The proofs convinced Detective Reese that the parallelogram postulates were true. Opposite angles in a parallelogram are congruent and adjacent angles are supplementary. Furthermore, a parallelogram is a rectangle if and only if its diagonals are congruent. But how could Reese use this evidence to learn more about specific parallelograms. In other words, how could he use these postulates to solve parallelogram problems? Reese consulted a forensic audio-visual specialist to find out. Then, he entered into evidence this video.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
In this video, we will use Parallelogram Postulates to solve problems involving parallelograms. Three postulates regarding parallelograms are listed here: opposite angles are congruent, adjacent angles are supplementary, and diagonals in a rectangle (which is a special parallelogram) are congruent.
In parallelogram ABCD, we know the measure of angle CAD is 25 degrees… and the measure of angle ACD is 80 degrees… What is the measure of angle DCB? Let us not forget that opposite sides of a parallelogram are parallel, and its diagonal acts like a transversal that crosses parallel lines… Therefore we know the measure of angle BCA is equal to the measure of CAD, 25 degrees because alternate interior angles of parallel lines are congruent. By adding together the measure of angle BCA and ACD we see the measure of angle DCB is 110 degrees...
Now that we know the measure of angle DCB, we can easily find the measure of angle CBA. These two angles are adjacent, or next to, one another in the parallelogram, therefore they are supplementary according to the second postulate from above. Supplementary angles sum to 180 degrees, so 110 plus the measure of angle CBA equals 180 degrees… Simple subtraction shows the measure of CBA is 70 degrees.
Now we turn our attention to a special parallelogram, rectangle EFGH. We are given dimensions of three segment lengths and are asked to find the length of segment EG. Since EG and FH are both diagonals of a rectangle, we know they are congruent by the third postulate from above. By adding the lengths of segment FK and KH we see that the total length of FH is 25 inches. Therefore the length of segment EG is also 25 inches...
Finally, let's determine the perimeter of rectangle EFGH. The perimeter is the total length around the outer edge of the rectangle. We know the two side lengths to be 7 inches, but the lengths of the top and bottom are currently unknown. However, we can find these lengths if you recognize the right triangle that shares the side length and diagonal of the rectangle. Using the Pythagorean equation, we can find the missing side length... The bottom and top of the rectangle both measure 24 inches in length. Therefore the perimeter is the sum of two sevens and two twenty-fours, or sixty-two inches.
Detective Reese was satisfied that his investigation was complete. He understood the parallelogram postulates, how to prove them, and how to apply them. He was now ready to conclude his investigation.
Question
Rectangle FGHK has diagonals FH and GK. If half of FH = 10 inches, how long is GK?