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.

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:

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.

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.

60 120

120

6 6

50°

130°

Rectangle Rhombus Square

Properties Properties Properties

Test your prior knowledge and try to fill in the chart with properties of the following quadrilaterals:

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

Use properties of diagonals of rhombuses, rectangles and squares.

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

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

A rhombus is a parallelogramwith four congruent sides

A rectangle is a parallelogram with four right angles.

A square is a parallelogram with four congruent sides and four right angles.

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.

RectanglesRhombuses

parallelograms

rhombuses rectangles

squares

Some examples of a rhombus

Decide whether the statement is always, sometimes, or never true.

a. A rhombus is a rectangle.b. A parallelogram is a rectangle.

RectanglesRhombuses

parallelograms


squares

Decide whether the statement is always, sometimes, or never true.a. 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.

RectanglesRhombuses

parallelograms


squares

Decide whether the statement is always, sometimes, or never true.

b. 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 aren’t.

RectanglesRhombuses

parallelograms


squares

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.

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.

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

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.

5y - 6

2y + 3

P

S

Q

R

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.

BC

DA

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.

BC

DA

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.

A B

CD

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.

Statements:1. ABCD is a rhombus2. AB ≅ CB3. AX ≅ CX4. BX ≅ DX5. ∆AXB ≅ ∆CXB6. AXB ≅ CXB7. AC BD

Reasons:1. Given

X

A

D

B

C


Reasons:1. Given2. Given

X

A

D

B

C


Reasons:1. Given2. Given3. Def. of . Diagonals

bisect each other.

X

A

D

B

C



bisect each other.4. Def. of . Diagonals

bisect each other.

X

A

D

B

C




bisect each other.5. SSS congruence

post.

X

A

D

B

C




bisect each other.5. SSS congruence

post.6. CPCTC

X

A

D

B

C




bisect each other.5. SSS congruence post.6. CPCTC7. Congruent Adjacent s

X

A

D

B

C

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)

B(b, 0)

C(0, - a)

D(- b, 0)

Find the lengths of the sides of ABCD. Use the distance formula (See – you’re never going to get rid of this)

AB=√(b – 0)2 + (0 – a)2 = √b2 + a2

BC= √(0 - b)2 + (– a - 0)2 = √b2 + a2

CD= √(- b – 0)2 + [0 - (– a)]2 = √b2 + a2

DA= √[(0 – (- b)]2 + (a – 0)2 = √b2 + a2

A(0, a)

B(b, 0)

C(0, - a)

D(- b, 0)

All the side lengths are equal, so ABCD is a rhombus.

CARPENTRY. You are building a rectangular frame for a theater set.

a. First, you nail four pieces of wood together as shown at the right. What is the shape of the frame?

b. 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.

4 feet

4 feet

6 feet 6 feet

a. 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.

4 feet

4 feet

6 feet 6 feet

b. 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.

4 feet

4 feet

6 feet 6 feet

You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

1. Take out a piece of notebook paper and make a hot dog fold over from the right side over to the pink line.

2. Now, divide the right hand section into 5 sections by drawing 4 evenly spaced lines.

The fold crease

3. Use scissors to cut along your drawn line, but ONLY to the crease!

4. Write QUADRILATERALS down the left hand side

The fold crease

5. Fold over the top cut section and write PARALLELOGRAM on the outside.

The fold crease

6. Reopen the fold.

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.

* Fold over the second cut section and write RECTANGLE on the outside.* Reopen the fold.






* On the left hand section, draw a rectangle.






* On the right hand side, list all of the properties of a rectangle.

1. Special parallelogram.

2. Has 4 right angles

3. Diagonals are congruent.

* Fold over the third cut section and write RHOMBUS on the outside.

* Reopen the fold.









* On the left hand section, draw a rhombus.






* 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

* Fold over the third cut section and write SQUARE on the outside.

* Reopen the fold.













* On the left hand section, draw a square.






* On the right hand side, list all of the properties of a square.

* 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

Name the figure described.

