Transcript 6-4 Rectangles

Rhombuses, Rectangles, and
Squares
Section 6.4
Objectives
Use properties of special types of
parallelograms.
Key Vocabulary
Rhombus
Rectangle
Square
Theorems
Corollaries
Rhombus Corollary
Rectangle Corollary
Square Corollary
6.10 Diagonals of a Rhombus
6.13 Diagonals of a Rectangle
Special Parallelograms
In this section we will study 3 special
types of parallelograms.
Rhombus
Rectangle
Square
Rhombuses Or Rhombi
What makes a quadrilateral a rhombus?
Rhombuses Or Rhombi
A rhombus is an
equilateral
parallelogram.
Four congruent
sides.
Rhombus Corollary
If a quadrilateral has four congruent
sides, then it is a rhombus.
A
D
B
C
If AB  BC  CD  DA,
then ABCD is a rhombus.
Rhombi
Since rhombi are parallelograms, they
have all the properties of a
parallelogram.
Opposite sides are || and ≅.
Opposite s are ≅.
Consecutive s are supplementary.
Diagonals bisect each other.
In addition, rhombi have one other
property, which is a theorem.
Theorem 6.10
The diagonals of
a rhombus are
perpendicular.
If ABCD is a
rhombus then
AC BD.
B
A
C
D
Properties of a RHOMBUS
A
1. Two pairs of parallel sides.
2. All sides are congruent.
B
D
3. Diagonals are NOT congruent
4. Diagonals bisect each other
5. Diagonals form a right angle
C
6. Consecutive angles are supplementary.
m A + m B = 180°
m B + m C = 180°
m C + m D = 180°
m D + m A = 180°
Example 1
The diagonals of rhombus WXYZ intersect
at V. If WX = 8x – 5 and WZ = 6x + 3, find x.
Example 1
WX  WZ
WX = WZ
8x – 5 = 6x + 3
2x – 5 = 3
2x = 8
x = 4
Answer: x = 4
By definition, all sides of a
rhombus are congruent.
Definition of congruence
Substitution
Subtract 6x from each
side.
Add 5 to each side.
Divide each side by 4.
Your Turn:
ABCD is a rhombus. If
BC = 4x – 5 and CD = 2x + 7,
find x.
A. x = 1
B. x = 3
C. x = 4
D. x = 6
Example 2
Use rhombus LMNP to find the value of y if
N
Example 2
The diagonals of a rhombus are
perpendicular.
Substitution
Add 54 to each side.
Take the square root of each side.
Answer: The value of y can be 12 or –12.
N
Your Turn:
Use rhombus ABCD and
the given information to
find the value of each
variable.
Answer: 8 or –8
Rectangles
What makes a quadrilateral a
rectangle?
Rectangles
A rectangle is an
equiangular
parallelogram.
All angles are
congruent.
What must each
angle be then?
Four right angles.
Rectangle Corollary
If a quadrilateral
has four right
angles, then it is a
rectangle.
Rectangles
Since rectangles are parallelograms, they
have all their properties:
Opposite sides are || and ≅.
Opposite s are ≅.
Consecutive s are supplementary.
Diagonals bisect each other.
In addition, rectangles have their own
special property, which leads us to our next
theorem.
Theorem 6.11
Diagonals of a Rectangle are
congruent.
Review: Properties of Rectangles
1) All four s are right s (def. of rectangle).
2) Opposite sides are || and ≅ (prop. of
parallelogram).
3) Opposite s are ≅ (prop. of parallelogram).
4) Consecutive s are supplementary (prop.
of parallelogram).
5) Diagonals bisect each other (prop. of
parallelogram).
6) Diagonals are ≅ (theorem 6.11).
Example 3:
Quadrilateral RSTU is a rectangle. If
find x.
and
Example 3:
The diagonals of a rectangle are congruent,
Definition of congruent segments
Substitution
Subtract 6x from each side.
Add 4 to each side.
Answer: 8
Your Turn:
Quadrilateral EFGH is a rectangle. If
find x.
Answer: 5
and
Example 4
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.
Example 4
Since JKLM is a rectangle, it is a
parallelogram. The diagonals of
a parallelogram bisect each
other, so LN = JN.
LN = 6.5 feet
JN + LN = JL
Segment Addition
LN + LN = JL
Substitution
2LN = JL
2(6.5) = JL
13 = JL
Simplify.
Substitution
Simplify.
Example 4
JL  KM
If a is a rectangle,
diagonals are .
JL = KM
Definition of congruence
13 = KM
Substitution
Answer: KM = 13 feet
Your Turn:
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
Example 5
Quadrilateral RSTU is a rectangle. If mRTU =
8x + 4 and mSUR = 3x – 2, find x.
Example 5
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RTU = 8x + 4
mSUR = 3x – 2
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.
Example 5
11x = 88
x = 8
Answer: x = 8
Subtract 2 from each
side.
Divide each side by 11.
Your Turn:
Quadrilateral EFGH is a rectangle. If mFGE =
6x – 5 and mHFE = 4x – 5, find x.
A. x = 1
B. x = 3
C. x = 5
D. x = 10
Example 6a:
Quadrilateral LMNP is a rectangle. Find x.
Example 6a:
Angle Addition Theorem
Substitution
Simplify.
Subtract 10 from each side.
Divide each side by 8.
Answer: 10
Example 6b:
Quadrilateral LMNP is a rectangle. Find y.
Example 6b:
Since a rectangle is a parallelogram, opposite sides are parallel. So,
alternate interior angles are congruent.
Alternate Interior Angles Theorem
Substitution
Simplify.
Subtract 2 from each side.
Divide each side by 6.
Answer: 5
Your Turn:
Quadrilateral EFGH is a rectangle.
a. Find x.
b. Find y.
Answer: 7
Answer: 11
Squares
What makes a quadrilateral a square?
Definition: Square
A square is a parallelogram with four
congruent sides and four right angles.
Squares
A square is a
regular
parallelogram.
All angles are
congruent
All sides are
congruent
Square Corollary
If a quadrilateral has
four congruent
sides and four right
angles, then it is a
square.
Venn Diagram
Shows the relationships between some
members of the parallelogram family.
Properties of a SQUARE
A
B
1. Two pairs of parallel sides.
2. All sides are congruent.
3. All angles are right.
4. Diagonals are congruent
D
C
5. Diagonals bisect each other
6. Diagonals form a right angle
7. Opposite angles are congruent.
8. Consecutive angles are supplementary.
m A + m B = 180°
m B + m C = 180°
m C + m D = 180°
m D + m A = 180°
EX. 7
D
5X+5
G
DEFG is a square
DG = 5X + 5
EF = 7X – 19
E
7X – 19
F
Find the value for X and the lenght of the
side.
Since all sides are congruent:
5X + 5 = 7X – 19
-5X
-5X
5 = 2X – 19
+19
+19
24 = 2X
2
2
Now since all sides are congruent,
we need to find the length of just
one side:
DG = 5X + 5
= 5(12 ) + 5
= 60 + 5
= 65
X=12
The length of the side is 65.
Your Turn:
K
9X – 3
H
HIJK is a square
KH =9X – 3
IJ = 6X + 24
J
6X + 24
I
Find the value for X and the lenght of the
side.
Since all sides are congruent:
9X – 3 = 6X + 24
-6X
-6X
3X – 3 = 24
+3 +3
3x = 27
3
3
Now since all sides are congruent,
we need to find the length of just
one side:
KH = 9X – 3
= 9( 9 ) – 3
= 81 – 3
= 78
X=9
The length of the side is 78.
Summary of Properties
Parallelogram, Rectangle,
Rhombus, and Square
Quadrilateral Relationships
1. Opposite sides parallel.
2. Opposite sides congruent.
3. Opposite angles are congruent.
4. Consecutive ∠s are supplementary.
5. Diagonals bisect each other.
1. Has 4 Congruent sides.
2. Diagonals are perpendicular.
1. Has 4 right angles.
2. Diagonals are congruent.
1. 4 congruent sides
2. 4 congruent (right) ∠s
Characteristics
Both pairs of opposite sides parallel
Parallelogram
Rectangle
Rhombus
Square
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Diagonals are congruent
Both pairs of opposite sides congruent
x
At least one right angle
Both pairs of opposite angles congruent
Diagonals are perpendicular
All sides are congruent
Consecutive angles congruent
Diagonals bisect each other
Consecutive angles supplementary
x
x
x
x
x
x
Review Problems
Example 1
In the diagram, ABCD is a rectangle.
a. Find AD and AB.
b. Find mA, mB, mC, and mD.
SOLUTION
a. By definition, a rectangle is a parallelogram, so
ABCD is a parallelogram. Because opposite sides
of a parallelogram are congruent, AD = BC = 5 and
AB = DC = 8.
b. By definition, a rectangle has four right angles, so
mA = mB = mC = mD = 90°.
Your Turn:
In the diagram, PQRS is a rhombus.
Find QR, RS, and SP.
ANSWER
QR = 6, RS = 6, SP = 6
Example 2
Use the information in the diagram
to name the special quadrilateral.
SOLUTION
The quadrilateral has four right angles. So, by the
Rectangle Corollary, the quadrilateral is a rectangle.
Because all of the sides are not the same length, you
know that the quadrilateral is not a square.
Your Turn:
Use the information in the diagram to name the
special quadrilateral.
1.
ANSWER
rhombus
ANSWER
square
2.
Example 3
ABCD is a rhombus. Find the value of x.
SOLUTION
By Theorem 6.10, the diagonals of a rhombus are
perpendicular. Therefore, BEC is a right angle, so
∆BEC is a right triangle.
By the Corollary to the Triangle Sum Theorem, the
acute angles of a right triangle are complementary.
So, x = 90 – 60 = 30.
Example 4
a. You nail four pieces of wood together
to build a four-sided frame, as shown.
What is the shape of the frame?
b. The diagonals measure 7 ft 4 in. and
7 ft 2 in. Is the frame a rectangle?
SOLUTION
a. The frame is a parallelogram because both pairs of
opposite sides are congruent.
b. The frame is not a rectangle because the diagonals
are not congruent.
Your Turn:
Find the value of x.
1. rhombus ABCD
ANSWER
90
2. rectangle EFGH
ANSWER
12
3. square JKLM
ANSWER
45
Match the properties of a quadrilateral
1. The diagonals are
A.Parallelogram
congruent B,D
B. Rectangle
2. Both pairs of opposite
C.Rhombus
sides are congruent A,B,C,D D.Square
3. Both pairs of opposite
sides are parallel A,B,C,D
4. All angles are congruent B,D
5. All sides are congruent C,D
Decide if the statement is sometimes, always, or never
true.
1. A rhombus is equilateral.
Always
2. The diagonals of a rectangle are ⊥.
Sometimes
3. The opposite angles of a rhombus are supplementary.
4. A square is a rectangle.
Sometimes
Always
5. The diagonals of a rectangle bisect each other.
Always
6. The consecutive angles of a square are supplementary.
Always
Joke Time
Why did the geometry student get so excited
after they finished a jigsaw puzzle in only 6
months?
Because on the box it said from 2-4 years.
Why did the geometry student climb the chainlink fence?
To see what was on the other side.
How did the geometry student die drinking
milk?
The cow fell on them.
Assignment
Pg. 328 - 330 #1 – 31 odd