1. **Stating the problem:** We have a triangle with a base of length 4 and angles 25.71° and 51.43°. We want to find the height $h$ perpendicular to the base and verify the calculations. 2. **Given information:** - Base $=4$ - Angle at base's right corner $=25.71^\circ$ - Angle formed by height and hypotenuse $=51.43^\circ$ 3. **Check angle sum:** Since the triangle's angles must sum to 180°, the third angle is $$180^\circ - 25.71^\circ - 51.43^\circ = 102.86^\circ$$ 4. **Using tangent to find height $h$:** From the angle $25.71^\circ$ at the base's right corner, the tangent relates height and base segment: $$\tan 25.71^\circ = \frac{h}{4} \implies h = 4 \tan 25.71^\circ$$ Calculate: $$h = 4 \times 0.4817 = 1.9268$$ 5. **Using tangent at $51.43^\circ$ angle:** Given $\tan 51.43^\circ = \frac{1}{2} = 0.5$ (approximation), then $$h = 2 \times \tan 51.43^\circ$$ Calculate $\tan 51.43^\circ$: $$\tan 51.43^\circ \approx 1.2341$$ So, $$h = 2 \times 1.2341 = 2.4682$$ 6. **Comparing the two values of $h$:** - From $25.71^\circ$ angle: $h \approx 1.9268$ - From $51.43^\circ$ angle: $h \approx 2.4682$ These two values differ, so the calculations or assumptions may be inconsistent. 7. **Conclusion:** The height $h$ cannot be both values simultaneously. The statement "$h = 2(1.234142.49)$" and "$h=4.86758$" appears to be a miscalculation or typo. **Therefore, the correct height using the tangent of $25.71^\circ$ is approximately $1.93$, and using the tangent of $51.43^\circ$ is approximately $2.47$. The original calculations are not fully correct.**