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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.
Rectangle
1. 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.
If a parallelogram is a
rectangle, then its
diagonals are congruent.
Proof
D
E
G
F
Given : Parallelog ram DEFG is a rectangle
with diagonals DF and GE
Prove : DF  GE
Statements
1. DEFG is a rectangle
2. DEFG is a parallelogram
Reasons
1. Given
2. All rect. are parallelogram
3. DGF and EFG are right angles
3. Def of rectangle
4. ∆DGF and ∆EFG are rt triangles
4. Def of rt triangle
5. DG  EF
5. Opp sides parallel congru
6. GF  GF
6. Reflexive
7. DGF  EFG
8. DF  GE
7. Leg-leg
8. CPCTC
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
LN + LN = JL
2LN = JL
2(6.5) = JL
13 = JL
JL  KM
JL = KM
13 = KM
Segment Addition
Substitution
Simplify.
Substitution
Simplify.
If a is a rectangle, diagonals .
Definition of congruence
Substitution
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. 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
11x = 88
x = 8
Add like terms.
Subtract 2 from each side.
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.
Step 2
Answer:
Determine whether the diagonals of
are congruent.
JKLM
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