A carpenter must ensure that a large window frame is a rectangle. If the corners are labled E,F,G, and H, the carpenter knows that EF=51 inches, FG=28 inches, GH=51 inches, and EH=28 inches. The carpenter could measure the diagonals EG and FH and verify that they are the same length, nut choose another approach. After verifying that angle E is a right angle the carpenter insists that EFGH must be a true rectangle. Finish the reasoning outlined below to justify the carpenter's logic.
1)EFGH is a parallelogram since__________________________.
2)consecutive angles are supplementary by Therom______, so angles F,G, and H are also right angles.
3)Therfore, by the ______________ Corollary, EFGH is a _________________.
1) EFGH is a parallelogram since opposite sides EF and GH are equal in length (51 inches) and also opposite sides FG and EH are equal in length (28 inches). This is one of the properties of a parallelogram - opposite sides are congruent.
2) Consecutive angles are supplementary by the theorem known as "the sum of the interior angles of a quadrilateral." This theorem states that the sum of the interior angles of a quadrilateral is equal to 360 degrees. Since angle E is a right angle (90 degrees), the other three angles, F, G, and H, must also be right angles. This is because the sum of the angles F, G, H, and E must be equal to 360 degrees.
3) Therefore, by the "converse of the definition of a rectangle" corollary, EFGH is a rectangle. This corollary states that if a quadrilateral has four right angles, then it is a rectangle. Since we have established that angles F, G, and H are right angles, along with angle E being a right angle, we can conclude that EFGH is a true rectangle based on this corollary.