unknown?
4. How does the Law of Cosines apply to this problem?
5. Can we calculate the perimeter of the shape?

Tip:

In problems involving non-right triangles, always check if the Law of Sines or Law of Cosines is useful before solving for unknown angles or sides!