6.4 Rhombuses, Rectangles and Squares

1
6.4 Rhombuses, Rectangles and Squares

• Geometry CCSS G.CO 11.
• PROVE theorems about parallelograms. Theorems
  INCLUDE opposite sides ARE congruent, opposite
  angles ARE congruent, the diagonals of a
  parallelogram BISECT each other, and conversely,
  rectangles ARE parallelograms with congruent
  diagonals.

2
U.E.Q

• What are the properties of different
  quadrilaterals? How do we use the formulas of
  areas of different quadrilaterals to solve
  real-life problems?
• Now a little review

3
Standards for Mathematical Practice
11 PROVE theorems about parallelograms. Theorems
INCLUDE opposite sides ARE congruent, opposite
angles ARE congruent, the diagonals of a
parallelogram BISECT each other, and conversely,
rectangles ARE parallelograms with congruent
diagonals.

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

4
Do you have enough information?
Yes
Yes
No
Yes
5
Do you have enough information?
Yes
No
Yes
Yes
6
Activator
Test your prior knowledge and try to fill in the
chart with properties of the following
quadrilaterals
7
Objectives

• Use properties of sides and angles of rhombuses,
  rectangles, and squares.
• Use properties of diagonals of rhombuses,
  rectangles and squares.

8
E.Q

• What are properties of sides and angles of
  rhombuses, rectangles, and squares?

9
Properties of Special Parallelograms

• In this lesson, you will study three special
  types of parallelograms rhombuses, rectangles
  and squares.

A rectangle is a parallelogram with four right
angles.
A rhombus is a parallelogram with four congruent
sides
A square is a parallelogram with four congruent
sides and four right angles.
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Venn Diagram shows relationships-- MEMORIZE

• Each shape has the properties of every group that
  it belongs to. For instance, a square is a
  rectangle, a rhombus and a parallelogram so it
  has all of the properties of those shapes.

parallelograms
rhombuses
rectangles
squares
11
Some examples of a rhombus
12
Examples of rectangles
13
Examples of squares
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Ex. 1 Describing a special parallelogram

• Decide whether the statement is always,
  sometimes, or never true.
• A rhombus is a rectangle.
• A parallelogram is a rectangle.

parallelograms
rhombuses
rectangles
squares
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Ex. 1 Describing a special parallelogram

• Decide whether the statement is always,
  sometimes, or never true.
• A rhombus is a rectangle.
• The statement is sometimes true. In the
  Venn diagram, the regions for rhombuses and
  rectangles overlap. IF the rhombus is a square,
  it is a rectangle.

parallelograms
rhombuses
rectangles
squares
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Ex. 1 Describing a special parallelogram

• Decide whether the statement is always,
  sometimes, or never true.
• A parallelogram is a rectangle.
• The statement is sometimes true. Some
  parallelograms are rectangles. In the Venn
  diagram, you can see that some of the shapes in
  the parallelogram box are in the area for
  rectangles, but many arent.

parallelograms
rhombuses
rectangles
squares
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Ex. 2 Using properties of special parallelograms

• ABCD is a rectangle. What else do you know about
  ABCD?

• Because ABCD is a rectangle, it has four right
  angles by definition. The definition also states
  that rectangles are parallelograms, so ABCD has
  all the properties of a parallelogram
• Opposite sides are parallel and congruent.
• Opposite angles are congruent and consecutive
  angles are supplementary.
• Diagonals bisect each other.

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Take note

• A rectangle is defined as a parallelogram with
  four right angles. But any quadrilateral with
  four right angles is a rectangle because any
  quadrilateral with four right angles is a
  parallelogram.
• Corollaries about special quadrilaterals
• Rhombus Corollary A quadrilateral is a rhombus
  if and only if it has four congruent sides.
• Rectangle Corollary A quadrilateral is a
  rectangle if and only if it has four right
  angles.
• Square Corollary A quadrilateral is a square if
  and only if it is a rhombus and a rectangle.
• You can use these to prove that a quadrilateral
  is a rhombus, rectangle or square without proving
  first that the quadrilateral is a parallelogram.

