Download Study Guide

.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
>ÃÈvÞÊÌÀˆ>˜}iÃÊ>˜`Êwʘ`ʓi>ÃÕÀiÃʜvÊ̅iˆÀÊ>˜}ið
6œV>LՏ>ÀÞ
!TRIANGLEISAPOLYGONWITHTHREESIDES
!SCALENETRIANGLEHASNOCONGRUENTSIDES
!NISOSCELESTRIANGLEHASATLEASTTWOCONGRUENTSIDES
!NEQUILATERALTRIANGLEHASTHREECONGRUENTSIDES
!NACUTETRIANGLEHASTHREEACUTEANGLES
!RIGHTTRIANGLEHASONERIGHTANGLE
!NOBTUSETRIANGLEHASONEOBTUSEANGLE
7HENTHESIDESOFAPOLYGONAREEXTENDEDOTHERANGLESAREFORMED
4HEORIGINALANGLESARETHEINTERIORANGLES4HEANGLESTHATFORMLINEAR
PAIRSWITHTHEINTERIORANGLESARETHEEXTERIORANGLES
--" Ê{°£
!NEQUIANGULARTRIANGLEHASTHREECONGRUENTANGLES
4HEOREM4RIANGLE3UM4HEOREM4HESUMOFTHEMEASURESOFTHE
INTERIORANGLESOFATRIANGLEIS
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
4HEOREM%XTERIOR!NGLE4HEOREM4HEMEASUREOFANEXTERIOR
ANGLEOFATRIANGLEISEQUALTOTHESUMOFTHEMEASURESOFTHETWO
NONADJACENTINTERIORANGLES
#OROLLARYTOTHE4RIANGLE3UM4HEOREM4HEACUTEANGLESOFARIGHT
TRIANGLEARECOMPLEMENTARY
8*Ê£
>ÃÈvÞÊÌÀˆ>˜}iÃÊLÞÊÈ`iÃÊ>˜`ÊLÞÊ>˜}iÃÊ
>ÃÈvÞÊ̅iÊÌÀˆ>˜}iÊLÞʈÌÃÊÈ`iÃÊ>˜`ÊLÞʈÌÃÊ>˜}ið
Ê >°
L°
-œṎœ˜
Ê >° 4RIANGLE$%&HASONEOBTUSEANGLEANDNOCONGRUENTSIDES
3ONg$%&ISANOBTUSESCALENETRIANGLE
Ê L° 4RIANGLE!"#HASONERIGHTANGLEANDTWOCONGRUENTSIDES
3ONg!"#ISARIGHTISOSCELESTRIANGLE
'EOMETRY
#HAPTER2ESOURCE"OOK
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
>ÃÈvÞÊ̅iÊÌÀˆ>˜}iÊLÞʈÌÃÊÈ`iÃÊ>˜`ÊLÞʈÌÃÊ>˜}ið
£°
Ó°
ΰ
ˆ˜`Ê>˜}iʓi>ÃÕÀiÃ
--" Ê{°£
8*ÊÓ
>° &INDMŽ"!#ANDMŽ"#!
L° &INDMŽ"#$ANDMŽ!"#
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
-œṎœ˜
>° XzzzzXzzzz
5SE#OROLLARYTOTHE4RIANGLE
3UM4HEOREM
3OLVEFORX
Xzz
3OMŽ"#!zzXzzzz+zzzzAND
MŽ"!#zzXzzzz+zzzz
L° XzzzzXzz
5SE%XTERIOR!NGLE4HEOREM
Xzz
3OLVEFORX
3OMŽ"#$zzXzzzz+zzzzAND
MŽ!"#zzXzzzz
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊÓ
{° &INDMŽ!"$ANDMŽ"$#
'EOMETRY
#HAPTER2ESOURCE"OOK
x° &INDMŽ#!"ANDMŽ#"!
