1. **Problem 1:** Find the measure of each numbered angle (1, 2, 3) in the diamond-shaped quadrilateral with a given angle of 45°. 2. The diamond shape is a rhombus, so all sides are equal and opposite angles are equal. 3. The diagonals of a rhombus bisect each other at right angles (90°). 4. Angle 1 is at the intersection of the diagonals, so angle 1 = 90°. 5. Angle 3 is adjacent to the 45° angle at the top vertex. Since the top vertex angle is split by the diagonal, angle 3 = 45°. 6. Angle 2 is at the right vertex, opposite to the top vertex angle, so angle 2 = 45°. --- 7. **Problem 2:** Find the measure of each numbered angle (1, 2) in the irregular pentagon with a given angle of 55°. 8. The right angle at the top-left corner is 90°. 9. The sides with one tick are equal, and the sides with two ticks are equal. 10. Using the properties of the figure and given angles, angle 1 and angle 2 can be found by applying the triangle angle sum rule and isosceles triangle properties. 11. Angle 1 = 35° and angle 2 = 90°. **Final answers:** - Problem 1: angle 1 = 90°, angle 2 = 45°, angle 3 = 45°. - Problem 2: angle 1 = 35°, angle 2 = 90°.