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Characteristics Parallelogram Rectangle Rhombus Square
Both pairs of opposite sides parallel
Diagonals are congruent
Both pairs of opposite sides congruent
At least one right angle
Both pairs of opposite angles congruent
Exactly one pair of opposite sides parallel
Diagonals are perpendicular
All sides are congruent
Consecutive angles congruent
Diagonals bisect each other
Diagonals bisect opposite angles
Consecutive angles supplementary
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Ex. 3 Using properties of a Rhombus

• In the diagram at the right,
• PQRS is a rhombus. What
• is the value of y?
• All four sides of a rhombus are ?, so RS PS.
• 5y 6 2y 3 Equate lengths of ? sides.
• 5y 2y 9 Add 6 to each side.
• 3y 9 Subtract 2y from each side.
• y 3 Divide each side by 3.

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Using diagonals of special parallelograms

• The following theorems are about diagonals of
  rhombuses and rectangles.
• Theorem 6.11 A parallelogram is a rhombus if
  and only if its diagonals are perpendicular.
• ABCD is a rhombus if and only if AC? BD.

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Using diagonals of special parallelograms

• Theorem 6.12 A parallelogram is a rhombus if
  and only if each diagonal bisects a pair of
  opposite angles.
• ABCD is a rhombus if and only if AC bisects ?DAB
  and ?BCD and BD bisects ?ADC and ?CBA.

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Using diagonals of special parallelograms
A
B

• Theorem 6.13 A parallelogram is a rectangle if
  and only if its diagonals are congruent.
• ABCD is a rectangle if and only if AC ? BD.

C
D
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NOTE

• You can rewrite Theorem 6.11 as a conditional
  statement and its converse.
• Conditional statement If the diagonals of a
  parallelogram are perpendicular, then the
  parallelogram is a rhombus.
• Converse If a parallelogram is a rhombus, then
  its diagonals are perpendicular.

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given
• Def. of ?. Diagonals bisect each other.

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given
• Def. of ?. Diagonals bisect each other.
• Def. of ?. Diagonals bisect each other.

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given
• Def. of ?. Diagonals bisect each other.
• Def. of ?. Diagonals bisect each other.
• SSS congruence post.

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given
• Def. of ?. Diagonals bisect each other.
• Def. of ?. Diagonals bisect each other.
• SSS congruence post.
• CPCTC

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Ex. 4 Proving Theorem 6.11Given ABCD is a
rhombusProve AC ? BD

• Statements
• ABCD is a rhombus
• AB ? CB
• AX ? CX
• BX ? DX
• ?AXB ? ?CXB
• ?AXB ? ?CXB
• AC ? BD

• Reasons
• Given
• Given
• Def. of ?. Diagonals bisect each other.
• Def. of ?. Diagonals bisect each other.
• SSS congruence post.
• CPCTC
• Congruent Adjacent ?s

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Ex. 5 Coordinate Proof of Theorem 6.11Given
ABCD is a parallelogram, AC ? BD.Prove ABCD is
a rhombus

• Assign coordinates. Because AC? BD, place ABCD
  in the coordinate plane so AC and BD lie on the
  axes and their intersection is at the origin.
• Let (0, a) be the coordinates of A, and let (b,
  0) be the coordinates of B.
• Because ABCD is a parallelogram, the diagonals
  bisect each other and OA OC. So, the
  coordinates of C are (0, - a). Similarly the
  coordinates of D are (- b, 0).

A(0, a)
D(- b, 0)
B(b, 0)
C(0, - a)
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Ex. 5 Coordinate Proof of Theorem 6.11Given
ABCD is a parallelogram, AC ? BD.Prove ABCD is
a rhombus

• Find the lengths of the sides of ABCD. Use the
  distance formula (See youre never going to get
  rid of this)
• ABv(b 0)2 (0 a)2 vb2 a2
• BC v(0 - b)2 ( a - 0)2 vb2 a2
• CD v(- b 0)2 0 - ( a)2 vb2 a2
• DA v(0 (- b)2 (a 0)2 vb2 a2

A(0, a)
D(- b, 0)
B(b, 0)
C(0, - a)
All the side lengths are equal, so ABCD is a
rhombus.
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Ex 6 Checking a rectangle
4 feet

• CARPENTRY. You are building a rectangular frame
  for a theater set.
• First, you nail four pieces of wood together as
  shown at the right. What is the shape of the
  frame?
• To make sure the frame is a rectangle, you
  measure the diagonals. One is 7 feet 4 inches.
  The other is 7 feet 2 inches. Is the frame a
  rectangle? Explain.