Answer Key
Lesson 4.1
Study Guide
1. right scalene 2. equiangular equilateral
3. obtuse isosceles 4. m∠ ABD 5 908, m∠ BDC 5 608
5. m∠ CAB 5 808, m∠ CBA 5 108
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
`i˜ÌˆvÞÊVœ˜}ÀÕi˜ÌÊwÊ}ÕÀið
6œV>LՏ>ÀÞ
)NTWOCONGRUENTlGURESALLTHEPARTSOFONElGUREARECONGRUENTTO
THECORRESPONDINGPARTSOFTHEOTHERlGURE)NCONGRUENTPOLYGONS
THISMEANSTHATTHECORRESPONDINGSIDESANDTHECORRESPONDINGANGLES
ARECONGRUENT
4HEOREM4HIRD!NGLES4HEOREM)FTWOANGLESOFONETRIANGLEARE
CONGRUENTTOTWOANGLESOFANOTHERTRIANGLETHENTHETHIRDANGLESARE
CONGRUENT
8*Ê£
1ÃiÊ«Àœ«iÀ̈iÃʜvÊVœ˜}ÀÕi˜ÌÊwÊ}ÕÀiÃ
˜Ê̅iÊ`ˆ>}À>“]Ê
ÊÊ°
Ê >° &INDTHEVALUEOFX
L° &INDTHEVALUEOFY
-œṎœ˜
] ]
>° 9OUKNOWTHAT!%z&*z
!%zz*&
zzXzz
zzX
zzX
L° 9OUKNOWTHATŽ$Ž)
MŽ$zzMŽ)
zzYzz
zzYzz
zzY
zzY
--" Ê{°Ó
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
˜Ê̅iÊ`ˆ>}À>“]ÊN
ÊÊN°
Ê £° &INDTHEVALUEOFX
Ê Ó° &INDTHEVALUEOFY
'EOMETRY
#HAPTER2ESOURCE"OOK
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
˜Ê̅iÊ`ˆ>}À>“]Ê
ÊÊ°
Ê Î° &INDTHEVALUEOFX
Ê {° &INDTHEVALUEOFY
8*ÊÓ
1ÃiÊ̅iÊ/…ˆÀ`ʘ}iÃÊ/…iœÀi“
ˆ˜`ʓŽ°
-œṎœ˜
--" Ê{°Ó
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊÓ
ˆ˜`Ê̅iÊÛ>ÕiʜvÊÝ°
x°
È°
'EOMETRY
#HAPTER2ESOURCE"OOK
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
)NTHEDIAGRAMŽ!Ž$ANDŽ"Ž%3OBYTHE4HIRD!NGLES4HEOREM
Ž#Ž&"YTHE4RIANGLE3UM4HEOREMMŽ#zzzzzzzz
3OMŽ#zzMŽ&zzBYTHEDElNITIONOFCONGRUENTANGLES
Answer Key
Lesson 4.2
Study Guide
1. 3 2. 5 3. 7 4. 11 5. 65 6. 2
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
1ÃiÊ̅iÊÈ`iʏi˜}̅ÃÊ̜ʫÀœÛiÊÌÀˆ>˜}iÃÊ>ÀiÊVœ˜}ÀÕi˜Ì°
6œV>LՏ>ÀÞ
0OSTULATE3IDE3IDE3IDE333#ONGRUENCE0OSTULATE)FTHREE
SIDESOFONETRIANGLEARECONGRUENTTOTHREESIDESOFASECONDTRIANGLE
THENTHETWOTRIANGLESARECONGRUENT
8*Ê£
1ÃiÊ̅iÊ---Ê
œ˜}ÀÕi˜ViÊ*œÃÌՏ>Ìi
*ÀœÛiÊ̅>ÌÊNÊÊN °Ê
-œṎœ˜
4HEMARKSONTHEDIAGRAMSHOWTHAT
] ] ] ]
] ]
*+z-,z+,z,.zAND*,z-.z
3OBYTHE333#ONGRUENCE0OSTULATENgg*+,Ng-,.
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
iVˆ`iÊ܅i̅iÀÊ̅iÊVœ˜}ÀÕi˜ViÊÃÌ>Ìi“i˜ÌʈÃÊÌÀÕi°ÊÝ«>ˆ˜ÊޜÕÀÊ
Ài>ܘˆ˜}°
Ê
Ó° Ng879Ng7:9
Ê Î° Ng234Ng654
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
£° Ng!"$Ng#$"
{° Ng&'(Ng*('
Ê Ê
--" Ê{°Î
x° Ng012Ng243
Ê
È° Ng*+,Ng-0.
