Determine if it is possible to construct each of the following. If the construction is not possible, explain why. a. A square, given one side b. A rectangle, given one diagonal c. A triangle with two obtuse angles d. A parallelogram with exactly three right angles
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Sure! Let’s dive into these geometric constructions and see what's possible. For a square given one side, you can definitely construct it! Simply use a compass and straightedge to draw the given side, then create perpendicular lines at each endpoint using the construction of a right angle. The intersections of these lines will give you the remaining two vertices, and voilà! You’ve got a square! Now, onto the rectangle given one diagonal—this is also possible. By marking the endpoints of the diagonal, you can use the property of rectangles (which have right angles) to create perpendicular lines at each endpoint. The intersection points of these lines, spaced equally from the diagonal’s length, will give you the remaining two vertices of the rectangle! Now, for the construction of a triangle with two obtuse angles, we hit a snag! Recall that the sum of angles in any triangle must total 180 degrees. If two angles are obtuse (i.e., greater than 90 degrees), their sum would already be over 180 degrees, leaving no possibility for a third angle. So, this construction is not feasible! Lastly, a parallelogram with exactly three right angles is another trickster! If a parallelogram has three right angles, then it must also have a fourth angle that complements the sum to 360 degrees. Since each angle in a parallelogram must have either a right angle to equal 180 degrees, if three angles are right angles, the last one must also be a right angle, making the figure a rectangle. Therefore, this specific construction can't exist as described!
