The exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles. Let the unknown interior angle be x.

Let the unknown interior angle be $x$.

$$125^\circ = 48^\circ + x$$

$$x = 125^\circ - 48^\circ$$ $$x = 77^\circ$$

The other non-adjacent interior angle is $\boxed{\text{}77^\circ\text{}}$

2. A regular polygon has an interior angle of $144^\circ$.

a) Explain why this polygon must have 10 sides.

The interior angle and exterior angle at any vertex of a polygon are supplementary.

$$\text{Exterior angle} = 180^\circ - \text{Interior angle}$$ $$\text{Exterior angle} = 180^\circ - 144^\circ$$ $$\text{Exterior angle} = 36^\circ$$

If $n$ is the number of sides, then each exterior angle is $\frac{360^\circ}{n}$.

$$\frac{360^\circ}{n} = 36^\circ$$

$$n = \frac{360^\circ}{36^\circ}$$ $$n = 10$$

Therefore, the polygon must have 10 sides.

b) Justify your reasoning using the formula for the sum of interior angles: Sum = $(n-2) \times 180^\circ$, where $n$ is the number of sides.

$$144n = (n-2) \times 180$$ $$144n = 180n - 360$$

$$360 = 180n - 144n$$ $$360 = 36n$$ $$n = \frac{360}{36}$$ $$n = 10$$

The polygon has $\boxed{\text{}10 \text{ sides}\text{}}$

3. Two angles are supplementary. One angle measures $67^\circ$. What is the measure of the other angle?

Let the unknown angle be $x$.

$$67^\circ + x = 180^\circ$$

$$x = 180^\circ - 67^\circ$$ $$x = 113^\circ$$

The measure of the other angle is $\boxed{\text{}113^\circ\text{}}$

4. A right-angled triangle has sides of 5 cm and 12 cm. Using the Pythagorean theorem, calculate the length of the hypotenuse.

Let $a = 5$ cm and $b = 12$ cm.

$$5^2 + 12^2 = c^2$$

$$169 = c^2$$ $$c = \sqrt{169}$$ $$c = 13 \text{ cm}$$

The length of the hypotenuse is $\boxed{\text{}13 \text{ cm}\text{}}$

5. A ladder is leaning against a wall. The ladder is 10 metres long, and the base of the ladder is 6 metres away from the wall.

a) Calculate the height the ladder reaches on the wall.

The ladder is the hypotenuse ($c$), the distance from the wall is one leg ($b$), and the height the ladder reaches on the wall is the other leg ($a$).
Using the Pythagorean theorem: $a^2 + b^2 = c^2$.
Given: $c = 10$ m, $b = 6$ m. We need to find $a$.

$$a^2 + 6^2 = 10^2$$

$$a^2 = 100 - 36$$ $$a^2 = 64$$

$$a = \sqrt{64}$$ $$a = 8 \text{ m}$$

The height the ladder reaches on the wall is $\boxed{\text{}8 \text{ m}\text{}}$