Section 6.4 Rectangles

A rectangle is a parallelogram with four right angles. By definition, a rectangle has the following properties.  All four angles are right angles.  Opposite sides are parallel and congruent.  Opposite angles are congruent.  Consecutive angles are supplementary.  Diagonals bisect each other.

In addition, the diagonals of a rectangle are congruent.

Example 1: 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. LN = JN because… JN + LN=______ because… LN + LN=______ because… 2LN=______ because… 2(6.5) = ______ because….. diagonals of a parallelogram bisect each other JL Segment Addition JL Substitution JL Simplify JL Substitution 13 = JL, since the diagonals of a rectangle are congruent, then JL = KM, so KM = 13 feet.

Example 1: b) Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. If a figure is a rectangle, then the diagonals are congruent and bisect each other. If FH = 15, then GE = 15 and GJ will be half of that, which is 7.5.

Example 2: a) Quadrilateral RSTU is a rectangle. If m  RTU = (8x + 4)  and m  SUR = (3x – 2) , solve for x. m  TUR = 90  because … PT  PU because …  RTU   SUT because … m  RTU = m  SUT because … m  SUT + m  SUR = 90  because … m  RTU + m  SUR = 90  because … 8x x – 2 = 90  because…. You can use properties of rectangles along with algebra to find missing values. a rectangle has four right angles the diagonals of a rectangle bisect each other ∆PTU is isosceles, so the base  ’s are  definition of  segments Angle Addition Substitution

11x + 2 = 90  Add like terms 11x = 88  Subtract 2 from each side x = 8Divide each side by 11

Example 2: b) Quadrilateral EFGH is a rectangle. If m  FGE = (6x – 5)  and m  HFE = (4x – 5) , solve for x.

The converse of Theorem 6.13 is also true.

Example 3: 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. Because AB  CD and DA  BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle.

Example 4: 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. You can also use the properties of rectangles to prove that a quadrilateral positioned on a coordinate plane is a rectangle given the coordinates of the vertices. Use the Distance Formula to determine whether JKLM is a parallelogram by determining if opposite sides are congruent. Since opposite sides of a quadrilateral have the same measure, they are congruent. So, quadrilateral JKLM is a parallelogram.

Determine whether the diagonals of JKLM are congruent. Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle.