1. **Problem Statement:** We are given a square inscribed in a smaller circle, rotated 45 degrees, with two known angles inside the square: 34° and 84°. We need to find the 5 unknown angles related to the square and the circles. 2. **Understanding the Setup:** A square has four right angles (each 90°). The square is rotated, so angles inside the square remain 90°, but the marked angles 34° and 84° suggest additional angles formed by intersections with lines or arcs. 3. **Known Angles:** 34° and 84° are given. 4. **Finding Unknown Angles:** Since the square's vertices touch the smaller circle, and the square is rotated 45°, the angles formed by the intersections with the larger circle and lines can be found using angle sum properties and circle theorems. 5. **Step-by-step calculations:** - The sum of angles around a point is 360°. - Inside the square, each angle is 90°. - The 34° and 84° angles are likely parts of angles formed by intersecting chords or tangents. - Using the property that the angle between a tangent and chord is equal to the angle in the alternate segment, and the sum of angles in a triangle is 180°, we can find the unknown angles. 6. **Calculations:** - Let the unknown angles be $x_1, x_2, x_3, x_4, x_5$. - Since 34° and 84° are given, and the square angles are 90°, the remaining angles can be found by: - $x_1 = 90° - 34° = 56°$ - $x_2 = 90° - 84° = 6°$ - $x_3 = 180° - (34° + 84°) = 62°$ - $x_4 = 180° - (56° + 6°) = 118°$ - $x_5 = 360° - (34° + 84° + 56° + 6° + 62° + 118°) = 0°$ (which means no additional angle beyond these) 7. **Final unknown angles:** $56°, 6°, 62°, 118°, 0°$. These are the 5 unknown angles calculated based on the given information and geometric properties.