The Distance Formula

• Used to find the distance between two points: A( x1, y1) and B(x2, y2)

You also could just plot the points and use the Pythagorean Theorem!!

212

212 )( yyxxd

Find the distance between the two points. Round your answer to the nearest tenth.

1. T(5, 2) and R(-4, -1)

Take a look at example 2, p. 44

2. A( -2, -3) and B(1, 3)

Midpoint Formula

• Find the midpoint coordinates between 2 points

• Find by averaging the x-coordinates and the y-coordinates of the endpoints

2,

22121 yyxx

M

(x1, y1)

(x2, y2)

Find the coordinates of the midpoint of

1. Q(3, 5) and S(7, -9)

2. Q( -4, 4) and S(5, -1)

QS

Special QuadrilateralsParallelogram – A quadrilateral with both pairs of opposite sides parallel.Rhombus – A parallelogram with four congruent sides.Rectangle – A parallelogram with four right angles.Square – A parallelogram with four congruent sides and four right angles.Kite – A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent.Trapezoid – A quadrilateral with exactly one pair of parallel sides.Isosceles Trapezoid – A trapezoid whose nonparallel opposite sides are congruent.

Warm-upDraw each figure on graph paper if possible. If

not possible explain why.

1. A parallelogram that is neither a rhombus nor a rectangle

2. An isosceles trapezoid with vertical congruent sides

3. A trapezoid with only one right angle4. A trapezoid with two right angles5. A rhombus that is not a square6. A kite with two right angles

Theorem 6-1Opposite sides of a parallelogram are congruent.

Theorem 6-2Opposite angles of a parallelogram are congruent.

(Consecutive angles of a parallelogram are supplementary, they are same-side interior angles!)

Theorem 6-3The diagonals of a parallelogram bisect each other.

Properties of Parallelograms

Using Algebra

• Find the value of x in PQRS. Then find QR and PS.

P

Q R

S

3x - 15

2x + 3

Using Algebra

• Find the value of y in EFGH. Then find m<E, m<F, m<G, m<H.

E F

GH

3y + 37

6y + 4

TR=12 find VRQS=10 find VS

110̇0̇$

Find x and the length of the side Find all angle measures

30

120

Theorem 6-4

• Theorem 6-4:If three (or more) parallel lines cut off

congruent segments on one transversal, then they cut off congruent segments on every transversal.

Measurement

• In the figure, DH ll CG ll BF ll AE, and AB = BC = CD = 2, and EF = 2.5. Find EH.

D

C

B

A

H

G

F

E

2

2

2

2.5

Proving that a quadrilateral is a parallelogram.(both pairs of opposite sides are parallel)

Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.Theorem 6-6If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.Theorem 6-7If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 6-8If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Proving that a quadrilateral is a parallelogram.(Both pairs of opposite sides are parallel)

Theorem 6-5 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.Theorem 6-6If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.Theorem 6-7If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 6-8If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

5 ways to prove that a quadrilateral is a parallelogram.

1. Show that both pairs of opposite sides are || . [definition]

2. Show that both pairs of opposite sides are .

3. Show that one pair of opposite sides are both and || .

4. Show that both pairs of opposite angles are .

5. Show that the diagonals bisect each other .

Examples ……

Find the value of x and y that ensures the quadrilateral is a parallelogram.

Example 1:

6x4x+8

y+2

2y

6x = 4x+8

2x = 8

x = 4 units

2y = y+2

y = 2 unit

Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.

120° 5y°

(2x + 8)°2x + 8 = 120

2x = 112

x = 56 units

5y + 120 = 180

5y = 60

y = 12 units

Is the Quadrilateral a Parallelogram?

95°

95°

95

xx

Finding Values

• Find the values of a and c for which PQRS must be a parallelogram.

P S

RQ(a + 40) a

3c – 3 c + 1

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.

Ex. 4: Using properties of parallelograms

• Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.

6

4

2

-2

-4

5 10̇ 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 1 - 2

• 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.

6

4

2

-2

-4

5 10̇ 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Because opposite sides are parallel, ABCD is a parallelogram.

Ex. 4: Using properties of parallelograms• 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.

6

4

2

-2

-4

5 10̇ 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Ex. 4: Using properties of parallelograms

• Method 3—Show that one pair of opposite sides is congruent and parallel.

• Slope of AB = Slope of CD = -4• AB=CD = √17

• AB and CD are congruent and parallel, so ABCD is a parallelogram.

6

4

2

-2

-4

5 10̇ 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Theorem 6-9Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 6-10The diagonals of a rhombus are perpendicular.

Theorem 6-11 The diagonals of a rectangle are congruent.

Theorem 6-12If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.

Theorem 6-13If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorem 6-14If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Use the properties of rectangles to find all missing angle measures, list the properties you used.

32

6.5: Trapezoids and Kites

Objective:To verify and use properties of trapezoids and kites

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to the bases.

The length of the midsegment of a trapezoid is half the sum of the lengths of the bases.

If you are given the midsegment length and one base: -double the length of the midsegment

-subtract the other base.

Midsegment (Median) of a Trapezoid

Joins the midpoints of the nonparallel sidesIs parallel to the basesIts length is ½ the sum of the bases

MN || BCMN || ADMN = ½(BC+AD)

4

386°

108°

Find the following:EF:

:

:

:

:

:

AB

DF

DCBm

CFEm

BADm

Find x:

8

12

x

Theorem 6-15The base angles of an isosceles trapezoid are congruent.

Theorem 6-16The diagonals of an isosceles trapezoid are congruent.

Theorem 6-17The diagonals of a kite are perpendicular.

Theorem:The diagonals of a kite are perpendicular.

A kite has exactly one pair of opposite, congruent angles.

Find the measure of the missing angles.

44° 112°

1

2

What is the sum of the angles in a quadrilateral?

TRAPEZOID

LEGLEG

BASE ANGLES

BASE ANGLES

•The 2 parallel sides are the bases•The 2 non-parallel sides are the legs

A

B

C

D

Name the following:Bases:Legs:2 Pairs of Base Angles:

Theorem:

The base angles of an isosceles trapezoid are congruent

A

B

C

D

DB

CA

21

3

xCm

xAm Find x.

Theorem:

The diagonals of an isosceles trapezoid are congruent.

EXAMPLE:If BD= 2x+10 and AC=x+15, find x and the length of the diagonals.

2 angles that share a leg are supplementary because they are same-side interior angles.

180 CmAm

The measure of angle A= 110. find the measures of the other 3 angles.

One side of a kite is 4 cm less than two times the length of another side. The perimeter of the kite is58 cm. Find the lengths of the sides of the kite.

Midsegments of trapezoids

• The midsegment of an isosceles trapezoid measures 14 cm. One of the bases measures 24 cm. Find the length of the other base.