Lesson 6-4

Rectangles

• rectangle

• Recognize and apply properties of rectangles.

• Determine whether parallelograms are rectangles.

Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)

Rectangle

• Def—A //ogram with 4 Right Angles

Properties of a Rectangle

• Rectangle Diagonals are • (Also has all the properties of a //ogram.)

–Opposite sides –Opposite angles –Consecutive angles supplementary

–Diagonals bisect each other

Given ABCD is a Rectangle, list everything that must be true.

CDAB // BCAD //

BCAD

90DmCmBmAm

BDAC

CDAB

A B

CD

E

#5: Diagonals bisect each other.

#4: Consec. s are Supp.

#3: Opp. s are

#2: Opp. Sides are

#1: Diagonals are

Def: 4 rt. s

//ogram: Opp. Sides //

ECAE EBDE

Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x.

Diagonals of a Rectangle

Answer: 8

Diagonals of a Rectangle

The diagonals of a rectangle are congruent,

Definition of congruent segments

Substitution

Subtract 6x from each side.

Add 4 to each side.

Diagonals of a rectangle are .

A. A

B. B

C. C

D. D

A. x = –1

B. x = 3

C. x = 5

D. x = 10

Quadrilateral EFGH is a rectangle. If FH = 5x + 4 and GE = 7x – 6, find x.

Angles of a Rectangle

Quadrilateral LMNP is a rectangle. Find x.

Angles of a Rectangle

Answer: 10

Angle Addition Postulate

Substitution

Simplify.

Subtract 10 from each side.Divide each side by 8.

1. A

2. B

3. C

4. D

A. 6

B. 7

C. 9

D. 14

Quadrilateral EFGH is a rectangle. Find x.

Reminder

• Perpendicular lines have opposite reciprocal slopes.

– Prove the sides of a quadrilateral are perpendicular and you have proven it is a rectangle.

Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0), and D(–1, –2). Determine whether ABCD is a rectangle using the Slope Formula.

Rectangle on a Coordinate Plane

Method 1: Use the Slope Formula, to see if

opposite sides are parallel and consecutive sides are

perpendicular.

Rectangle on a Coordinate Plane

= Slopes // lines

Opp. Reciprocal Slopes lines

//ogram with 4 right angles Rectangle

Rectangle on a Coordinate Plane

Method 2: Use the Distance Formula,

to determine whether opposite sides are congruent.

Rectangle on a Coordinate Plane

Opp. Sides //ogram

Find the length of the diagonals.

//ogram w/ Diagonals Rectangle

1. A

2. B

3. C

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 using the Distance Formula.

A. A

B. B

C. C

D. D

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

HomeworkHomework

• pg 344:pg 344:

1, 2, 7, 8, 10, 13-21, 1, 2, 7, 8, 10, 13-21, 27-2927-29