1. A quadrilateral that is both a rhombus and a rectangle.

2. A quadrilateral with exactly one pair of parallel sides.

3. A parallelogram with perpendicular diagonals

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

What are some properties of trapezoids and kits?

Use properties of trapezoids. Use properties of kites.

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.

base

base

legleg

A B

D C

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

Theorem 6.14 If a trapezoid is

isosceles, then each pair of base angles is congruent.

A ≅ B, C ≅ D

A B

D C

Theorem 6.15 If a trapezoid has a

pair of congruent base angles, then it is an isosceles trapezoid.

ABCD is an isosceles trapezoid

A B

D C

Theorem 6.16 A trapezoid is

isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC ≅ BD.

A B

D C

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.

midsegment

B C

DA

The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.

MN║AD, MN║BC MN = ½ (AD + BC)

NM

A D

CB

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?

Use the midsegment theorem for trapezoids.

DG = ½(EF + CH)=½ (8 + 20) = 14”

C

D

E

D

G

F

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

Theorem 6.18 If a quadrilateral is

a kite, then its diagonals are perpendicular.

AC BD

B

C

A

D

Theorem 6.19 If a quadrilateral is

a kite, then exactly one pair of opposite angles is congruent.

A ≅ C, B ≅ D

B

C

A

D

WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths.

WX = √202 + 122 ≈ 23.32 XY = √122 + 122 ≈ 16.97 Because WXYZ is a kite,

WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97

12

1220

12

U

X

Z

W Y

Find mG and mJ in the diagram at the right.SOLUTION:GHJK is a kite, so G ≅ J and mG = mJ.2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360° 2(mG) = 168°Simplify mG = 84° Divide each side by 2.

So, mJ = mG = 84°

J

G

H K132° 60°

Quadrilaterals4-sided polygons

Trapezoids Parallelograms Kites1. one pair of opposite sides 1. 2 pairs of opposite sides 1. two pairs of

2. 2 pairs of opposite sides consecutive sides2. midsegment parallel to both bases 3. 2 pairs of opposite angles 2. diagonals are

and average length of the bases 4. consecutive angles are supplementary5. diagonals bisect each other 3. one pair of opposite

angles

Isosceles Trapezoids Rhombus Rectangle1. legs are congruent 1. 4 sides 1. 4 right angles

2. base angles are congruent 2. diagonals are 2. diagonals are congruent

3. diagonals are congruent 3. diagonals bisect a pair ofopposite angles

Square

1. combination of rectangle and rhombus

Geometry

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.

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.

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.

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 ll’ogram with perpendicular diagonals.

Identify special quadrilaterals based on limited information.

Prove that a quadrilateral is a special type of quadrilateral, such as a rhombus or trapezoid.

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

Quadrilateral

Kite Parallelogram Trapezoid

Rhombus Rectangle

Square

Isosceles trapezoid

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

Parallelogram

Opposite sides are congruent.

Rhombus

All sides are congruent.

Opposites sides are .≅

All sides are congruent.

Legs are congruent.

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

H G

FE

A

DB

C

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 FG║AC and EG║AC, so FG║EH. Similar reasoning shows that EF║HG.

So by definition, EFGH is a parallelogram.

H G

FE

A

DB

C

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.

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)

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 LM=NK=MN=KL, KLMN is a rhombus.

8

6

4

2

-2

5 10 15

L(-2, 3)K(2, 5) = 4.47 cm

L(-2, 3)M(2, 1) = 4.47 cm

M(2, 1)N(6, 3) = 4.47 cm

K(2, 5)N(6, 3) = 4.47 cm

N(6, 3)

M(2, 1)

L(-2, 3)

K(2, 5)

What type of quadrilateral is ABCD? Explain your reasoning.

AD

C

B

60°120°

120°

60°

What type of quadrilateral is ABCD? Explain your reasoning.

Solution: A and D are supplementary, but A and B are not. So, AB║DC, but AD is not parallel to BC. By definition, ABCD is a trapezoid. Because base angles are congruent, ABCD is an isosceles trapezoid

AD

C

B

60°120°

120°

60°

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.

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.

A

B

C

D

N

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.

6.4 Rhombuses, Rectangles and Squares

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