Given parallelogram ABCD and diagonal t Prove that triangles ABC and CDA are congruent (Theorem 5.14) and opposite angles and sides are congruent (Theorem 5.15)

Statements Reasons

ml

pn

31

B

m

l

p

n

1

4

2

3

A

C D

t

24

ACAC

ADCCBA

CDACBA

ABCDBCAD ,

3241 mmmm

41 mmBADm

32 mmBCDm

BADBCD

Def of a Parallelogram

Def of a Parallelogram

5.5

5.5

reflexive

ASA

CP

CP

APE

AAP

AAP

Substitution

B

A

C

D

m

l

B

A C

D

If the transversals are

perpendicular and l and m are

parallel…

How would we prove Theorem 5.17 If both pairs of opposite sides of a quadrilateral are congruent then it is a parallelogram.

Are lines AB and CD parallel?

Are segments AB and CD congruent?

Parallel lines are everywhere equidistant

B

A

C

D

b

a

c

d

Theorem 5.18 If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. If you were trying to prove this you could start with the sum of all of the angles.

Theorem 5.19 If a quadrilateral has two sides that are parallel and congruent, then it is a parallelogram. Try to prove this using SAS

B

A

C

D

>

>

|

|

odcba 360obaba 360

oba 36022 oba 180obc 180

B

A

C

D

Theorem 5.20 The diagonals of a parallelogram bisect each other.

B

A

C

D CDEABE that proveFirst

E

E |

| ||

|| Theorem 5.21 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

5.19. Theorem use and that proveFirst CEBAED

Some Applications

Linkages

Forces and Resultant Vectors -the combination of two forces acting on an object.

B

A

C

D

Theorem 5.17 If both pairs of opposite sides of a quadrilateral are congruent then it is a parallelogram.

ram.parallelog a is ABCD ralQuadrilate :Prove

ADBC and CDBA :Given

Statements Reasons

CDBA Given

ADBC Given

ACAC Reflexive

ΔCDAΔABC SSS

ACBAD C CP

CABACD CP

ADBC

DCAB

5.1

5.1

ramParallelog a is ABCD ramParallelog a of Def

Theorem 5.20 The diagonals of a parallelogram bisect each other.

B

A

C

D eachotherbisect AC and BD :Prove

ram.parallelog a is ABCD ralQuadrilate :Given

E

Statements Reasons

CEDBEA

ramparallelog a is ABCD Given

CDBA ramparallelog a of Def

5.5CDEABE

Theorem s' Vertical

CDBA 15.5

ΔDECΔBEA AAS

ECAE and EDBE CP

eachotherbisect BD and AC BisectorSegment a of Definition

Since the rhombus is a parallelogram, what can we say about the diagonals?

Theorem 5.22 Every Rhombus is a parallelogram.

A

B C

D

A

B C

D

E

(Both pairs of opposite sides are congruent)

The four triangles in the diagram are congruent by… SSS

The diagonals are perpendicular by…

The Congruent Supplements Theorem

So you can say the opposite angles are bisected by… CP

Or Theorem 4.9

A

B C

D

E

If the diagonals of a parallelogram are perpendicular, why can we say it’s a rhombus?

If the opposite angles of a parallelogram are bisected, why can we say it’s a rhombus?

Opposite angles are congruent (5.15) and bisected (given). The diagonals are bisected (5.20), so the triangles are congruent by SAS and the sides are congruant by CP

A

B C

D

E

The diagonals are bisected (5.20) and the trianlges are congruent by SAS, so all the sides are congruent by CP.

Number 46 from Section 5.3 asks what will happen to a cue ball, shot at a 45 degree angle on a pool table with dimensions of 9 by 4.5 feet.

Now, what if we changed the dimensions of the pool table to 9 by 6 feet with no side pockets.

At home, use the Paper Pool Applet in the Unit 5 notes to experiment with a variety of table dimensions and begin recording your results in a table. Will the ball always land in a pocket? Are there any dimensions for which the ball will not land in a pocket? Can you generalize a rule for determining how many bounces it will take for any size pool table? Extensions: What impact does changing the starting point or angle of departure have on your conclusions.

