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12/02/2014 Conditions for Parallelograms. Grade 9 ASP. Warm Up Justify each statement. 1. 2. Evaluate each expression for x = 12 and y = 8.5. 3. 2 x + 7 4. 16 x – 9 5. (8 y + 5)°. Reflex Prop. of . Conv. of Alt. Int. s Thm. 31. 183. 73°. Objective.

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slide3

Warm Up

Justify each statement.

1.

2.

Evaluate each expression for x = 12 and

y = 8.5.

3. 2x + 7

4. 16x –9

5.(8y + 5)°

Reflex Prop. of 

Conv. of Alt. Int. s Thm.

31

183

73°

slide4

Objective

Prove that a given quadrilateral is a parallelogram.

slide5

You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you can

use the definition of a parallelogram or the conditions below.

slide8

Example 1A: Verifying Figures are Parallelograms

Show that JKLM is a parallelogram for a = 3 and b = 9.

Step 1 Find JK and LM.

Given

LM = 10a + 4

JK = 15a – 11

Substitute and simplify.

LM = 10(3)+ 4 = 34

JK = 15(3) – 11 = 34

slide9

Example 1A Continued

Step 2 Find KL and JM.

Given

KL = 5b + 6

JM = 8b – 21

Substitute and simplify.

KL = 5(9) + 6 = 51

JM = 8(9) – 21 = 51

Since JK = LM and KL = JM, JKLM is a parallelogram by Theorem 6-3-2.

slide10

Example 1B: Verifying Figures are Parallelograms

Show that PQRS is a parallelogram for x = 10 and y = 6.5.

mQ = (6y + 7)°

Given

Substitute 6.5 for y and simplify.

mQ = [(6(6.5) + 7)]° = 46°

mS = (8y – 6)°

Given

Substitute 6.5 for y and simplify.

mS = [(8(6.5) – 6)]° = 46°

mR = (15x – 16)°

Given

Substitute 10 for x and simplify.

mR = [(15(10) – 16)]° = 134°

slide11

Example 1B Continued

Since 46° + 134° = 180°, R is supplementary to both Q and S. PQRS is a parallelogram by Theorem 6-3-4.

slide12

PQ = RS = 16.8, so

Therefore,

Check It Out! Example 1

Show that PQRS is a parallelogram for a = 2.4 and b = 9.

mQ = 74°, and mR = 106°, so Q and R are supplementary.

So one pair of opposite sides of PQRS are || and .

By Theorem 6-3-1, PQRS is a parallelogram.

slide13

Example 2A: Applying Conditions for Parallelograms

Determine if the quadrilateral must be a parallelogram. Justify your answer.

Yes. The 73° angle is supplementary to both its corresponding angles. By Theorem 6-3-4, the quadrilateral is a parallelogram.

slide14

Example 2B: Applying Conditions for Parallelograms

Determine if the quadrilateral must be a parallelogram. Justify your answer.

No. One pair of opposite angles are congruent. The other pair is not. The conditions for a parallelogram are not met.

slide15

Check It Out! Example 2a

Determine if the quadrilateral must be a parallelogram. Justify your answer.

Yes

The diagonal of the quadrilateral forms 2 triangles.

Two angles of one triangle are congruent to two angles of the other triangle, so the third pair of angles are congruent by the Third Angles Theorem.

So both pairs of opposite angles of the quadrilateral are congruent .

By Theorem 6-3-3, the quadrilateral is a parallelogram.

slide16

Check It Out! Example 2b

Determine if each quadrilateral must be a parallelogram. Justify your answer.

No. Two pairs of consective sides are congruent.

None of the sets of conditions for a parallelogram are met.

slide17

Helpful Hint

To say that a quadrilateral is a parallelogram by

definition, you must show that both pairs of opposite sides are parallel.

slide18

Example 3A: Proving Parallelograms in the Coordinate Plane

Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0).

Find the slopes of both pairs of opposite sides.

Since both pairs of opposite sides are parallel, JKLM is a parallelogram by definition.

slide19

AB and CD have the same slope, so . Since AB = CD, . So by Theorem 6-3-1, ABCD is a parallelogram.

Example 3B: Proving Parallelograms in the Coordinate Plane

Show that quadrilateral ABCD is a parallelogram by using Theorem 6-3-1.A(2, 3), B(6, 2), C(5, 0), D(1, 1).

Find the slopes and lengths of one pair of opposite sides.

slide20

Both pairs of opposite sides have the same slope so and by definition, KLMN is a parallelogram.

