1. **Problem 3.4:** Determine the angle $\theta$ formed by the line from the origin to the point $(3, -4)$ with the positive x-axis, correct to 2 decimal places. 2. The point $(3, -4)$ lies in the fourth quadrant, so $x=3$ and $y=-4$. 3. The angle $\theta$ can be found using the tangent function: $$\tan(\theta) = \frac{|y|}{x} = \frac{4}{3}$$ 4. Calculate $\theta$: $$\theta = \arctan\left(\frac{4}{3}\right)$$ 5. Using a calculator: $$\theta \approx 53.13^\circ$$ 6. Since the point is in the fourth quadrant, the angle $\theta$ is measured clockwise from the positive x-axis, so the angle is $-53.13^\circ$ or equivalently $306.87^\circ$ if measured counterclockwise. 7. The problem likely wants the acute angle between the positive x-axis and the hypotenuse, so the answer is: $$\boxed{53.13^\circ}$$ --- 1. **Problem 3.5.1:** Given a right triangle with hypotenuse length 2, a height of 1 drawn from the right angle vertex to the hypotenuse, and one base angle of $45^\circ$, find all missing angles and side lengths. 2. Since one base angle is $45^\circ$ and the triangle is right angled, the other base angle is also $45^\circ$ (because angles in a triangle sum to $180^\circ$). 3. This means the triangle is an isosceles right triangle with legs equal. 4. The hypotenuse is given as 2, so each leg length is: $$\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$ 5. The height of 1 is drawn from the right angle vertex to the hypotenuse, forming two smaller right triangles. 6. Using the properties of the triangle and the height, the height divides the hypotenuse into two segments. Let these be $d$ and $2 - d$. 7. Using the Pythagorean theorem on the smaller triangles and the height, the segments can be found, but since the problem only asks to fill missing angles and side lengths, the key missing side lengths are the legs $\sqrt{2}$ and the other angles $45^\circ$. **Final answers:** - Angle $\theta$ in problem 3.4 is $53.13^\circ$. - In problem 3.5.1, the missing angles are $45^\circ$ and $45^\circ$, and the missing side lengths (legs) are $\sqrt{2}$ each.