6-4 Rectangles You used properties of parallelograms and determined whether quadrilaterals were parallelograms. Recognize and apply properties of rectangles. Determine whether parallelograms are rectangles.

Rectangle Definition Rectangle 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. Diagonals are congruent. A rectangle is a parallelogram with four right angles.

Rectangle Draw a rectangle on your paper. Draw diagonals in your rectangle. Measure the diagonals. Are the diagonals congruent? Are the diagonals perpendicular? A parallelogram is a rectangle if and only if its diagonals are congruent.

If a parallelogram is a rectangle, then its diagonals are congruent. Proof D E G F Reasons Given All rect. are parallelogram Def of rectangle Def of rt triangle Opp sides parallel congru Reflexive Leg-leg CPCTC Statements DEFG is a rectangle DEFG is a parallelogram 4. ∆DGF and ∆EFG are rt triangles

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. 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 Simplify. 2(6.5) = JL Substitution 13 = JL Simplify. JL  KM If a is a rectangle, diagonals . JL = KM Definition of congruence 13 = KM Substitution

Quadrilateral EFGH is a rectangle 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

Quadrilateral RSTU is a rectangle Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x. 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. mSUT + mSUR = 90 Angle Addition mRTU + mSUR = 90 Substitution 8x + 4 + 3x – 2 = 90 Substitution 11x + 2 = 90 Add like terms. 11x = 88 Subtract 2 from each side. x = 8 Divide each side by 11.

Max is building a swimming pool in his backyard 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.

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. Since opposite sides of a quadrilateral have the same measure, they are congruent. So, quadrilateral JKLM is a parallelogram.

Step 2 Determine whether the diagonals of JKLM are congruent. Answer: Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle.

Rectangle 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. Diagonals are congruent.

6.4 Assignment Page 426, 10-18, 22-23, 26-31