Check It Out! Example 3

Use the definition of a parallelogram to show that the quadrilateral with vertices K(–3, 0), L(–5, 7), M(3, 5), and N(5, –2) is a parallelogram.

slide21

You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply.

slide22

Helpful Hint

To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions.

slide23

Example 4: Application

The legs of a keyboard tray are connected by a bolt at their midpoints, which allows the tray to be raised or lowered. Why is PQRS always a parallelogram?

Since the bolt is at the midpoint of both legs, PE = ER and SE = EQ. So the diagonals of PQRS bisect each other, and by Theorem 6-3-5, PQRS is always a parallelogram.

slide24

Since ABRS is a parallelogram, it is always true that .

Since AB stays vertical, RS also remains vertical no matter how the frame is adjusted.

Check It Out! Example 4

The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same?

Therefore the viewing never changes.

slide25

Lesson Quiz: Part I

1. Show that JKLM is a parallelogram for a = 4 and b = 5.

2. Determine if QWRT must be a parallelogram. Justify your answer.

JN = LN = 22; KN = MN = 10; so JKLM is a parallelogram by Theorem 6-3-5.

No; One pair of consecutive s are , and one pair of opposite sides are ||. The conditions for a parallelogram are not met.

slide26

Lesson Quiz: Part II

3. Show that the quadrilateral with vertices E(–1, 5), F(2, 4), G(0, –3), and H(–3, –2) is a parallelogram.

Since one pair of opposite sides are || and , EFGH is a parallelogram by Theorem 6-3-1.

slide27
L.O.
  • Application in Racing
  • Using Properties in Parallelograms to find Measures.
  • Parallelograms in Coordinate Planes.
  • Using Properties of Parallelograms in Proof.
slide28

Lesson Quiz

1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex.

nonagon; irregular; concave

2. Find the sum of the interior angle measures of a convex 11-gon.

1620°

3. Find the measure of each interior angle of a regular 18-gon.

4. Find the measure of each exterior angle of a regular 15-gon.

160°

24°

classwork and homework
CLASSWORK AND HOMEWORK

CLASSWORK

HOMEWORK

See Homework booklet. Week 3

  • (Pages 407 to 409 )

1, 2, 3 to 13, 14, 21 to 24, 25, 26, 27 to 30, 32 to 43, 46, 47, 50, 51, 52, 53.

slide30

Objectives

Prove and apply properties of parallelograms.

Use properties of parallelograms to solve problems.

slide31

Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.

slide32

Helpful Hint

Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.

slide33

A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .

slide36

In CDEF, DE = 74 mm,

DG = 31 mm, and mFCD = 42°. Find CF.

opp. sides

Example 1A: Properties of Parallelograms

CF = DE

Def. of segs.

CF = 74 mm

Substitute 74 for DE.

slide37

In CDEF, DE = 74 mm,

DG = 31 mm, and mFCD = 42°. Find mEFC.

cons. s supp.

Example 1B: Properties of Parallelograms

mEFC + mFCD = 180°

mEFC + 42= 180

Substitute 42 for mFCD.

mEFC = 138°

Subtract 42 from both sides.

slide38

In CDEF, DE = 74 mm,

DG = 31 mm, and mFCD = 42°. Find DF.

diags. bisect each other.

Example 1C: Properties of Parallelograms

DF = 2DG

DF = 2(31)

Substitute 31 for DG.

DF = 62

Simplify.

slide39

opp. sides

Check It Out! Example 1a

In KLMN, LM = 28 in.,

LN = 26 in., and mLKN = 74°. Find KN.

LM = KN

Def. of segs.

LM = 28 in.

Substitute 28 for DE.

slide40

opp. s 

Check It Out! Example 1b

In KLMN, LM = 28 in.,

LN = 26 in., and mLKN = 74°. Find mNML.

NML  LKN

mNML = mLKN

Def. of  s.

mNML = 74°

Substitute 74° for mLKN.

Def. of angles.

slide41

diags. bisect each other.

Check It Out! Example 1c

In KLMN, LM = 28 in.,

LN = 26 in., and mLKN = 74°. Find LO.

LN = 2LO

26 = 2LO

Substitute 26 for LN.

LO = 13 in.

Simplify.

slide42

opp. s 

Example 2A: Using Properties of Parallelograms to Find Measures

WXYZ is a parallelogram. Find YZ.

YZ = XW

Def. of  segs.

8a – 4 = 6a + 10

Substitute the given values.

Subtract 6a from both sides and add 4 to both sides.

2a = 14

a = 7

Divide both sides by 2.

YZ = 8a – 4 = 8(7) – 4 = 52

slide43

cons. s supp.

