6.3 tests for parallelograms

7/23/2019 6.3 Tests for Parallelograms

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Five-Minute Check (over Lesson 6–2)

CCSS

Then/Now

Theorems: Conditions or !"r"##e#o$r"ms

!roo: Theorem 6%&

'"m#e *: +denti, !"r"##e#o$r"ms

'"m#e 2: e"#-.or#d '"m#e: se !"r"##e#o$r"ms to !rovee#"tionshis

'"m#e 0: se !"r"##e#o$r"ms "nd 1#$er" to Find 3"#uesConcet Summ"r,: !rove th"t " 4u"dri#"ter"# +s " !"r"##e#o$r"m

'"m#e 5: !"r"##e#o$r"ms "nd Coordin"te eometr,

'"m#e 7: !"r"##e#o$r"ms "nd Coordin"te !roos



8ver Lesson 6–2

1%

9%

C%

____ ?



8ver Lesson 6–2

1%

9%

C%

?



8ver Lesson 6–2

1% A

9%

B

C% C

?



8ver Lesson 6–2

1n e"nd"#e $"te is m"de o "r"##e#o$r"ms th"th"ve "n$#es th"t ch"n$eme"sure "s the $"te is"dusted% .hich o the

o##owin$ st"tements is"#w",s true;

1% ∠ A ≅ ∠C and ∠B ≅ ∠D

9% ∠ A ≅ ∠B and ∠C ≅ ∠D

C%

<%



Content St"nd"rds

G.CO.11 Prove theorems about

parallelograms.G.GPE.4 Use coordinates to prove simplegeometric theorems algebraically.

M"them"tic"# !r"ctices

Construct viable arguments and criti!uethe reasoning o" others.

# $eason abstractly and !uantitatively.



%ou recogni&ed and applied properties o"parallelograms.

' $ecogni&e the conditions that ensure a

!uadrilateral is a parallelogram.' Prove that a set o" points "orms a

parallelogram in the coordinate plane.



+denti, !"r"##e#o$r"ms

<etermine whether the =u"dri#"ter"# is ""r"##e#o$r"m% >usti, ,our "nswer%

1nswer:Each pair o" opposite sides has the samemeasure. (here"ore) they are congruent.*" both pairs o" opposite sides o" a!uadrilateral are congruent) the !uadrilateralis a parallelogram.



1% 9oth "irs o o% sides ??%

9% 9oth "irs o o% sides %

C% 9oth "irs o o% s %

<% 8ne "ir o o% sides oth?? "nd %

.hich method wou#d rove the=u"dri#"ter"# is " "r"##e#o$r"m;



se !"r"##e#o$r"ms to !rove

e#"tionshis

M'C@1N+CS Scissor #itsA #ikethe #"torm #it shownA "recommon#, "#ied to too#sintended to #it he"v, items% +n the

di"$r"mA A C "nd B D%'#"in wh, the consecutive"n$#es wi## "#w",s esu#ement"r,A re$"rd#ess o thehei$ht o the #"torm%



se !"r"##e#o$r"ms to !rove

e#"tionshis

1nswer: +ince both pairs o" opposite angles o"!uadrilateral ABCD are congruent) ABCD isa parallelogram by (heorem ,.1-. (heorem,. states that consecutive angles o"

parallelograms are supplementary.(here"ore) m∠ A / m∠B 0 1- andm∠C / m∠D 0 1-. 2y substitution)m∠ A / m∠D 0 1- and m∠C / m∠B 0 1-.



The di"$r"m shows " c"r "ck used to r"ise " c"rrom the $round% +n the di"$r"mA AD BC "nd AB DC % 9"sed on this inorm"tionA whichst"tement wi## e trueA re$"rd#ess o the hei$ht o

the c"r "ck%1% A B

9% A C

C% AB BC

<% m A B m C *DE



se !"r"##e#o$r"ms "nd 1#$er" to Find 3"#ues

Find x "nd y so th"t the =u"dri#"ter"# is ""r"##e#o$r"m%

Opposite sides o" a parallelogram are congruent.




+ubstitution

3istributive Property

dd 1 to each side.

+ubtract x "rom each side.

AB = DC




1nswer: +o) 5hen x 0 6 and y 0 ) !uadrilateral ABCD is a parallelogram.

+ubstitution

3istributive Property

dd # to each side.

+ubtract y "rom each side.



1% m 2

9% m 0

C% m 6

<% m D

Find m so th"t the =u"dri#"ter"# is " "r"##e#o$r"m%



!"r"##e#o$r"ms "nd Coordin"te eometr,

C88<+N1T' '8M'T 4u"dri#"ter"# QRST h"s verticesQ (–*A 0)A R (0A *)A S (2A –0)A "ndT (–2A –*)% <etermine whether the=u"dri#"ter"# is " "r"##e#o$r"m%>usti, ,our "nswer , usin$ theS#oe Formu#"%

*" the opposite sides o" a !uadrilateral are parallel)

then it is a parallelogram.



!"r"##e#o$r"ms "nd Coordin"te eometr,

1nswer: +ince opposite sides have the same slope)QR 7ST and RS7TQ. (here"ore) QRST is aparallelogram by de"inition.



1% ,es

9% no

r"h =u"dri#"ter"# EFGH with vertices E (–2A 2)AF (2A E)A G (*A –7)A "nd H (–0A –2)% <etermine whetherthe =u"dri#"ter"# is " "r"##e#o$r"m%



!"r"##e#o$r"ms "nd Coordin"te !roos

.rite " coordin"te roo or the o##owin$st"tement%

*" both pairs o" opposite sides o" a !uadrilateral arecongruent) then the !uadrilateral is a parallelogram.

8 2egin by placing the verte9 A at the origin.

Ste * Position !uadrilateral ABCD on the coordinateplane such that AB ≅ DC and AD ≅ BC .

8 :et AB have a length o" a units. (hen B hascoordinates ;a) -<.




8 +o that the distance "rom D to C is also a units) letthe x =coordinate o" D be b and o" C be b / a.

8 +ince AD ≅ BC, position the endpoints o" DC so thatthey have the same y =coordinate) c .




Ste 2 Use your "igure to 5rite a proo".

iven: !uadrilateral ABCD) AB ≅ DC ) AD ≅ BC

!rove: ABCD is a parallelogram.

Coordin"te !roo:

2y de"inition) a !uadrilateral is a parallelogram i"opposite sides are parallel.

Use the +lope >ormula.




1nswer: +o) !uadrilateral ABCD is a parallelogrambecause opposite sides are parallel.

+ince AB and CD have the same slope and AD and BC have the same slope) AB7CD and AD7BC .

(he slope o" CD is -.

(he slope o" AB is -.



.hich o the o##owin$ c"n e used to rove thest"tement e#ow;*" a !uadrilateral is aparallelogram) then one pair o"opposite sides is both parallel and

congruent.1% AB 0 a units and DC 0 a units

slope o" AB 0 - and slope o"DC 0 -

9% AD 0 c units and BC 0 c units

slope o" and slope o"

6.3 tests for parallelograms

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