Ê
'EOMETRY
#HAPTER2ESOURCE"OOK
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
8*ÊÓ
œ˜}ÀÕi˜ÌÊÌÀˆ>˜}iÃʈ˜Ê>ÊVœœÀ`ˆ˜>ÌiÊ«>˜i
1ÃiÊ̅iÊ---Ê
œ˜}ÀÕi˜ViÊ*œÃÌՏ>ÌiÊ̜ÊŜÜÊÊ
̅>ÌÊN
ÊÊN
°
-œṎœ˜
5SETHE$ISTANCE&ORMULATOSHOWTHATCORRESPONDINGSIDESARETHESAMELENGTH
]]]
z zzzz
z
z
!"zzqzz
]
zzq zzz
zzq z
]
]]
#$zzq zzz
zzzzz
]
zzq zzz
]
zzq z
] ]
3O!"zz#$ANDHENCE!"z#$z
]]]
zzzzz
z
z
"#zzq zz
zzq zzz
zzq z
]
]
zzq zzz
]
zzq z
] ]
3O"#zz$%ANDHENCE"#z$%z
]]]
zzzzz
#!zzq zzz
]
zzq zzz
zzq z
]
]]
%#zzq zzzzzz
z
z
]]
zzq zz
z
z
]
zzq z
] ]
3O#!zz%#ANDHENCE#!z%#z
3OBYTHE333#ONGRUENCE0OSTULATEYOUKNOWTHATNg!"#Ng#$%
ÝiÀVˆÃiÊvœÀÊÝ>“«iÊÓ
Ê Ç° 0ROVETHATNg!"#Ng$%&Ê
--" Ê{°Î
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
]
]]
$%zzq zzzzzz
z
z
'EOMETRY
#HAPTER2ESOURCE"OOK
Answer Key
Lesson 4.3
Study Guide
1. Yes, the corresponding triangle sides are congruent
}
} }
}
2. No; WY À ZY, XY À WY
3. Yes, the corresponding triangle sides are congruent
4. Yes, the corresponding triangle sides are congruent
5. Yes, the corresponding triangle sides are congruent
}
} } }
}
} }
}
}
}
}
}
6. No; JK À MP, JL À MN 7. AB 5 DE 5 Ï 5 so AB > DE;
}
BC 5 EF 5 Ï 13 so BC > EF; CA 5 FD 5 2Ï5 so CA > FD; By the SSS Congruence Postulate, n ABC
> n DEF.
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
--" Ê{°{
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
1ÃiÊÈ`iÃÊ>˜`Ê>˜}iÃÊ̜ʫÀœÛiÊVœ˜}ÀÕi˜Vi°
6œV>LՏ>ÀÞ
)NARIGHTTRIANGLETHESIDESADJACENTTOTHERIGHTANGLEARECALLED
THELEGS
4HESIDEOPPOSITETHERIGHTANGLEISCALLEDTHEHYPOTENUSEOFTHE
RIGHTTRIANGLE
0OSTULATE3IDE!NGLE3IDE3!3#ONGRUENCE0OSTULATE)FTWO
SIDESANDTHEINCLUDEDANGLEOFONETRIANGLEARECONGRUENTTOTWO
SIDESANDTHEINCLUDEDANGLEOFASECONDTRIANGLETHENTHETWOTRIANGLES
ARECONGRUENT
4HEOREM(YPOTENUSE,EG#ONGRUENCE4HEOREM)FTHE
HYPOTENUSEANDALEGOFARIGHTTRIANGLEARECONGRUENTTOTHE
HYPOTENUSEANDALEGOFASECONDRIGHTTRIANGLETHENTHETWOTRIANGLES
ARECONGRUENT
8*Ê£
1ÃiÊ̅iÊ--Ê
œ˜}ÀÕi˜ViÊ*œÃÌՏ>Ìi
*ÀœÛiÊ̅>ÌÊN
ÊÊN°
-œṎœ˜
] ] ] ]
4HEMARKSONTHEDIAGRAMSHOWTHAT!"z$%z"#z%&zANDŽ"Ž%
3OBYTHE3!3#ONGRUENCE0OSTULATENg!"#Ng$%&
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
iVˆ`iÊ܅i̅iÀÊi˜œÕ}…ʈ˜vœÀ“>̈œ˜ÊˆÃÊ}ˆÛi˜Ê̜ʫÀœÛiÊ̅>ÌÊ̅iÊÌÀˆ>˜}iÃÊ
>ÀiÊVœ˜}ÀÕi˜ÌÊÕȘ}Ê̅iÊ--Ê
œ˜}ÀÕi˜ViÊ*œÃÌՏ>Ìi°
£° Ng014Ng213
Ó° Ng.+*N,+-
ΰ N789Ng:89
'EOMETRY
#HAPTER2ESOURCE"OOK
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
1ÃiÊ̅iÊÞ«œÌi˜ÕÃi‡i}Ê/…iœÀi“
7ÀˆÌiÊ>Ê«Àœœv°
] ] ]
--" Ê{°{
8*ÊÓ
] ] ]
')6%.!"z$#z"!z>!#z#$z>$"z
02/6%Ng!"#Ng$#"
-œṎœ˜
2EDRAWTHETRIANGLESSOTHEYARESIDEBYSIDE
WITHTHECORRESPONDINGPARTSINTHESAME
POSITION-ARKTHEGIVENINFORMATIONIN
THEDIAGRAM
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
3TATEMENTS
] ] ] ]
£° "!
