Vocabulary rectangle—a parallelogram with right angles

Concept 1

Example 1 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.

Example 1 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=JLSegment Addition LN + LN=JLSubstitution 2LN=JLSimplify. 2(6.5)=JLSubstitution 13=JLSimplify.

Example 1 Use Properties of Rectangles Answer: KM = 13 feet JL = KMDefinition of congruence 13 = KMSubstitution JL  KMIf a is a rectangle, diagonals are .

A.A B.B C.C D.D Example 1 A.3 feet B.7.5 feet C.9 feet D.12 feet Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Example 2 Use Properties of Rectangles and Algebra Quadrilateral RSTU is a rectangle. If m  RTU = 8x + 4 and m  SUR = 3x – 2, find x.

Example 2 Use Properties of Rectangles and Algebra m  SUT + m  SUR=90Angle Addition m  RTU + m  SUR=90Substitution 8x x – 2=90Substitution 11x + 2=90Add like terms. Since RSTU is a rectangle, it has four right angles. So, m  TUR = 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 m  RTU = m  SUT.

Example 2 Use Properties of Rectangles and Algebra Answer: x = 8 11x=88Subtract 2 from each side. x=8Divide each side by 11.

A.A B.B C.C D.D Example 2 A.x = 1 B.x = 3 C.x = 5 D.x = 10 Quadrilateral EFGH is a rectangle. If m  FGE = 6x – 5 and m  HFE = 4x – 5, find x.

Concept 2

Example 3 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.

Example 3 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.

A.A B.B C.C D.D Example 3 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.

Example 4 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 1Use the Distance Formula to determine whether JKLM is a parallelogram by determining if opposite sides are congruent.

Example 4 Rectangles and Coordinate Geometry Since opposite sides of a quadrilateral have the same measure, they are congruent. So, quadrilateral JKLM is a parallelogram.

Example 4 Rectangles and Coordinate Geometry Answer:Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle. Step 2Determine whether the diagonals of JKLM are congruent.

1.A 2.B 3.C Example 4 A.yes B.no C.cannot be determined 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.A B.B C.C D.D Example 4 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