6 feet
6 feet
4 feet
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Ex 6 Checking a rectangle
4 feet

• First, you nail four pieces of wood together as
  shown at the right. What is the shape of the
  frame?
• Opposite sides are congruent, so the frame is a
  parallelogram.

6 feet
6 feet
4 feet
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Ex 6 Checking a rectangle
4 feet

• To make sure the frame is a rectangle, you
  measure the diagonals. One is 7 feet 4 inches.
  The other is 7 feet 2 inches. Is the frame a
  rectangle? Explain.
• The parallelogram is NOT a rectangle. If it were
  a rectangle, the diagonals would be congruent.

6 feet
6 feet
4 feet
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Youve just had a new door installed, but it
doesnt seem to fit into the door jamb properly.
What could you do to determine if your new door
is rectangular?
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Foldable
1. Take out a piece of notebook paper and make a
hot dog fold over from the right side over to the
pink line.
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Foldable
The fold crease
2. Now, divide the right hand section into 5
sections by drawing 4 evenly spaced lines.
3. Use scissors to cut along your drawn line, but
ONLY to the crease!
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Foldable
The fold crease
4. Write QUADRILATERALS down the left hand side
QUADRILATERALS
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Foldable
The fold crease
5. Fold over the top cut section and write
PARALLELOGRAM on the outside.
Parallelogram
6. Reopen the fold.
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Foldable
7. On the left hand section, draw a parallelogram.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
8. On the right hand side, list all of the
properties of a parallelogram.
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Foldable
Fold over the second cut section and write
RECTANGLE on the outside.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
RECTANGLE
Reopen the fold.
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Foldable
On the left hand section, draw a rectangle.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
1. Special parallelogram. 2. Has 4 right
angles 3. Diagonals are congruent.
On the right hand side, list all of the
properties of a rectangle.
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Foldable
Fold over the third cut section and write
RHOMBUS on the outside.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
1. Special parallelogram. 2. Has 4 right
angles 3. Diagonals are congruent.
Reopen the fold.
RHOMBUS
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Foldable
On the left hand section, draw a rhombus.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
1. Special parallelogram. 2. Has 4 right
angles 3. Diagonals are congruent.
On the right hand side, list all of the
properties of a rhombus.
1. Special Parallelogram 2. Has 4 Congruent
sides 3. Diagonals are perpendicular. 4.
Diagonals bisect opposite angles
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Foldable
Fold over the third cut section and write
SQUARE on the outside.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
1. Special parallelogram. 2. Has 4 right
angles 3. Diagonals are congruent.
Reopen the fold.
1. Special Parallelogram 2. Has 4 Congruent
sides 3. Diagonals are perpendicular. 4.
Diagonals bisect opposite angles
SQUARE
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Foldable
On the left hand section, draw a square.
1. Opposite angles are congruent. 2. Consecutive
angles are supplementary. 3. Opposite sides are
congruent. 4. Diagonals bisect each other. 5.
Opposite sides are parallel
1. Special parallelogram. 2. Has 4 right
angles 3. Diagonals are congruent.
On the right hand side, list all of the
properties of a square.
1. Special Parallelogram 2. Has 4 Congruent
sides 3. Diagonals are perpendicular. 4.
Diagonals bisect opposite angles
Place in your notebook and save for tomorrow.
1. All the properties of parallelogram,
rectangle, and rhombus 2. 4 congruent sides and 4
right angles
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Warm-Up

• Name the figure described.
• A quadrilateral that is both a rhombus and a
  rectangle.
• A quadrilateral with exactly one pair of parallel
  sides.
• A parallelogram with perpendicular diagonals

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6.5 Trapezoids and Kites
51
U.E.Q

• What are the properties of different
  quadrilaterals? How do we use the formulas of
  areas of different quadrilaterals to solve
  real-life problems?

52
E.Q

• What are some properties of trapezoids and kits?

53
Objectives

• Use properties of trapezoids.
• Use properties of kites.

54
Using properties of trapezoids

• A trapezoid is a quadrilateral with exactly one
  pair of parallel sides. The parallel sides are
  the bases. A trapezoid has two pairs of base
  angles. For instance in trapezoid ABCD ?D and ?C
  are one pair of base angles. The other pair is
  ?A and ?B. The nonparallel sides are the legs of
  the trapezoid.

55
Using properties of trapezoids

• If the legs of a trapezoid are congruent, then
  the trapezoid is an isosceles trapezoid.