z>!#z#$z>$"z
Ó° Ž!ANDŽ$ARERIGHTANGLES
ΰ Ng!"#ANDNg$#"ARERIGHTTRIANGLES
] ]
( {° #"
z"#z
] ]
, x° !"
z$#z
È° Ng!"#Ng$#"
2EASONS
£° 'IVEN
Ó° $ElNITIONOF>LINES
ΰ $ElNITIONOFARIGHTTRIANGLE
{° 2EmEXIVE0ROPERTYOF#ONGRUENCE
x° 'IVEN
È° (,#ONGRUENCE4HEOREM
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊÓ
7ÀˆÌiÊ>Ê«Àœœv°
]
] ] ]
{° ')6%.!"z$"z"#z>!$z
02/6%Ng!"#Ng$"#
x° ')6%. MŽ*+,zzMŽ-,+zz
] ]
*,z-+z
] ]
02/6%*+z-,z
'EOMETRY
#HAPTER2ESOURCE"OOK
Answer Key
Lesson 4.4
Study Guide
1. Yes; You are given that two sides and the included angle of one triangle are congruent to two sides and
the included angle of another triangle.
2. Yes; ∠ JKN and ∠ MKL are congruent because they are vertical angles. So you have two sides and the
included angle of one triangle that are congruent to two sides and the included angle of another triangle.
3. No; You have two sides in nWXY that are congruent to two sides in n ZXY, but the angle in n ZXY is not
the included angle.
4.
Statements
} }
H 1. AB > DB
} }
2. BC ⊥ AD
3. ∠ ACB and ∠ DCB
are right angles.
4. n ABC and n DCB
are right triangles.
} }
L 5. BC > BC
6. n ABC > n DBC
Reasons
1. Given
2. Given
3. Def. of ⊥ lines
4. Def. of a right
triangle
5. Reflexive Property of Congruence
6. HL Congruence Theorem
5.
Statements
1. ∠ JKL and ∠ MLK
are right angles.
2. n JKL and n MLK
are right triangles.
} }
3. JL > MK
} }
4. KL > LK
5. n JKL > n MLK
} }
6. JK > ML
Reasons
1. Given
2. Def. of a right
triangle
3. Given
4. Reflexive Property of Congruence
5. HL Congruence Theorem
6. Corresponding parts of > triangles are congruent.
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
1ÃiÊÌܜʓœÀiʓi̅œ`ÃÊ̜ʫÀœÛiÊVœ˜}ÀÕi˜Við
6œV>LՏ>ÀÞ
!mOWPROOFUSESARROWSTOSHOWTHEmOWOFALOGICALARGUMENT
0OSTULATE!NGLE3IDE!NGLE!3!#ONGRUENCE0OSTULATE
)FTWOANGLESANDTHEINCLUDEDSIDEOFONETRIANGLEARECONGRUENTTOTWO
ANGLESANDTHEINCLUDEDSIDEOFASECONDTRIANGLETHENTHETWOTRIANGLES
ARECONGRUENT
8*Ê£
`i˜ÌˆvÞÊVœ˜}ÀÕi˜ÌÊÌÀˆ>˜}iÃ
>˜Ê̅iÊÌÀˆ>˜}iÃÊLiÊ«ÀœÛi˜ÊVœ˜}ÀÕi˜ÌÊ܈̅Ê̅iʈ˜vœÀ“>̈œ˜Ê}ˆÛi˜Êˆ˜Ê
̅iÊ`ˆ>}À>“¶ÊvÊÜ]ÊÃÌ>ÌiÊ̅iÊ«œÃÌՏ>ÌiʜÀÊ̅iœÀi“ÊޜÕÊܜՏ`ÊÕÃi°
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
>°
L°
--" Ê{°x
4HEOREM!NGLE!NGLE3IDE!!3#ONGRUENCE4HEOREM)FTWO
ANGLESANDANONINCLUDEDSIDEOFONETRIANGLEARECONGRUENTTOTWO
ANGLESANDTHECORRESPONDINGNONINCLUDEDSIDEOFASECONDTRIANGLE
THENTHETWOTRIANGLESARECONGRUENT
V°
-œṎœ˜
Ê >° 4HEVERTICALANGLESARECONGRUENTSOTHREEPAIRSOFANGLESARECONGRUENT4HERE
ISNOTENOUGHINFORMATIONTOPROVETHETRIANGLESARECONGRUENTBECAUSENOSIDES
AREKNOWNTOBECONGRUENT
Ê L° 4HEVERTICALANGLESARECONGRUENTSOTWOPAIRSOFANGLESANDTHEIRINCLUDED
SIDESARECONGRUENT4HETRIANGLESARECONGRUENTBYTHE!3!#ONGRUENCE
0OSTULATE
Ê V° 4WOPAIRSOFANGLESANDANONINCLUDEDPAIROFSIDESARECONGRUENT4HE
TRIANGLESARECONGRUENTBYTHE!!3#ONGRUENCE4HEOREM
'EOMETRY
#HAPTER2ESOURCE"OOK
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
>˜Ê̅iÊÌÀˆ>˜}iÃÊLiÊ«ÀœÛi˜ÊVœ˜}ÀÕi˜ÌÊ܈̅Ê̅iʈ˜vœÀ“>̈œ˜Ê}ˆÛi˜Êˆ˜Ê̅iÊ
`ˆ>}À>“¶ÊvÊÜ]ÊÃÌ>ÌiÊ̅iÊ«œÃÌՏ>ÌiʜÀÊ̅iœÀi“ÊޜÕÊܜՏ`ÊÕÃi°
8*ÊÓ
Ó°
ΰ
7ÀˆÌiÊ>ÊyʜÜÊ«Àœœv
]
]
˜Ê̅iÊ`ˆ>}À>“]ʎÊʎÊ>˜`ÊÊ
Ê{
z{
Ê Ê°zÊ
7ÀˆÌiÊ>ÊyʜÜÊ«ÀœœvÊ̜ÊŜÜÊN
ÊÊN
°
-œṎœ˜
] ]
')6%. Ž'Ž"#"{
z{'!
z
02/6%Ng'#!Ng"!#
] ]
z{'!
z
#"{
Ž"#!Ž'!#
'IVEN
Ž'Ž"
!LTERNATE)NTERIOR
N'#!N"!#
'IVEN
!!3#ONGRUENCE4HEOREM
] ]
!#z!#z 2EmEXIVE0ROPERTY
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊÓ
7ÀˆÌiÊ>ÊyʜÜÊ«ÀœœvÊ̜ÊŜÜÊ̅>ÌÊ̅iÊÌÀˆ>˜}iÃÊ>ÀiÊVœ˜}ÀÕi˜Ì°
{° ')6%.Ž013Ž213
Ž130Ž132
02/6%N013N213
'EOMETRY
#HAPTER2ESOURCE"OOK
Ž,-/Ž*./
02/6%Ng-*.Ng.,
x° ')6%.Ž/-.Ž/.-
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
--" Ê{°x
£°
Answer Key
Lesson 4.5
Study Guide
1. The vertical angles are congruent, so two pairs of angles and their included sides are congruent. The
triangles are congruent by the ASA Congruence Postulate.
2. Two pairs of angles and a non-included pair of sides are congruent. The triangles are congruent by the
AAS Congruence Theorem.
3. Two pairs of sides and a pair of angles are congruent. This is not enough information to prove that the
triangles are congruent.
4.
5.
