6.4 Rhombuses, Rectangles and Squares

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6.4 Rhombuses, Rectangles and Squares. Geometry CCSS: G.CO 11.

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### 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.

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:

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.

6

6

Do you have enough information?

Yes

Yes

No

Yes

50°

130°

Do you have enough information?

Yes

No

Yes

Yes

Activator

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

Objectives:

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

Use properties of diagonals of rhombuses, rectangles and squares.

E.Q

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

Properties of Special Parallelograms

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.

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

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

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

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

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 aren’t.

parallelograms

rhombuses

rectangles

squares

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.
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.
• 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.
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.

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.
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.
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.

D

C

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.

Statements:

• ABCD is a rhombus
• AB ≅ CB
• AX ≅ CX
• BX ≅ DX
• ∆AXB ≅ ∆CXB
• AXB ≅ CXB
• AC  BD

Reasons:

• Given

Statements:

• ABCD is a rhombus
• AB ≅ CB
• AX ≅ CX
• BX ≅ DX
• ∆AXB ≅ ∆CXB
• AXB ≅ CXB
• AC  BD

Reasons:

• Given
• Given

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.

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.

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.

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

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
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)

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 – 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)

D(- b, 0)

B(b, 0)

C(0, - a)

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

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

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

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

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?

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.

The fold crease

Foldable

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!

The fold crease

Foldable

4. Write QUADRILATERALS down the left hand side

The fold crease

Foldable

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

Parallelogram

6. Reopen the fold.

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.

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.

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.

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

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

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

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

### 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

### 6.5 Trapezoids and Kites

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?

E.Q

What are some properties of trapezoids and kits?

Objectives:
• 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. Using properties of trapezoids
Theorem 6.14

If a trapezoid is isosceles, then each pair of base angles is congruent.

A ≅ B, C ≅ D

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

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.

Trapezoid Theorems
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 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.

MN = ½ (AD + BC)

Theorem 6.17: Midsegment of a trapezoid
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?Ex. 3: Finding Midsegment lengths of trapezoids
A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Using properties of kites
Theorem 6.18

If a quadrilateral is a kite, then its diagonals are perpendicular.

AC  BD

Kite theorems
Theorem 6.19

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

A ≅ C, B ≅ D

Kite theorems
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

Ex. 4: Using the diagonals of a kite
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° = 360°Sum of measures of int. s of a quad. is 360°

2(mG) = 168°Simplify

mG = 84° Divide each side by 2.

So, mJ = mG = 84°

132°

60°

Geometry

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.

### 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.

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 ll’ogram with perpendicular diagonals.

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

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.Summarizing Properties of Quadrilaterals

Trapezoid

Kite

Parallelogram

Rhombus

Rectangle

Isosceles trapezoid

Square

• 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.

When you join the midpoints of the sides of any quadrilateral, what special quadrilateral is formed? Why?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 FG║AC and EG║AC, so FG║EH. Similar reasoning shows that EF║HG.

So by definition, EFGH is a parallelogram.

Ex. 2: Connecting midpoints of sides
• 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.

Ex. 3: Proving a quadrilateral is a rhombus

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

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.

Given: AN ≅ BN ≅ CN ≅ DN

Prove ABCD is a rectangle.

B

C

N

A

D

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.

Given: AN ≅ BN ≅ CN ≅ DNProve: ABCD is a rectangle.