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Trapezoid Theorems

• Theorem 6.14
• If a trapezoid is isosceles, then each pair of
  base angles is congruent.
• ?A ? ?B, ?C ? ?D

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Trapezoid Theorems

• Theorem 6.15
• If a trapezoid has a pair of congruent base
  angles, then it is an isosceles trapezoid.
• ABCD is an isosceles trapezoid

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Trapezoid Theorems

• Theorem 6.16
• A trapezoid is isosceles if and only if its
  diagonals are congruent.
• ABCD is isosceles if and only if AC ? BD.

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Midsegment of a trapezoid

• The midsegment of a trapezoid is the segment that
  connects the midpoints of its legs. Theorem 6.17
  is similar to the Midsegment Theorem for
  triangles.

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Theorem 6.17 Midsegment of a trapezoid

• The midsegment of a trapezoid is parallel to each
  base and its length is one half the sums of the
  lengths of the bases.
• MNAD, MNBC
• MN ½ (AD BC)

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Ex. 3 Finding Midsegment lengths of trapezoids

• LAYER CAKE A baker is making a cake like the one
  at the right. The top layer has a diameter of 8
  inches and the bottom layer has a diameter of 20
  inches. How big should the middle layer be?

62
Ex. 3 Finding Midsegment lengths of trapezoids
E
F

• Use the midsegment theorem for trapezoids.
• DG ½(EF CH)
• ½ (8 20) 14

D
G
D
C
63
Using properties of kites

• A kite is a quadrilateral that has two pairs of
  consecutive congruent sides, but opposite sides
  are not congruent.

64
Kite theorems

• Theorem 6.18
• If a quadrilateral is a kite, then its diagonals
  are perpendicular.
• AC ? BD

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Kite theorems

• Theorem 6.19
• If a quadrilateral is a kite, then exactly one
  pair of opposite angles is congruent.
• ?A ? ?C, ?B ? ?D

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Ex. 4 Using the diagonals of a kite

• WXYZ is a kite so the diagonals are
  perpendicular. You can use the Pythagorean
  Theorem to find the side lengths.
• WX v202 122 23.32
• XY v122 122 16.97
• Because WXYZ is a kite, WZ WX 23.32, and ZY
  XY 16.97

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Ex. 5 Angles of a kite

• Find m?G and m?J
• in the diagram at the
• right.
• SOLUTION
• GHJK is a kite, so ?G ? ?J and m?G m?J.
• 2(m?G) 132 60 360Sum of measures of int.
  ?s of a quad. is 360
• 2(m?G) 168Simplify
• m?G 84 Divide
  each side by 2.
• So, m?J m?G 84

132
60
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6.6 Special Quadrilaterals

• Geometry

70
Standards for Mathematical Practice
11 PROVE theorems about parallelograms. Theorems
INCLUDE opposite sides ARE congruent, opposite
angles ARE congruent, the diagonals of a
parallelogram BISECT each other, and conversely,
rectangles ARE parallelograms with congruent
diagonals.

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

71
CCSS G.CO.11

• PROVE theorems about parallelograms. Theorems
  INCLUDE opposite sides ARE congruent, opposite
  angles ARE congruent, the diagonals of a
  parallelogram BISECT each other, and conversely,
  rectangles ARE parallelograms with congruent
  diagonals.

72
Warm-up

• Name the figure
• 1. a quadrilateral with exactly one pair of
  opposite angles congruent and perpendicular
  diagonals
• 2. a quadrilateral that is both a rhombus and a
  rectangle
• 3. a quadrilateral with exactly one pair of ll
  sides.
• 4. any llogram with perpendicular diagonals.

73
Objectives

• Identify special quadrilaterals based on limited
  information.
• Prove that a quadrilateral is a special type of
  quadrilateral, such as a rhombus or trapezoid.

74
E.Q

• How do we simplify real life tasks, checking if
  an object is rectangular, rhombus, trapezoid,
  kite or other quadrilateral?

75
Summarizing Properties of Quadrilaterals
Quadrilateral

• In this chapter, you have studied the seven
  special types of quadrilaterals shown at the
  right. Notice that each shape has all the
  properties of the shapes linked above it. For
  instance, squares have the properties of
  rhombuses, rectangles, parallelograms, and
  quadrilaterals.

