6.1 Classifying Quadrilaterals page 288

6.1 6.1 Classifying Classifying

QuadrilateralsQuadrilateralspage 288page 288

Obj 1:Obj 1: To define & classify To define & classify special types of special types of quadrilateralsquadrilaterals

And why…And why…• To use the properties of special

quadrilaterals with a kite, as in Example 3.

Seven important types of Seven important types of quadrilaterals …quadrilaterals …

1) Parallelogram-has both pairs of opposite sides parallel

2) Rhombus-has four congruent sides3) Rectangle-has four right angles4) Square-has four congruent sides and

four right angles5) Kite-has two pairs of adjacent sides

congruent and no opposite opposite sides congruent.

Continued….Continued….6. Trapezoid-has exactly one pair of

parallel sides. (you have same side interior angles)

7. Isosceles trapezoid-is a trapezoid whose non-parallel opposite sides are congruent

ABCD is a quadrilateral because it has four sides.

Judging by appearance, classify ABCD in as many

ways as possible.

It is a trapezoid because AB and DC appear parallel and AD and BC appear nonparallel.

Classifying QuadrilateralsClassifying Quadrilaterals

6-1

You try oneYou try one• Turn to page 289 and complete

check understanding 1 (top of page)

Classifying by Coordinate Classifying by Coordinate MethodMethod

• Do you remember the slope formula?

• Do you remember the distance formula that finds the distance between two points?

• Do you remember how to tell if two lines are parallel?

• Do you remember how to tell if two lines are perpendicular?

Determine the most precise name for the quadrilateral with vertices Q(–4, 4), B(–2, 9), H(8, 9), and A(10, 4).

Graph quadrilateral QBHA.

First, find the slope of each side.

slope of QB = slope of BH = slope of HA = slope of QA = 9 – 4 –2 – (–4)

52=

9 – 9 8 – (–2) = 0

4 – 9 10 – 8 = – 5

2 4 – 4 –4 – 10 = 0

BH is parallel to QA because their slopes are equal. QB is not parallel to HA because their slopes are not equal.

Classifying QuadrilateralsClassifying Quadrilaterals

6-1

Because QB = HA, QBHA is an isosceles trapezoid.

One pair of opposite sides are parallel, so QBHA is a trapezoid.

Next, use the distance formula to see whether any pairs of sides are congruent.

QB = ( –2 – ( –4))2 + (9 – 4)2 = 4 + 25 = 29

HA = (10 – 8)2 + (4 – 9)2 = 4 + 25 = 29

BH = (8 – (–2))2 + (9 – 9)2 = 100 + 0 = 10

QA = (– 4 – 10)2 + (4 – 4)2 = 196 + 0 = 14

Classifying QuadrilateralsClassifying Quadrilaterals(continued)

6-1

You try one You try one • Turn to page 289 and complete

check understanding 2 (bottom of page).

In parallelogram RSTU, m R = 2x – 10 and m S = 3x + 50. Find x.

Draw quadrilateral RSTU. Label R and S.

RSTU is a parallelogram. Given

Definition of parallelogram ST || RU

Classifying QuadrilateralsClassifying Quadrilaterals

m R + m S = 180 If lines are parallel, then interior angles on the same side of a transversal are supplementary.

6-1

(continued)

Subtract 40 from each side.5x = 140

5x + 40 = 180 Simplify.

x = 28 Divide each side by 5.

(2x – 10) + (3x + 50) = 180 Substitute 2x – 10 for m R and 3x + 50 for m S.

Classifying QuadrilateralsClassifying Quadrilaterals

6-1

You try oneYou try one• Turn to page 290 and complete

check understanding 3 (middle of page)

Summary 6.1Summary 6.1

What are the seven types of What are the seven types of quadrilaterals we have described quadrilaterals we have described today?today?

How do you tell if two lines are How do you tell if two lines are parallel?parallel?

How do you tell if two lines are How do you tell if two lines are perpendicular?perpendicular?

6.2 Properties of 6.2 Properties of Parallelograms (page 294)Parallelograms (page 294)

• Obj 1: to use relationships among sides & among angles of parallelograms

• Obj 2: to use relationships involving diagonals of parallelograms & transversals

• You can use what you know about parallel lines & transversals to prove some theorems about parallelograms

• Theorem 6.1 p. 294---Opposite sides of a parallelogram are congruent

•

Theorems Continued…Theorems Continued…• Theorem 6.2 page 295 –Opposite angles

of a parallelogram are congruent

• Theorem 6.3 page 296—the diagonals of a parallelogram bisect each other

• Theorem 6.4 page 297—If three of more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transveral.

Use KMOQ to find m O.

Q and O are consecutive angles of KMOQ, so they are supplementary.

Definition of supplementary anglesm O + m Q = 180

Substitute 35 for m Q.m O + 35 = 180

Subtract 35 from each side.m O = 145

Properties of ParallelogramsProperties of ParallelogramsGEOMETRY LESSON 6-2GEOMETRY LESSON 6-2

6-2

Find the value of x in ABCD. Then find m A.

2x + 15 = 135 Add x to each side.

2x = 120 Subtract 15 from each side.

x = 60 Divide each side by 2.

x + 15 = 135 – x Opposite angles of a are congruent.

Properties of ParallelogramsProperties of Parallelograms

Substitute 60 for x. m B = 60 + 15 = 75

Consecutive angles of a parallelogram are supplementary.

m A + m B = 180

Subtract 75 from each side.m A = 105

m A + 75 = 180 Substitute 75 for m B.

GEOMETRY LESSON 6-2GEOMETRY LESSON 6-2

6-2

Find the values of x and y in

KLMN.x = 7y – 16 The diagonals of a parallelogram

bisect each other.2x + 5 = 5y

2(7y – 16) + 5 = 5y Substitute 7y – 16 for x in the second equation to solve for y.

14y – 32 + 5 = 5y Distribute.

14y – 27 = 5y Simplify.

–27 = –9y Subtract 14y from each side.

3 = y Divide each side by –9.

x = 7(3) – 16 Substitute 3 for y in the first equation to solve for x.

x = 5 Simplify.So x = 5 and y = 3.

6-2

Summary 6.2Summary 6.2 What are the properties of

parallelograms?Theorem 6.1-Theorem 6.2-Theorem 6.3-Theorem 6.4-

HomeworkHomework6.1 page 290 2-26 E, 37-42