Example 2B: Using Properties of Parallelograms to Find Measures

WXYZ is a parallelogram. Find mZ.

mZ + mW = 180°

(9b + 2)+ (18b –11) = 180

Substitute the given values.

Combine like terms.

27b – 9 = 180

27b = 189

b = 7

Divide by 27.

mZ = (9b + 2)° = [9(7) + 2]° = 65°

slide44

diags. bisect each other.

Check It Out! Example 2a

EFGH is a parallelogram.

Find JG.

EJ = JG

Def. of  segs.

3w = w + 8

Substitute.

2w = 8

Simplify.

w = 4

Divide both sides by 2.

JG = w + 8 = 4 + 8 = 12

slide45

diags. bisect each other.

Check It Out! Example 2b

EFGH is a parallelogram.

Find FH.

FJ = JH

Def. of  segs.

4z – 9 = 2z

Substitute.

2z = 9

Simplify.

z = 4.5

Divide both sides by 2.

FH = (4z – 9) + (2z) = 4(4.5) – 9 + 2(4.5) = 18

slide46

Remember!

When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices.

slide47

L

K

J

Example 3: Parallelograms in the Coordinate Plane

Three vertices of JKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M.

Since JKLM is a parallelogram, both pairs of opposite sides must be parallel.

Step 1 Graph the given points.

slide48

Step 2 Find the slope of by counting the units from K to L.

    • The rise from 2 to 6 is 4.
    • The run of –2 to 2 is 4.

L

K

J

Example 3 Continued

  • Step 3 Start at J and count the
  • same number of units.
    • A rise of 4 from –8 is –4.
    • A run of 4 from 3 is 7. Label (7, –4) as vertex M.

M

slide49

Step 4 Use the slope formula to verify that

L

K

M

J

Example 3 Continued

The coordinates of vertex M are (7, –4).

slide50

Q

S

P

Check It Out! Example 3

Three vertices of PQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R.

Since PQRS is a parallelogram, both pairs of opposite sides must be parallel.

Step 1 Graph the given points.

slide51

Step 2 Find the slope of by counting the units from P to Q.

    • The rise from –2 to 4 is 6.
    • The run of –3 to –1 is 2.

Q

S

P

Check It Out! Example 3 Continued

R

  • Step 3 Start at S and count the
  • same number of units.
    • A rise of 6 from 0 is 6.
    • A run of 2 from 5 is 7. Label (7, 6) as vertex R.
slide52

Step 4 Use the slope formula to verify that

R

Q

S

P

Check It Out! Example 3 Continued

The coordinates of vertex R are (7, 6).

slide53

Example 4A: Using Properties of Parallelograms in a Proof

Write a two-column proof.

Given: ABCD is a parallelogram.

Prove:∆AEB∆CED

slide54

3. diags. bisect

each other

2. opp. sides 

Example 4A Continued

Proof:

1. ABCD is a parallelogram

1. Given

4. SSS Steps 2, 3

slide55

Example 4B: Using Properties of Parallelograms in a Proof

Write a two-column proof.

Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear.

Prove:H M

slide56

2.H and HJN are supp.

M and MJK are supp.

2. cons. s supp.

Example 4B Continued

Proof:

1.GHJN and JKLM are

parallelograms.

1. Given

3.HJN  MJK

3. Vert. s Thm.

4.H  M

4. Supps. Thm.

slide57

Check It Out! Example 4

Write a two-column proof.

Given: GHJN and JKLM are parallelograms.

H and M are collinear. N and K are collinear.

Prove: N  K

slide58

2.N and HJN are supp.

K and MJK are supp.

2. cons. s supp.

Check It Out! Example 4 Continued

Proof:

1.GHJN and JKLM

are parallelograms.

1. Given

3.HJN  MJK

3. Vert. s Thm.

4.N  K

4. Supps. Thm.

slide59

Lesson Quiz: Part I

In PNWL, NW = 12, PM = 9, and

mWLP = 144°. Find each measure.

1.PW2. mPNW

18

144°

slide61

Lesson Quiz: Part II

QRST is a parallelogram. Find each measure.

2.TQ3. mT

71°

28

slide62

Lesson Quiz: Part III

5. Three vertices of ABCD are A (2, –6), B (–1, 2), and C(5, 3). Find the coordinates of vertex D.

(8, –5)

slide63

1.RSTU is a parallelogram.

1. Given

3.R  T

4. ∆RSU∆TUS

4. SAS

2. cons. s 

3. opp. s 

Lesson Quiz: Part IV

6. Write a two-column proof.

Given:RSTU is a parallelogram.

Prove: ∆RSU∆TUS