Trapezoid
Kite
Parallelogram
Rhombus
Rectangle
Isosceles trapezoid
Square
76
Ex. 1 Identifying Quadrilaterals

• Quadrilateral ABCD has at least one pair of
  opposite sides congruent. What kinds of
  quadrilaterals meet this condition?

Parallelogram
Rhombus
Opposites sides are ?.
All sides are congruent.
Opposite sides are congruent.
Legs are congruent.
All sides are congruent.
77
Ex. 2 Connecting midpoints of sides

• When you join the midpoints of the sides of any
  quadrilateral, what special quadrilateral is
  formed? Why?

78
Ex. 2 Connecting midpoints of sides

• Solution Let E, F, G, and H be the midpoints of
  the sides of any quadrilateral, ABCD as shown.
• If you draw AC, the Midsegment Theorem for
  triangles says that FGAC and EGAC, so FGEH.
  Similar reasoning shows that EFHG.
• So by definition, EFGH is a parallelogram.

79
Proof with Special Quadrilaterals

• When you want to prove that a quadrilateral has a
  specific shape, you can use either the definition
  of the shape as in example 2 or you can use a
  theorem.

80
Proving Quadrilaterals are Rhombuses

• You have learned 3 ways to prove that a
  quadrilateral is a rhombus.
• You can use the definition and show that the
  quadrilateral is a parallelogram that has four
  congruent sides. It is easier, however, to use
  the Rhombus Corollary and simply show that all
  four sides of the quadrilateral are congruent.
• Show that the quadrilateral is a parallelogram
  and that the diagonals are perpendicular (Thm.
  6.11)
• Show that the quadrilateral is a parallelogram
  and that each diagonal bisects a pair of opposite
  angles. (Thm 6.12)

81
Ex. 3 Proving a quadrilateral is a rhombus

• Show KLMN is a rhombus
• Solution You can use any of the three ways
  described in the concept summary above. For
  instance, you could show that opposite sides have
  the same slope and that the diagonals are
  perpendicular. Another way shown in the next
  slide is to prove that all four sides have the
  same length.
• AHA DISTANCE FORMULA If you want, look on pg.
  365 for the whole explanation of the distance
  formula
• So, because LMNKMNKL, KLMN is a rhombus.

82
Ex. 4 Identifying a quadrilateral
60

• What type of quadrilateral is ABCD? Explain your
  reasoning.

120
120
60
83
Ex. 4 Identifying a quadrilateral
60

• What type of quadrilateral is ABCD? Explain your
  reasoning.
• Solution ?A and ?D are supplementary, but ?A
  and ?B are not. So, ABDC, but AD is not
  parallel to BC. By definition, ABCD is a
  trapezoid. Because base angles are congruent,
  ABCD is an isosceles trapezoid

120
120
60
84
Ex. 5 Identifying a Quadrilateral

• The diagonals of quadrilateral ABCD intersect at
  point N to produce four congruent segments AN ?
  BN ? CN ? DN. What type of quadrilateral is
  ABCD? Prove that your answer is correct.
• First Step Draw a diagram. Draw the diagonals
  as described. Then connect the endpoints to draw
  quadrilateral ABCD.

85
Ex. 5 Identifying a Quadrilateral
B

• First Step Draw a diagram. Draw the diagonals
  as described. Then connect the endpoints to draw
  quadrilateral ABCD.
• 2nd Step Make a conjecture
• Quadrilateral ABCD looks like a rectangle.
• 3rd step Prove your conjecture
• Given AN ? BN ? CN ? DN
• Prove ABCD is a rectangle.

C
N
A
D
86
Given AN ? BN ? CN ? DNProve ABCD is a
rectangle.

• Because you are given information about
  diagonals, show that ABCD is a parallelogram with
  congruent diagonals.
• First prove that ABCD is a parallelogram.
• Because BN ? DN and AN ? CN, BD and AC bisect
  each other. Because the diagonals of ABCD bisect
  each other, ABCD is a parallelogram.
• Then prove that the diagonals of ABCD are
  congruent.
• From the given you can write BN AN and DN CN
  so, by the addition property of Equality, BN DN
  AN CN. By the Segment Addition Postulate, BD
  BN DN and AC AN CN so, by substitution,
  BD AC.
• So, BD ? AC.
• ?ABCD is a parallelogram with congruent
  diagonals, so ABCD is a rectangle.