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### 6.4 Rhombuses, Rectangles and Squares

### Warm-Up

### 6.5 Trapezoids and Kites

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:

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.
- 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

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.

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 6.11Given: 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 6.11Given: 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 6.11Given: 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 6.11Given: 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 6.11Given: 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 6.11Given: 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 6.11Given: 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.

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

Foldable5. 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

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

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?

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

If the legs of a trapezoid are congruent, then the trapezoid is an isosceles 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 TheoremsTheorem 6.15

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

ABCD is an isosceles trapezoid

Trapezoid TheoremsTheorem 6.16

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

ABCD is isosceles if and only if AC ≅ BD.

Trapezoid TheoremsThe 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║AD, MN║BC

MN = ½ (AD + BC)

Theorem 6.17: Midsegment of a trapezoidLAYER 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

Use the midsegment theorem for trapezoids.

DG = ½(EF + CH)=

½ (8 + 20) = 14”

Ex. 3: Finding Midsegment lengths of trapezoidsE

F

D

G

D

C

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

Theorem 6.19

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

A ≅ C, B ≅ D

Kite theoremsWXYZ 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 kiteEx. 5: Angles of a kite

- 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°

132°

60°

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