1. **Problem statement:** Given a right triangle ABC with right angle at B, side BC = 7, angle $\beta = 70^\circ$, and angle $\alpha = 20^\circ$, find the missing sides AB and AC and verify the angles. 2. **Known facts:** - Triangle ABC is right angled at B, so $\angle B = 90^\circ$. - Angles in a triangle sum to $180^\circ$, so $\alpha + \beta + 90^\circ = 180^\circ$. - Given $\alpha = 20^\circ$ and $\beta = 70^\circ$, which sum to 90°, consistent with the right angle at B. 3. **Sides and angles:** - Side BC is opposite $\alpha$. - Side AB is opposite $\beta$. - Side AC is the hypotenuse. 4. **Use trigonometric ratios:** - $\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} \Rightarrow AC = \frac{BC}{\sin(\alpha)}$ - $\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} \Rightarrow AB = AC \times \cos(\alpha)$ 5. **Calculate AC:** $$ AC = \frac{7}{\sin(20^\circ)} $$ Calculate $\sin(20^\circ) \approx 0.342020143$: $$ AC = \frac{7}{0.342020143} \approx 20.467 $$ 6. **Calculate AB:** $$ AB = 20.467 \times \cos(20^\circ) $$ Calculate $\cos(20^\circ) \approx 0.939692621$: $$ AB = 20.467 \times 0.939692621 \approx 19.217 $$ 7. **Summary of results:** - $AB \approx 19.217$ - $AC \approx 20.467$ - $\alpha = 20^\circ$ - $\beta = 70^\circ$ - $\angle B = 90^\circ$ (given) All values are rounded to 3 decimal places as requested.