A

B C

D

.AC BD and ramparallelog a is ABCD :Prove

rhombus. a is ABCD ralQuadrilate :Given

Prove Theorems 5.22 and 5.23

Statements Reasons

AD CDBCAB Rhombus a of Definition

ramParallelog a is ABCD 5.17

eachotherbisect BD and AC 5.20

E

EDBE BisectorSegment a of Definition

ECEC Reflexive

ΔDECΔBEC SSS

DECBEC CP

anglestraight a is BED Given

arysupplement are DEC and BEC anglestraight a of Definition

anglesright are DEC and BEC Theorem sSupplementCongruent

ACBD of Definition

A

B C

D

.AC BD and ramparallelog a is ABCD :Prove

rhombus. a is ABCD ralQuadrilate :Given

Prove Theorems 5.22 and 5.23

Statements Reasons

AD CDBCAB Rhombus a of Definition

ramParallelog a is ABCD 5.17

eachotherbisect BD and AC 5.20

E

EDBE BisectorSegment a of Definition

ECEC Reflexive

ΔDECΔBEC SSS

BCE DCE CP

is an angle bisectorCE Def of an angle bisector

is isoscelesBCD Definition of an isosceles triangle

is the bisector of BDCE 4.9

CE BD Definition of bisector

What is the definition of a rectangle?

Theorem 5.26 Every Rectangle is a parallelogram

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Theorem 5.27 A parallelogram with one right angle is a rectangle? How would you prove this Theorem? (Use Theorem 5.6)

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> >

(By Theorem 5.18 – the opposite angles are congruent)

>

Theorem 5.28 The diagonals of a rectangle are congruent.

A D

B C

DCAABD gettingby thisProve

Single house architecture from Tamizo Architects Group.

>

>

Theorem 5.30 The base angles of an isosceles trapezoid are congruent.

Introduce a segment through C parallel to segment AB to form a parallelogram and classify triangle ECD.

A

B C

D E

>

> A

B C

D E

Is the converse of this theorem true? If the base angles of a trapezoid are congruent, are the sides opposite them congruent?

Introduce a segment CE such that angle CED is congruent to angle CDE.

Doorways of Machu Picchu

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>>

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> >

Theorem 5.27 A parallelogram with one right angle is a rectangle

rectangle. a is ABCD :Prove

right a is BAD and ABCD ramParallelog :Given

Statements Reasons

A D

B C

ADBA

BCBA

CDBA and ADBC

angleright a is BAD

angleright a is ABC

ramparallelog a is ABCD

CDBC

angleright a is BCD

ADCD

angleright a is ADC

rectangle a is ABCD

Given

Given

ramParallelog a of Definition

of Definition

6.5

of Definition

6.5

of Definition

6.5

of Definition

rectangle a of Definition

>

Theorem 5.28 The diagonals of a rectangle are congruent.

A D

B C

BDAC :Prove

rectangle a is BCD :Given

A

Statements Reasons

rectangle a is ABCD

ramparallelog a is ABCD

DCAB 15.5

ADAD

CDABAD

anglesright areCDA and BAD

CDABAD

ACBD

Given

26.5

Reflexive

rectangle a of Definition

Theorem AngleRight

SAS

CP

Theorem 5.29 If a parallelogram has congruent diagonals, then it is a rectangle.

Statements Reasons

ramparallelog a is ABCD

DCAB

ADAD

ΔCDAΔBAD

CDABAD

arysupplement areCDA and BAD

anglesright areCDA and BAD

rectangle a is ABCD

given

Theorem 5.15

Reflexive

SSS

CP

Same-side Interior Angles

Congruent Supplements Theorem

Theorem 5.27

BDAC given

>

A D

B C

1

2

3

Given: and transversal t

Prove: 1 2 are supplementary

l m

and

Statements Reasons

5.7

Given

1 = m 3m

2 and 3 form a straight angle

2 + m 3 =180om

2 + m 1 =180om

1 and 2 are supplementary

Def. of a Straight Angle

Substitution

Def. of Supplementary

3

1

2

4

Given: and transversal t

Prove: 1 2 are supplementary

l m

and

Statements Reasons

3 and 4 are supplementary

3 + m 4 =180om

1 3

2 4

1 + m 2 =180om

1 and 2 are supplementary

5.8

Def. of Supplementary

Vertical Angles Thm.

Vertical Angles Thm.

Substitution

Def. of Supplementary

A B D E

C

Given : CBCD

Prove : ABCCDE

Statements Reasons

CBDCDB

ABC and CBD form a straight

mABC+mCBD 180o

mCDB+mCDE 180o

mCDB+mCDE mABC mCBD

mCDE mABC

CDEABC

Base Angles Thm.

Given

Def. of a Straight

Def. of a Straight

Substitution

APE

Def. of Angles

CDB and CDE form a straight

Given

A B D E

C

Statements Reasons

CBDCDB

ABC and CBD are supplementary

CDB and CDE are supplementary

CDEABC

Base Angles Thm.

Def. of a Straight

Def. of a Straight

Supplements Theorem

Given : CBCD

Prove : ABCCDE

ABC and CBD form a straight

Given

CDB and CDE form a straight

Given