Quadrilaterals -...

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Quadrilaterals

Unit 6

Objectives

• To identify any quadrilateral, by name, as

specifically as you can, based on its

characteristics

Quadrilateral

• a quadrilateral is a polygon with 4 sides.

Specific Quadrilaterals

• There are several specific types of

quadrilaterals. They are classified based

on their sides or angles.

2 pairs adjacent

congruent sides

1 pair parallel sides2 pairs parallel

4 right angles4 congruent sidesnon-parallel sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A quadrilateral simply has 4 sides –

no other special

requirements.

Examples of Quadrilaterals

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A parallelogram

has two pairs of

parallel sides.

Parallelogram

• Two pairs of parallel sides

• opposite sides are actually congruent.

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A rhombus is a parallelogram

that has four congruent sides.

Rhombus

• Still has two pairs of parallel sides; with

opposite sides congruent.

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A rectangle has

four right angles.

Rectangle

• Still has two pairs of parallel sides; with

opposite sides congruent.

• Has four right angles

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A square is a specific case of both a

rhombus AND a rectangle, having four

right angles and 4 congruent sides.

Square

• Still has two pairs of parallel sides.

• Has four congruent sides

• Has four right angles

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

A trapezoid has

only one pair of

parallel sides.

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

An isosceles

trapezoid is a

trapeziod with the

non-parallel sides

congruent.

Trapezoid

• has one pair of parallel sides.

Isosceles trapezoid trapezoids

(Each of these examples shown has top and bottom sides parallel.)

2 pairs adjacent

congruent sides

are congruent

Quadrilaterals

Parallelogram

Rectangle

Rhombus

Square

Isosceles

Trapezoid

An kite is a quadrilateral

with NO parallel sides but 2

pairs of adjacent congruent

sides.

Example of a Kite

Practice Pg. 290-292

# 1-12 all, 13, 17, 19, 23

#29-34

#36 can be turned in for 5 extra credit points!

Properties of Parallelograms

In this lesson . . .

And the rest of the unit, you will study special

quadrilaterals. A parallelogram is a quadrilateral

with both pairs of opposite sides parallel.

When you mark diagrams of quadrilaterals, use

matching arrowheads to indicate which sides are

parallel. For example, in the diagram to the

right, PQ║RS and QR║SP. The symbol

PQRS is read “parallelogram PQRS.”

Theorems about parallelograms

• 6.1—If a

quadrilateral is a

parallelogram,

then its opposite

sides are

congruent.

►PQ≅RS and

SP≅QR P

• 6.2—If a quadrilateral is a parallelogram, then its opposite angles are congruent.

P ≅ R and

Q ≅ S P

• 6.3—If a

quadrilateral is a

parallelogram, then

its consecutive

angles are

supplementary (add

up to 180°).

mP +mQ = 180°,

mQ +mR = 180°,

mR + mS = 180°,

mS + mP = 180°

• 6.4—If a

quadrilateral is a

parallelogram,

then its

diagonals bisect

each other.

QM ≅ SM and

PM ≅ RM P

Ex. 1: Using properties of

Parallelograms

• FGHJ is a

parallelogram. Find

the unknown length.

Explain your

reasoning.

Parallelograms • FGHJ is a

parallelogram. Find the unknown length. Explain your reasoning.

SOLUTION:

a. JH = FG Opposite sides of a are ≅.

JH = 5 Substitute 5 for FG.

Parallelograms • FGHJ is a

parallelogram. Find the unknown length. Explain your reasoning.

SOLUTION:

a. JH = FG Opposite sides of a are ≅.

JH = 5 Substitute 5 for FG.

b. b. JK = GK Diagonals of a

bisect each other.

JK = 3 Substitute 3 for GK

Ex. 2: Using properties of parallelograms

PQRS is a parallelogram.

Find the angle measure.

a. mR = mP Opposite angles of a are ≅.

mR = 70° Substitute 70° for mP.

a. mR = mP Opposite angles of a are ≅.

mR = 70° Substitute 70° for mP.

b. mQ + mP = 180° Consecutive s of a are

supplementary.

mQ + 70° = 180° Substitute 70° for mP.

mQ = 110° Subtract 70° from each side.

Ex. 3: Using Algebra with Parallelograms

PQRS is a

parallelogram. Find

the value of x.

mS + mR = 180°

3x + 120 = 180

3x = 60

x = 20

Consecutive s of a □ are

supplementary.

Substitute 3x for mS and 120 for

Subtract 120 from each side.

Divide each side by 3.

R 3x° 120°

Ex. 4: Proving Facts about Parallelograms

Given: ABCD and AEFG are

parallelograms.

Prove 1 ≅ 3.

