Part 1: Calculate w, x, y, z to 2 significant figures.

We use the tangent function, as $w$ is the adjacent side and 10 is the opposite side to the $78^\circ$ angle.

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan(78^\circ) = \frac{10}{w}$$ $$w = \frac{10}{\tan(78^\circ)}$$ $$w \approx \frac{10}{4.7046}$$ $$w \approx 2.1255$$

Rounding to 2 significant figures:

$$w \approx \boxed{2.1}$$

We use the sine function, as 10 is the opposite side and $x$ is the hypotenuse to the $78^\circ$ angle.

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\sin(78^\circ) = \frac{10}{x}$$ $$x = \frac{10}{\sin(78^\circ)}$$ $$x \approx \frac{10}{0.9781}$$ $$x \approx 10.2239$$

Rounding to 2 significant figures:

$$x \approx \boxed{10}$$

For triangle b):

We use the sine function, as $z$ is the opposite side and 7 is the hypotenuse to the $22^\circ$ angle.

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\sin(22^\circ) = \frac{z}{7}$$ $$z = 7 \times \sin(22^\circ)$$ $$z \approx 7 \times 0.3746$$ $$z \approx 2.6222$$

Rounding to 2 significant figures:

$$z \approx \boxed{2.6}$$

We use the cosine function, as $y$ is the adjacent side and 7 is the hypotenuse to the $22^\circ$ angle.

$$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\cos(22^\circ) = \frac{y}{7}$$ $$y = 7 \times \cos(22^\circ)$$ $$y \approx 7 \times 0.9272$$ $$y \approx 6.4904$$

Rounding to 2 significant figures:

$$y \approx \boxed{6.5}$$

The identity states that $\sin A = \cos (90^\circ - A)$.
Therefore, we can rewrite $\cos 53^\circ$ as $\sin (90^\circ - 53^\circ)$.

$$\cos 53^\circ = \sin (90^\circ - 53^\circ)$$ $$\cos 53^\circ = \sin (37^\circ)$$

Given $\sin \theta = \cos 53^\circ$, substitute the result from Step 1:

$$\sin \theta = \sin 37^\circ$$

Therefore,

$$\theta = \boxed{37^\circ}$$