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

A rectangle is a parallelogram with four right angles.

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.

Rectangle1. Draw a rectangle on your paper.2. Draw diagonals in your rectangle.3. Measure the diagonals. Are the diagonals

congruent?4. Are the diagonals perpendicular?

A parallelogram is a rectangle if and only if its diagonals are congruent.

GEDF

DEFGGiven

anddiagonalswith

rectangleaisramParallelog:

ProofIf a parallelogram is a rectangle, then its diagonals are congruent.

GEDFProve :Statements

Reasons1. Given

2. All rect. are parallelogram

3. Def of rectangle

4. Def of rt triangle

5. Opp sides parallel congru

6. Reflexive

7. Leg-leg

8. CPCTC

1. DEFG is a rectangle

2. DEFG is a parallelogramanglesrightareand.3 EFGDGF

4. ∆DGF and ∆EFG are rt trianglesEFDG .5GFGF .6

EFGDGF .7GEDF .8

D E

FG

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 AdditionLN + LN = JL Substitution

2LN = JL Simplify.2(6.5) = JL Substitution

13 = JL Simplify.

JL = KM Definition of congruence13 = KM Substitution

JL KM If a is a rectangle, diagonals .

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.

Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

mSUT + mSUR = 90 Angle AdditionmRTU + mSUR = 90 Substitution

8x + 4 + 3x – 2 = 90 Substitution11x + 2 = 90 Add like terms.

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.

11x = 88 Subtract 2 from each side.x = 8 Divide each side by 11.

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.

Answer: Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle.

Step 2 Determine whether the diagonals of JKLMare congruent.

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