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• rectangle—a parallelogram with right angles Use Properties of Rectangles CONSTRUCTION A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM. Use Properties of Rectangles Since JKLM is a rectangle, it is a parallelogram. The diagonals of a parallelogram bisect each other, so LN = JN. JN + LN = JL Segment Addition LN + LN = JL Substitution 2LN = JL 2(6.5) = JL 13 = JL Simplify. Substitution Simplify. Use Properties of Rectangles JL KM If a is a rectangle, diagonals are . JL = KM Definition of congruence 13 = KM Substitution Answer: KM = 13 feet Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. A. 3 feet B. 7.5 feet C. 9 feet D. 12 feet A. B. C. D. A B C D Use Properties of Rectangles and Algebra Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x. Use Properties of Rectangles and Algebra Since RSTU is a rectangle, it has four right angles. So, mTUR = 90. The diagonals of a rectangle bisect each other and are congruent, so PT PU. Since triangle PTU is isosceles, the base angles are congruent so RTU SUT and mRTU = mSUT. mSUT + mSUR = 90 Angle Addition mRTU + mSUR = 90 Substitution 8x + 4 + 3x – 2 = 90 Substitution 11x + 2 = 90 Add like terms. Use Properties of Rectangles and Algebra 11x = 88 x = 8 Answer: x = 8 Subtract 2 from each side. Divide each side by 11. Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x. A. x = 1 B. x = 3 C. x = 5 D. x = 10 A. B. C. D. A B C D Proving Rectangle Relationships ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular. Proving Rectangle Relationships Since AB = CD, DA = BC, and AC = BD, AB CD, DA BC, and AC BD. Answer: Because AB CD and DA BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle. Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90? A. Since opp. sides are ||, STUR must be a rectangle. B. Since opp. sides are , STUR must be a rectangle. C. Since diagonals of the are , STUR must be a rectangle. D. STUR is not a rectangle. A. B. C. D. A B C D Rectangles and Coordinate Geometry Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula. Step 1 Use the Distance Formula to determine whether JKLM is a parallelogram by determining if opposite sides are congruent. Rectangles and Coordinate Geometry Since opposite sides of a quadrilateral have the same measure, they are congruent. So, quadrilateral JKLM is a parallelogram. Rectangles and Coordinate Geometry Step 2 Determine whether the diagonals of are congruent. JKLM Answer: Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle. Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). Determine whether WXYZ is a rectangle by using the Distance Formula. A. yes B. no C. cannot be determined 1. 2. 3. A B C Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). What are the lengths of diagonals WY and XZ? A. B. 4 C. 5 D. 25 A. B. C. D. A B C D