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6.4 Rhombuses, Rectangles and Squares. Geometry NCSCOS: 1.02; 2.02; 2.03. 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: . 6. 6.

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6 4 rhombuses rectangles and squares

6.4 Rhombuses, Rectangles and Squares

Geometry NCSCOS: 1.02; 2.02; 2.03


U e q
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:


6 4 rhombuses rectangles and squares

6

6

Do you have enough information?

Yes

Yes

No

Yes


6 4 rhombuses rectangles and squares

50°

130°

Do you have enough information?

Yes

No

Yes

Yes


Activator
Activator

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


Objectives
Objectives:

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

Use properties of diagonals of rhombuses, rectangles and squares.


6 4 rhombuses rectangles and squares
E.Q

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


Properties of special parallelograms
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
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
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 parallelogram1
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 parallelogram2
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
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
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.


Ex 3 using properties of a rhombus
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
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 parallelograms1
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 parallelograms2
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


6 4 rhombuses rectangles and squares
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.


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd
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


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd1
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


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd2
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.


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd3
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.


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd4
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.


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd5
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


Ex 4 proving theorem 6 11 given abcd is a rhombus prove ac bd6
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


Ex 5 coordinate proof of theorem 6 11 given abcd is a parallelogram ac bd prove abcd is a rhombus
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 11 given abcd is a parallelogram ac bd prove abcd is a rhombus1
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
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 rectangle1
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 rectangle2
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


6 4 rhombuses rectangles and squares

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
Foldable 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.


Foldable1

The fold crease seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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!


Foldable2

The fold crease seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

Foldable

4. Write QUADRILATERALS down the left hand side

QUADRILATERALS


Foldable3

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

The fold crease

Foldable

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

Parallelogram

6. Reopen the fold.


Foldable4

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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.


Foldable5

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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.


Foldable6

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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.


Foldable7

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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


Foldable8

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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


Foldable9

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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


Foldable10

QUADRILATERALS seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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

Warm-Up seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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

6.5 Trapezoids and Kites seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

Geometry NCSCOS:1.02; 2.02; 2.03


U e q1
U.E.Q seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

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


6 4 rhombuses rectangles and squares
E.Q seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

What are some properties of trapezoids and kits?


Objectives1
Objectives: seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

  • Use properties of trapezoids.

  • Use properties of kites.


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.

Using properties of trapezoids


Using properties of trapezoids1

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

Using properties of trapezoids


Trapezoid theorems

Theorem 6.14 is an isosceles trapezoid.

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

A ≅ B, C ≅ D

Trapezoid Theorems


Trapezoid theorems1

Theorem 6.15 is an isosceles trapezoid.

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

ABCD is an isosceles trapezoid

Trapezoid Theorems


Trapezoid theorems2

Theorem 6.16 is an isosceles trapezoid.

A trapezoid is isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC ≅ BD.

Trapezoid Theorems


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.

Midsegment of a trapezoid


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.

MN║AD, MN║BC

MN = ½ (AD + BC)

Theorem 6.17: Midsegment of a trapezoid


Ex 3 finding midsegment lengths of trapezoids

LAYER CAKE its length is one half the sums of the lengths of the bases. 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


Ex 3 finding midsegment lengths of trapezoids1

Use the midsegment theorem for trapezoids. its length is one half the sums of the lengths of the bases.

DG = ½(EF + CH)=

½ (8 + 20) = 14”

Ex. 3: Finding Midsegment lengths of trapezoids

E

F

D

G

D

C


Using properties of kites

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

Using properties of kites


Kite theorems

Theorem 6.18 congruent sides, but opposite sides are not congruent.

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

AC  BD

Kite theorems


Kite theorems1

Theorem 6.19 congruent sides, but opposite sides are not congruent.

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

A ≅ C, B ≅ D

Kite theorems


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 = √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
Ex. 5: Angles of a kite use the Pythagorean Theorem to find the side lengths.

  • 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°