1. ABCD is a □. AEFG is a ▭. 2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3

1. Given

parallelograms.

Prove 1 ≅ 3.

1. ABCD is a □. AEFG is a □.

2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3

1. Given

2. Opposite s of a ▭ are

parallelograms.

Prove 1 ≅ 3.

1. ABCD is a □. AEFG is a □.

2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3

1. Given

2. Opposite s of a ▭ are

3. Transitive prop. of

congruence.

Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram.

Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .

2. Draw BD.

3. AB ║CD, AD ║ CB.

4. ABD ≅ CDB, ADB ≅ CBD

5. DB ≅ DB

6. ∆ADB ≅ ∆CBD

7. AB ≅ CD, AD ≅ CB

1. Given

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

2. Through any two points, there exists exactly one line.

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

3. Definition of a parallelogram

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

4. Alternate Interior s Thm.

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

5. Reflexive property of congruence

6. ASA Congruence Postulate

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

6. ASA Congruence Postulate

7. CPCTC

Ex. 6: Using parallelograms in real life

FURNITURE DESIGN. A drafting

table is made so that the legs can

be joined in different ways to

change the slope of the drawing

surface. In the arrangement

below, the legs AC and BD do not

bisect each other. Is ABCD a

parallelogram?

1. ABCD is a .

2. Draw BD.

5. DB ≅ DB

1. Given

Ex. 6: Using parallelograms in real life

FURNITURE DESIGN. A drafting

table is made so that the legs can

be joined in different ways to

change the slope of the drawing

surface. In the arrangement

below, the legs AC and BD do not

bisect each other. Is ABCD a

parallelogram?

ANSWER: NO. If ABCD were a

parallelogram, then by Theorem

6.5, AC would bisect BD and BD

would bisect AC. They do not, so

it cannot be a parallelogram.

Practice!

Pg.297-300

#2-16 even

#18,20,22

#24-33 all

#53-55

6.3 Proving Quadrilaterals

are Parallelograms

Theorems

Theorem 6.5: If the

diagonals of a

quadrilateral

bisect each other,

then the

quadrilateral is a

parallelogram. ABCD is a parallelogram.

Theorems

Theorem 6.6: If both

pairs of opposite

sides of a

quadrilateral are

congruent, then the

quadrilateral is a

parallelogram.

ABCD is a parallelogram.

Theorems

Theorem 6.7: If both

pairs of opposite

angles of a

quadrilateral are

congruent, then the

quadrilateral is a

parallelogram.

Theorems

Theorem 6.8: If an

angle of a

quadrilateral is

supplementary

to both of its

consecutive

angles, then the

quadrilateral is a

parallelogram.

(180 – x)° x°

Ex. 1: Proof of Theorem 6.6

Statements:

1. AB ≅ CD, AD ≅ CB.

2. AC ≅ AC

3. ∆ABC ≅ ∆CDA

4. BAC ≅ DCA, DAC ≅ BCA

5. AB║CD, AD ║CB.

6. ABCD is a

Reasons:

1. Given

Statements:

2. AC ≅ AC

6. ABCD is a

Reasons:

1. Given

2. Reflexive Prop. of Congruence

Statements:

2. AC ≅ AC

6. ABCD is a

Reasons:

1. Given

3. SSS Congruence Postulate

Statements:

2. AC ≅ AC

6. ABCD is a

Reasons:

1. Given

4. CPCTC

Statements:

2. AC ≅ AC

6. ABCD is a

Reasons:

1. Given

4. CPCTC

5. Alternate Interior s Converse

Statements:

2. AC ≅ AC

6. ABCD is a

Reasons:

1. Given

4. CPCTC

5. Alternate Interior s Converse

6. Def. of a parallelogram.

Ex. 2: Proving Quadrilaterals are

Parallelograms

• As the sewing box below is opened, the

trays are always parallel to each other.

2.75 in. 2.75 in.

Ex. 2: Proving Quadrilaterals are

Parallelograms

• Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.

2.75 in. 2.75 in.

Another Theorem ~

• Theorem 6.10—If one pair of opposite

sides of a quadrilateral are congruent and

parallel, then the quadrilateral is a

parallelogram.

• ABCD is a

parallelogram.

Given: BC║DA, BC ≅ DA

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

5. ∆BAC ≅ ∆DCA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

3. Reflexive Property

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

4. Given

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

4. Given

5. SAS Congruence Post.

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

4. Given

6. CPCTC

Prove: ABCD is a

Statements:

1. BC ║DA

2. DAC ≅ BCA

3. AC ≅ AC

4. BC ≅ DA

6. AB ≅ CD

7. ABCD is a

Reasons:

1. Given

2. Alt. Int. s Thm.

4. Given

6. CPCTC

7. If opp. sides of a quad. are ≅, then it is a .

Objective 2: Using Coordinate Geometry

• When a figure is in the coordinate plane,

you can use the Distance Formula (see—it

never goes away) to prove that sides are

congruent and you can use the slope

formula (see how you use this again?) to

prove sides are parallel.

