# Task

In this task, you will create arguments and provide reasoning for the properties of geometric figures illustrated in tangram puzzles. The tangram puzzle is a group of seven polygons that can be rearranged into different orientations to create other shapes. These polygons can be rearranged into a square, with no gaps or overlapping of the individual polygons, as shown in the figure below:Figure 1. Given: ● are large, congruent right isosceles triangles. ● are small, congruent right isosceles triangles. ● is a right isosceles triangle. ● is a square. ● is a parallelogram. ● is a square with dimensions of 1 unit by 1 unit (i.e., the entire area of the square is 1 unit2 ). Note: The right angle for each triangle can be determined by inspection. All line segments that appear straight are straight (e.g., ̅̅̅ is straight, with no bend at F). There are no gaps or overlapping figures. The tangram can be rearranged into each of the following polygons:Figure 2. Requirements: A. Determine the dimensions and area of each of the seven individual pieces from the square arrangement in Figure 1. (rearranging the pieces is not allowed). 1. Explain with full geometric justification, how you determined the dimensions of each piece. Note: You do not need to explain how you determined the area of each piece. Note: You cannot make midpoint assumptions (e.g., B is the midpoint between A and C). B. Determine the angle measures of each of the seven individual pieces from the square arrangement in Figure 1. (rearranging the pieces is not allowed). 1. Explain with full geometric justification, how you determined the angle measures of each piece. C. Analyze the properties of the polygons in Figure 2. by doing the following: 1. Identify the two given polygons, using the most specific classification possible (e.g., right scalene triangle). a. Prove that each tangram construction represents the polygon you have identified, using the side and angle measurements you verified in parts A and B. Note: You must geometrically prove the full classification of each polygon you have identified (e.g., a right scalene triangle must be shown to be both a right triangle and a scalene triangle). Your proofs must include a full justification of each statement you make.

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