1. **Stating the problem:** We have a right-angled triangle with sides 3 m, 4 m, and 5 m. We want to find the sum of the two non-right angles (a). 2. **Formula and rules:** In any triangle, the sum of all angles is $180^\circ$. In a right triangle, one angle is $90^\circ$. The sum of the other two angles is therefore: $$\text{Sum of other angles} = 180^\circ - 90^\circ = 90^\circ$$ 3. **Intermediate work:** Since the triangle is right-angled at $C$, angles at $A$ and $B$ add up to $90^\circ$. 4. **Answer:** The sum of the two non-right angles is $90^\circ$. --- **Next, part b:** 1. **Problem:** Check if a triangle with sides 2 m, 3 m, and 4 m is right-angled. 2. **Formula:** Use the Pythagorean theorem: For sides $a$, $b$, and hypotenuse $c$, if $a^2 + b^2 = c^2$, the triangle is right-angled. 3. **Work:** Longest side $c=4$, other sides $a=2$, $b=3$. Calculate: $$a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13$$ $$c^2 = 4^2 = 16$$ Since $13 \neq 16$, the triangle is not right-angled. 4. **Answer:** Rói is correct; the triangle with sides 2, 3, and 4 is not right-angled. --- **Part c:** 1. **Problem:** Find the lengths of the shorter sides in a right triangle with hypotenuse 6 m. 2. **Formula:** Use the Pythagorean theorem: $a^2 + b^2 = c^2$ with $c=6$. 3. **Work:** We want integer or simple values for $a$ and $b$ such that: $$a^2 + b^2 = 6^2 = 36$$ Examples: - $3^2 + \sqrt{27}^2 = 9 + 27 = 36$ - $4^2 + 2^2 = 16 + 4 = 20$ (no) - $\sqrt{12}^2 + \sqrt{24}^2 = 12 + 24 = 36$ 4. **Answer:** Possible side lengths are $a=3$ m and $b=\sqrt{27} \approx 5.2$ m, or other pairs satisfying $a^2 + b^2 = 36$. --- **Summary:** - a) Sum of other angles is $90^\circ$. - b) Triangle with sides 2, 3, 4 is not right-angled. - c) For hypotenuse 6, shorter sides satisfy $a^2 + b^2 = 36$.