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
8*Ê£
1ÃiÊVœ˜}ÀÕi˜ÌÊÌÀˆ>˜}iÃÊ̜ʫÀœÛiÊVœÀÀi뜘`ˆ˜}Ê«>ÀÌÃÊ
Vœ˜}ÀÕi˜Ì°
`i˜ÌˆvÞÊVœ˜}ÀÕi˜ÌÊÌÀˆ>˜}iÃ
Ý«>ˆ˜Ê…œÜÊޜÕÊV>˜ÊÕÃiÊ̅iÊ}ˆÛi˜Êˆ˜vœÀ“>̈œ˜ÊÊ
>˜`ÊVœ˜}ÀÕi˜ÌÊÌÀˆ>˜}iÃ̜ʫÀœÛiÊ̅iÊÃÌ>Ìi“i˜Ì°
] ] ] ]
')6%. !"z{{$%
z!"z$%z
]
02/6% #ISTHEMIDPOINTOF"%z
-œṎœ˜
)FYOUCANSHOWTHATNg!"#Ng$%#YOUWILLKNOWTHAT#ISTHEMIDPOINT
]
OF"%z&IRSTCOPYTHEDIAGRAMANDMARKTHEGIVENINFORMATION4HENADDTHE
INFORMATIONTHATYOUCANDEDUCE)NTHISCASEŽ"Ž%ANDŽ!Ž$BYTHE
!LTERNATE)NTERIOR!NGLES4HEOREM
-ARKGIVENINFORMATION
!DDDEDUCEDINFORMATION
--" Ê{°È
4WOANGLEPAIRSANDTHEINCLUDEDSIDESARECONGRUENTSOBYTHE!3!#ONGRUENCE
0OSTULATENg!"#N$%#"ECAUSECORRESPONDINGPARTSOFCONGRUENTTRIANGLES
] ]
]
ARECONGRUENT"#z#%z"YTHEDElNITIONOFMIDPOINT#ISTHEMIDPOINTOF"%z
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊ£
/iÊ܅ˆV…ÊÌÀˆ>˜}iÃÊޜÕÊV>˜ÊŜÜÊ>ÀiÊVœ˜}ÀÕi˜Ìʈ˜ÊœÀ`iÀÊ̜ʫÀœÛiÊ̅iÊ
ÃÌ>Ìi“i˜Ì°Ê7…>ÌÊ«œÃÌՏ>ÌiʜÀÊ̅iœÀi“ÊܜՏ`ÊޜÕÊÕÃi¶
] ]
£° *+z,+z
Ó° Ž201Ž423
'EOMETRY
#HAPTER2ESOURCE"OOK
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
8*ÊÓ
*>˜Ê>Ê«Àœœvʈ˜ÛœÛˆ˜}Ê«>ˆÀÃʜvÊÌÀˆ>˜}iÃ
1ÃiÊ̅iÊ}ˆÛi˜Êˆ˜vœÀ“>̈œ˜Ê̜ÊÜÀˆÌiÊ>Ê
«>˜ÊvœÀÊ>Ê«Àœœv°
')6%. ŽŽŽŽ
02/6% Ng$%&Ng$#"
-œṎœ˜
)NNg$%&ANDNg$#"YOUKNOWŽŽ)FYOUCANSHOWTHATŽ%$&Ž#$"
] ]
AND%$z#$zYOUCANUSETHE3!3#ONGRUENCE0OSTULATE
"ECAUSEŽ%$&ANDŽ#$"AREVERTICALANGLESŽ%$&Ž#$"BYTHE
6ERTICAL!NGLES4HEOREM
] ]
4OPROVETHAT%$z#$zYOUCANlRSTPRIVETHATNg!%$Ng!#$9OUAREGIVEN
] ]
ŽŽANDŽŽ!$z!$zBYTHE2EmEXIVE0ROPERTY9OUCANUSETHE
!3!#ONGRUENCE0OSTULATETOPROVETHATNg!%$Ng!#$
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÊÓ
1ÃiÊ̅iÊ`ˆ>}À>“Ê>˜`Ê̅iÊ}ˆÛi˜Êˆ˜vœÀ“>̈œ˜Ê̜ÊÜÀˆÌiÊ>Ê«>˜Ê
vœÀÊ>Ê«Àœœv°
] ] ] ]
] ] ]
ΰ ')6%. !"
