1. 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.

2. 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.

4. 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.

5. 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

7. 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

8. 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

9. 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 x – 2 = 90 Substitution
11x + 2 = 90 Add like terms.
11x = 88 Subtract 2 from each side.
x = 8 Divide each side by 11.

10. 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.

11. 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.

12. 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.

13. 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.

14. 6.4 Assignment
Page 426, 10-18, 22-23, 26-31