1. Let's state the problem: We are given two right triangles with angles $61^\circ 30'$ and $56^\circ 20'$, and a vertical segment (flagpole) measuring 20 feet. We want to analyze or find relationships involving these angles and measurements in the context of the triangles. 2. Convert angles from degrees and minutes to decimal degrees to simplify calculations: - $61^\circ 30' = 61 + \frac{30}{60} = 61.5^\circ$ - $56^\circ 20' = 56 + \frac{20}{60} = 56.3333^\circ$ 3. Recognize that these angles are likely related as complementary or parts of adjacent triangles, since both are acute angles near the right angle vertices. 4. Given the vertical segment (flagpole) is 20 feet, this can be used as an opposite or adjacent side relative to these angles depending on triangle orientation. 5. If we denote the height of the building as $h$, and use trigonometric ratios (e.g., tangent or sine) with the angles to relate $h$ and the flagpole height 20 feet, we can find unknown lengths in the triangles. For instance, for the triangle with angle $61.5^\circ$: $$ \tan 61.5^\circ = \frac{\text{opposite}}{\text{adjacent}} $$ If the flagpole is the "opposite" side, then: $$ \text{adjacent} = \frac{20}{\tan 61.5^\circ} $$ 6. Calculate the tangent values: - $\tan 61.5^\circ \approx 1.804$ (approximate) - $\tan 56.3333^\circ \approx 1.507$ (approximate) 7. Calculate the adjacent side for each triangle: - For $61.5^\circ$ angle: $$ \text{adjacent} \approx \frac{20}{1.804} \approx 11.09 \text{ feet} $$ - For $56.3333^\circ$ angle: $$ \text{adjacent} \approx \frac{20}{1.507} \approx 13.27 \text{ feet} $$ 8. The vertical height of the building $h$ plus 20 feet of the flagpole can thus be related to these lengths depending on which side the flagpole is on. Final answer: The flagpole is 20 feet tall; using trigonometry with the given angles, the adjacent sides in the triangles are approximately 11.09 feet and 13.27 feet respectively.