z%"z&"z#"z
] ]
02/6% "'z"$z
]
{° ')6%.23z34z05z01z
Ž504Ž102
02/6%Ng045Ng021
--" Ê{°È
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
0LANFOR0ROOF 5SETHE!3!#ONGRUENCE0OSTULATETOPROVETHATNg!%$Ng!#$
] ]
4HENSTATETHAT$%z$#zBECAUSECORRESPONDINGPARTSOFCONGRUENTTRIANGLESARE
CONGRUENT5SETHE!3!#ONGRUENCE0OSTULATETOPROVETHATNg$%&Ng$#"
'EOMETRY
#HAPTER2ESOURCE"OOK
Answer Key
Lesson 4.6
Study Guide
}
}
}
}
1. If you can show that n JKM > n LKM, then you will know that JK > LK. Since KM > KM by the
reflexive property, then n JKM > n LKM by the SAS Congruence Postulate. Because corresponding parts of
} }
congruent triangles are congruent, JK > LK.
2. If you can show that n PQR > n RST, then you will know that ∠ RPQ > ∠ TRS. Because RQ i TS,
∠ PRQ > ∠ RTS by the Corresponding Angles Postulate. By the AAS Congruence Theorem, n PQR >
n RST. Because corresponding parts of congruent triangles are congruent, ∠ RPQ > ∠ TRS. 3. Use the
SAS Congruence Postulate to prove that n ABF > n EBC. Then state that ∠ AFB > ∠ ECB because they are
corresponding parts of congruent triangles. ∠ CBD and ∠ FBG are congruent because they are vertical
} }
angles. Use the ASA Congruence Postulate to prove that BG > BD.
} }
4. Use the SAS Congruence Postulate to prove that n PTS > n PRS. Then state that PT > PR because they
are corresponding parts of congruent triangles. Use the SAS Congruence Postulate to prove that n PTU >
n PRQ.
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`i
&ORUSEWITHPAGESn
"
1ÃiÊ̅iœÀi“ÃÊ>LœÕÌʈÜÃViiÃÊ>˜`ÊiµÕˆ>ÌiÀ>ÊÌÀˆ>˜}ið
6œV>LՏ>ÀÞ
7HENANISOSCELESTRIANGLEHASEXACTLYTWOCONGRUENTSIDESTHESETWO
SIDESARETHELEGS4HEANGLEFORMEDBYTHELEGSISTHEVERTEXANGLE
4HETHIRDSIDEISTHEBASEOFTHEISOSCELESTRIANGLE4HETWOANGLES
ADJACENTTOTHEBASEARECALLEDBASEANGLES
4HEOREM"ASE!NGLES4HEOREM)FTWOSIDESOFATRIANGLEARE
CONGRUENTTHENTHEANGLESOPPOSITETHEMARECONGRUENT
4HEOREM#ONVERSEOF"ASE!NGLES4HEOREM)FTWOANGLESOFA
TRIANGLEARECONGRUENTTHENTHESIDESOPPOSITETHEMARECONGRUENT
#OROLLARYTOTHE"ASE!NGLES4HEOREM)FATRIANGLEISEQUILATERAL
THENITISEQUIANGULAR
#OROLLARYTOTHE#ONVERSEOF"ASE!NGLES4HEOREM)FATRIANGLEIS
EQUIANGULARTHENITISEQUILATERAL
`i˜ÌˆvÞÊVœ˜}ÀÕi˜ÌÊ>˜}iÃ
]
]
˜Ê̅iÊ`ˆ>}À>“]ÊÊ,/ÊzÊÊ-/Ê°zÊ >“iÊÌܜÊVœ˜}ÀÕi˜ÌÊ
>˜}ið
-œṎœ˜
] ]
24z34zSOBYTHE"ASE!NGLES4HEOREMŽ2Ž3
8*ÊÓ
ˆ˜`ʓi>ÃÕÀiÃʈ˜Ê>ÊÌÀˆ>˜}i
ˆ˜`ÊÊ>˜`Ê
ʈ˜Ê̅iÊÌÀˆ>˜}iÊ>ÌÊ̅iÊÀˆ}…Ì°
-œṎœ˜
--" Ê{°Ç
4HEDIAGRAMSHOWSTHATNg!"#ISEQUIANGULAR4HEREFORE
BYTHE#OROLLARYTOTHE#ONVERSEOF"ASE!NGLES4HEOREM
Ng!"#ISEQUILATERAL3O!"zz"#zz!#zz
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÃÊ£Ê>˜`ÊÓ
1ÃiÊ̅iʈ˜vœÀ“>̈œ˜Êˆ˜Ê̅iÊ`ˆ>}À>“Ê̜Êwʘ`Ê̅iÊ
}ˆÛi˜ÊÛ>Õið
Ê £° &IND79
Ê Ó° &INDMŽ789
'EOMETRY
#HAPTER2ESOURCE"OOK
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
8*Ê£
.AMEˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ ,%33/.