• Show that A(2, -1), B(1,

3), C(6, 5) and D(7,1)

are the vertices of a

parallelogram.

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Ex. 4: Using properties of parallelograms • Method 1—Show that opposite

sides have the same slope, so they are parallel.

• Slope of AB. – 3-(-1) = - 4

• Slope of CD. – 1 – 5 = - 4

7 – 6

• Slope of BC. – 5 – 3 = 2

6 - 1 5

• Slope of DA. – - 1 – 1 = 2

2 - 7 5

• AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Because opposite sides are

parallel, ABCD is a

parallelogram.

• Method 2—Show that

opposite sides have the

same length.

• AB=√(1 – 2)2 + [3 – (- 1)2] = √17

• CD=√(7 – 6)2 + (1 - 5)2 = √17

• BC=√(6 – 1)2 + (5 - 3)2 = √29

• DA= √(2 – 7)2 + (-1 - 1)2 = √29

• AB ≅ CD and BC ≅ DA.

Because both pairs of opposites

sides are congruent, ABCD is a

parallelogram.

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

• Method 3—Show that

one pair of opposite

sides is congruent and

parallel.

• Slope of AB = Slope of CD

• AB=CD = √17

• AB and CD are congruent

and parallel, so ABCD is a

parallelogram.

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Practice!!

Pg. 307-308

#1-17 all

#26-29 all

#32-34 all

Geometry

6.4 Special Parallelograms

Rectangle

• A quadrilateral with four right angles.

Why is a rectangle a parallelogram?

Both Pairs of Opp. Angles are Congruent

Rhombus

• A quadrilateral with four congruent sides.

Why is a rhombus a parallelogram?

Both Pairs of Opp. Sides are Congruent

Square

• A quadrilateral with four congruent sides

and four right angles.

Why is a rhombus a parallelogram?

Both Pairs of Opp. Sides are Congruent

Both Pairs of Opp. Angles are Congruent

Review: Rectangles, Rhombuses, and

Squares all Share these Properties of a

Parallelogram… 1) Opp. Sides are //

2) Opp. Angles are congruent

3) Opp. Sides are congruent

4) Diagonals Bisect Each Other

In addition, rectangles, rhombuses, and squares all have their own special properties. These are the focus of this lesson.

Theorem: The diagonals of a

Rectangle are Congruent

Draw two congruent intersecting lines that bisect each

other.

Connect the corners. You drew a rectangle.

Theorem: The diagonals of a

rhombus are perpendicular.

Draw two lines that bisect each other & are perpendicular.

Connect the corners. You have drawn a rhombus.

Theorem: Each diagonal of a rhombus

bisects two angles of the rhombus.

Draw a rhombus and its diagonals.

You bisected all four angles.

Theorem: The midpoint of the hypotenuse of a

right triangle is equidistant from all three vertices.

Draw a right triangle and put a point at the midpoint of the

hypotenuse.

Draw a line from that point to the vertex of the right angle.

All three distances are equal.

Theorem: If an angle of a parallelogram is a right

angle, then the parallelogram is a rectangle.

Draw one right angle.

Draw the two other sides parallel to the opposite side.

You have drawn a rectangle.

Why is it a rectangle?

Opp. Angles of a Parallelogram

Are congruent

Parallel lines imply SS Int. angles

are supplementary.

Theorem: If two consecutive sides of a

parallelogram are congruent, then the

parallelogram is a rhombus. Draw two congruent sides of an angle.

Draw the two other sides parallel

to the opposite sides.

You have drawn a rhombus.

Why is it a rhombus?

Opp. Sides of a Parallelogram are congruent.

Given Quad. WXYZ is a rectangle. Complete the statements with numbers.

Make sure your + and – are clear!

3. If TX = 4.5, then WY = _____.

4. If WY = 3a + 16 and ZX = 5a – 18, then a = _____,

WY = _____ and ZX = _____.

5. If m<TWZ = 70, then m<TZW = _____ and

m<WTZ = _____.

7. If m<4 = 25, then m<5 = _____.

8. If m<DAB = 130, then m<ADC = _____.

9. If m<4 = 3x – 2 and m<5 = 2x + 7,

then x = ____, m<4 = ____, and m<5 =____.

11. If m<2 = 3y + 9 and m<4 = 2y – 4,

then y = _____, m<2 = _____, and m<4 = ____.

Given Quad. ABCD is a rhombus. Complete the statements with numbers.

