In the diagram below, M is the mid-point of XZ and |MZ| = |MX|. a) Calculate |MZ| b) Hence write down the value of |XZ| and Calculate θ

In the right-angled triangle $\triangle YZM$, we know the adjacent side $|YZ| = 6$ cm and the angle $\angle MYZ = 32^\circ$. We want to find the opposite side $|MZ|$. We use the tangent function.

$$\tan(\angle MYZ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|MZ|}{|YZ|}$$

$$\tan(32^\circ) = \frac{|MZ|}{6 \text{ cm}}$$ $$|MZ| = 6 \times \tan(32^\circ)$$ $$|MZ| \approx 6 \times 0.624869$$ $$|MZ| \approx 3.749214 \text{ cm}$$

Rounding to two decimal places:

$$|MZ| \approx \boxed{\text{3.75 cm}}$$

Since $M$ is the mid-point of $XZ$, the length of $XZ$ is twice the length of $MZ$.

$$|XZ| = 2 \times |MZ|$$ $$|XZ| = 2 \times 3.749214 \text{ cm}$$ $$|XZ| \approx 7.498428 \text{ cm}$$

Rounding to two decimal places:

$$|XZ| \approx \boxed{\text{7.50 cm}}$$

Consider the larger right-angled triangle $\triangle YZX$. We know the opposite side $|XZ| \approx 7.498428$ cm and the adjacent side $|YZ| = 6$ cm for the angle $\angle XYZ$.

$$\tan(\angle XYZ) = \frac{|XZ|}{|YZ|}$$ $$\tan(\angle XYZ) = \frac{7.498428 \text{ cm}}{6 \text{ cm}}$$ $$\tan(\angle XYZ) \approx 1.249738$$

$$\angle XYZ = \arctan(1.249738)$$ $$\angle XYZ \approx 51.329^\circ$$

From the diagram, $\angle XYZ = \angle XYM + \angle MYZ$.
So, $\angle XYZ = \theta + 32^\circ$.

$$\theta = \angle XYZ - 32^\circ$$ $$\theta \approx 51.329^\circ - 32^\circ$$ $$\theta \approx 19.329^\circ$$

Rounding to one decimal place:

$$\theta \approx \boxed{\text{19.3}^\circ}$$