$ATEˆˆˆˆˆˆˆˆˆˆˆˆ
-ÌÕ`ÞÊՈ`iÊ CONTINUED
&ORUSEWITHPAGESn
8*ÊÎ
1ÃiʈÜÃViiÃÊ>˜`ÊiµÕˆ>ÌiÀ>ÊÌÀˆ>˜}iÃ
˜Ê̅iÊ`ˆ>}À>“]ʓŽÊʙä°Êˆ˜`Ê̅iÊÛ>ÕiÃÊ
œvÊÝÊ>˜`ÊÞ°
-œṎœ˜
-/*Ê£
-/*ÊÓ
8*Ê{
INDTHEVALUEOFX"ECAUSEN$%'IS
&
EQUILATERALITISALSOEQUIANGULARAND
MŽ'$%zzMŽ$%'zzX3OBYTHE
4RIANGLE3UM4HEOREMXzz
ANDXzz
] ]
INDTHEVALUEOFY"ECAUSEŽ'%&Ž'&%'%z'&zBYTHE#ONVERSEOF
&
"ASE!NGLES4HEOREMSO'%zz"ECAUSENg$%'ISEQUILATERAL
$%zz$'zz'%zz"ECAUSEMŽ$%&zzNg$%&ISARIGHTTRIANGLE
]
5SINGTHE0YTHAGOREAN4HEOREMYzzq zzzz
z
-œÛiÊ>ʓՏ̈‡ÃÌi«Ê«ÀœLi“
1ÃiÊ̅iÊ`ˆ>}À>“Ê̜Ê>˜ÃÜiÀÊ̅iʵÕiÃ̈œ˜Ã°
Ê >° 7HATCONGRUENCEPOSTULATECANYOUUSETO
PROVETHATNg!"#Ng!%$
Ê V° 3HOWTHATNg!"$Ng!%#
-œṎœ˜
]
]
]
]
Ê >° 9OUCANSEETHAT!"z!%zAND"#z%$z"YTHE"ASE!NGLES4HEOREMYOU
KNOWTHATŽ"Ž%3OBYTHE3!3#ONGRUENCE0OSTULATENg!"#Ng!%$
Ê L° "ECAUSECORRESPONDINGPARTSOFCONGRUENTTRIANGLESARECONGRUENTYOU
KNOWTHATŽ!#"Ž!$%ANDBYTHE#ONGRUENT3UPPLEMENTS4HEOREM
Ž!#$Ž!$#3OMŽ!$#zzMŽ!#$zzAND
MŽ#!$zzzzzzzzANDNg!#$ISEQUIANGULAR
Ê V° &ROMPARTBYOUKNOWTHATNg!#$ISEQUIANGULAR3OŽ!$"Ž!#%AND
THEREFORENg!"$Ng!%#BYTHE!!3#ONGRUENCE0OSTULATE
ÝiÀVˆÃiÃÊvœÀÊÝ>“«iÃÊÎÊ>˜`Ê{
Ê Î° &INDTHEVALUESOFXANDYINTHEDIAGRAM
ATTHERIGHT
Ê {° )N%XAMPLEABOVESHOWTHATN!"$N!%#
--" Ê{°Ç
#OPYRIGHT¥BY-C$OUGAL,ITTELLADIVISIONOF(OUGHTON-IFmIN#OMPANY
Ê L° %XPLAINWHYNg!#$ISEQUIANGULAR
USINGTHE333#ONGRUENCE0OSTULATE
'EOMETRY
#HAPTER2ESOURCE"OOK
Answer Key
Lesson 4.7
Study Guide
1. 6 2. 608 3. x 5 75; y 5 21
4. From part (b) you know that n ACD is equiangular. By the Corollary to the Converse of Base Angles
} }
} }
}
Theorem, n ACD is equilateral, and AD > AC. Because BC > ED, then BC 1 CD 5 ED 1 DC, and BD 5
}
EC. Therefore, n ABD > n AEC by the SSS Congruence Postulate.