Given Quad. JKLM is a square. Complete the statements with numbers.

14. If JL =18, then MK = _____, JX = _____, and XK = _____.

15. m<MJK = _____, m<MXJ = _____ and m<KLJ = _____.

Property Parallelogram Rectangle Rhombus Square

5) Diags. Bisect

each other X X X X

6) Diags. Are

conguent X X

X 7) Diags. Are

Perpendicular

8) A diagonal

Bisects 2 angles

Review

Practice!!

Pg. 315-317

#1-9 all

#10,12,14

#22 &24

Lesson 6-5 Trapezoids and Kites

Definition

Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Perpendicular Diagonals of a Kite

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

Non-Vertex Angles of a Kite

If a quadrilateral is a kite, then non-vertex angles are congruent

A C, B D

Vertex diagonals bisect vertex angles

If a quadrilateral is a kite then the vertex

diagonal bisects the vertex angles.

Vertex diagonal bisects the non-vertex diagonal

If a quadrilateral is a kite then the vertex diagonal

bisects the non-vertex diagonal

Definition-a quadrilateral with exactly one pair of parallel sides.

Leg Leg

Base A B

C D ›

Trapezoid

<A + <C = 180

<B + <D = 180

C D ›

Leg Angles are Supplementary

Property of a Trapezoid

Isosceles Trapezoid

Definition - A trapezoid with congruent legs.

Isosceles Trapezoid - Properties

1) Base Angles Are Congruent

2) Diagonals Are Congruent

Example

PQRS is an isosceles trapezoid. Find m P, m Q and mR.

m R = 50 since base angles are congruent

mP = 130 and mQ = 130 (consecutive angles

of parallel lines cut by a transversal are )

Find the measures of the angles in trapezoid

m< A = 132

m< B = 132

m< D = 48

Find BE

AC = 17.5, AE = 9.6

Example

Find the side lengths of the kite.

Example Continued

WX = 4 34

likewise WZ = 4 34

XY =12 2

likewise ZY =12 2

We can use the Pythagorean Theorem to

find the side lengths.

122 + 202 = (WX)2

144 + 400 = (WX)2

544 = (WX)2

122 + 122 = (XY)2

144 + 144 = (XY)2

288 = (XY)2

Find the lengths of the sides of the kite

Find the lengths of the sides of kite to the nearest tenth

Example 3

Find mG and mJ.

Since GHJK is a kite G J

So 2(mG) + 132 + 60 = 360

2(mG) =168

mG = 84 and mJ = 84

Try This!

RSTU is a kite. Find mR, mS and mT.

x +30 + 125 + 125 + x = 360

2x + 280 = 360

2x = 80

x = 40

So mR = 70, mT = 40 and mS = 125

Try These

legleg

D C base

legleg

1. If <A = 134, find m<D 2. m<C = x +12 and

m<B = 3x – 2, find x and the

measures of the 2 angles

m<D = 46

x = 42.5

m<C = 54.5

m<B = 125.5

Using Properties of Trapezoids

Find the area of this trapezoid.

When working with a trapezoid, the height may be measured

anywhere between the two bases. Also, beware of "extra"

information. The 35 and 28 are not needed to compute this area.

Area of trapezoid = 212

A = ½ * 26 * (20 + 42)

A = 806

Using Properties of Trapezoids

Find the area of a trapezoid

with bases of 10 in and 14

in, and a height of 5 in.

Example 2

Using Properties of Kites

Area Kite = one-half product of diagonals

BDACArea 2

Using Properties of Kites

Example 6

ABCD is a Kite.

a) Find the lengths of all

the sides.

b) Find the area of the Kite.

Venn Diagram:

http://teachers2.wcs.edu/high/rhs/staceyh/Geometry/Chapter%206%20Notes.ppt#435,22,6.2 – Properties of Parallelograms

Flow Chart:

Practice!!

Pg. 322-324

#1-16 all

#20-25 odd

#40 for 5 extra credit points

Placing Figures in the

Coordinate Plane

ABCD is a rectangle. A(0,0), B(5,0),

C(x,y) and D(0,4) give coordinates for

point C .

ABCD is a rectangle. A(0,0), B(m,0),

C(x,y) and D(0,n) give coordinates for

point C in terms of m and n..

Give coordinates for points W and Z

without using any new variables.

ABCD is a square. A(3,3) and B(-3,3) find the

coordinates for C and D in the third and fourth

quadrants. Can you find the other set of coordinates

in the first and second quadrants?

Plot the points to make a trapezoid.

A(0,0), B(6,0), C(4,5) and D(-2,5)

To Recap the main ideas…

Practice!!

Pg. 328-39

#1-13 all

#33 &34

Quadrilaterals -...

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