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A:
Summary of Main ldeas
TRIANGLES: What are the types of triangles and their properties? How do we prove
congruent triangles? How do we apply the properties of congruent triangles to determine
information about parts of a triangle? How do we use the special segments of triangles to
determine information about parts of a triangle?
Terms you should KNOW!
Unit 1 & 2 Terms
triangle
vertex
plus:
adjacent sides
opposite side
interior angles
corollary
scalene triangle
acute triangle
obtuse triangle
isosceles triangle
base
base angles
exterior angles
vertex angle
perpendicular bisector
angle bisector
legs
hypotenuse
right triangle
equilateral triangle
equiangular triangle
midsegment
corresponding angles
corresponding sides
equidistant
Be sure you can classify triangles by sides AND ang les.
Ct
fbtr,i 3r
s r a'cil(t\Q-
tulc
*Carlrrre
{i,,,ilafetaL
fiduI0
r^iq
$ciie're-o
*'*
t'$"''wj^ta{
f*
bJu
:*-
,t*'u,
Other lmportant Triangles Information
Right Triangles and lsosceles Triangle?_
B*
HYPotenuse: f'5C
Vertex:
r-.g'
Base
A15
r-eg: y'\L
6
Angle: A
Base Angle:
c
AZ
riLeg: f+ ti
;G
Leg:
eur",
L
f)L:
c
Gorollary to Triangle Sum Theorem
m/.A+ mlB + mlC
--
mZ.A +
-$q'
mlB :
'1a"
Third Angle Theorem
Exterior Ang le Theorern
_wLA +rnLh
B
A\
tfl/A =rfir-D
"X \o ,Xi--.-\o
Congruent Triangles and Figures
Congruent Triangles
SIDES
ANGLES
AB=SE
LA =ZD
ZB=L€
BC =
h-=
€:F
A
D
't/'l --,
*\\
t::-.'\
F,, .'''
Ll=tF
llL = DF
*\--.
=#.- --l,''''''''''''''''''''''''''i
Naming Gongruent Figures
Properties of Triangle Congruence
Given that ABCDEF = HJKLMI'I
Reflexive ProDertv:
LABC =
List all pairs ofcongruent sides:
m, ?
r--u_
_ E:,
)ii, Ifr
A Al;C'
Symmetric Property:
i unc =
't-f [u
f='t"". R
ki..j-r<
^AF"=H\J.
Llst all pairs ol'congruent
angles:
LAitn t$*= t-L
L fi-=L-T / E"' L
L(-'z t u Ltr'L t-xt
^DEF,th"'
A
.Dtf
:
AAO0
Transitive Property:
If MBC = A,DEF
14
ANd LDEF
= AJKL,thE"
A AdC g
AsK L
Proving Triangles Gongruent
We need at least 3 pairs of congruent corresponding parts to prove triangles congruent. The six possible
combinations are below with the four that don't work x-ed out.
Side-Side-Side
Side-Angle-Side
Three pairs of congruent sides
Two pairs of congruent sides and
one pair of congruent angles
(angle between sides)
hA
Angle-Side-Angle
Angle-Angle-Side
Two pairs of congruent angles and
one pair of congruent sides
(side between angles)
Two pairs of congruent angles and
one pair of congruent sides
(side not between angles)
f-\
,,f-\
n--- ,s:---
Base Angles Theorem
Gonverse to Base Angles Theorem
B
B
lfAB=BC
IflA=lC
'rJ
/n-l/'
then ./- 11 - t-t-
then I4b
,A r
rv
n
-
= f)U
Gorollary to Converse to Base Angles Theorem
Corollary to Base Angles Theorem
If a triangle
If a triansle is eouians.ular.
then it is
then
Hypotenuse-Leg (HL) Theorem
Perpendicular Bisector (J- bis.)
Theorem
t\
t\
t\t\
it i'"
il
e4u'il
ii
U aL
Converse to the Perpendicular
Bisector (l- bis.) Theorem
?
\\
P
L-\t-\
K-L
B-c
If O
Remember, to use HL you
MUST have
f'rcih{ / 5
C
Angle Bisector
(l
!
bis.) Theorem
is
I
bisector
of AB, then
AC=BC
Converse to the Angle Bisector
(Z bis.)Theorem
If
AD = BD, then D is on the
bisector of AB
I
Midsegment Theorem
R
/\
D+€
AC
Tf AD iS Z biS, Of IBAC
then D-B : DC
,
If
DB: DC,then
bis.
ADrs
I
Triangle Inequality Theorem
Ordering Sides
Ordering Angles
DEllACand DE=+AC
of IBAC
U
M
i.{
z-l
9
Lh
F
T
l+l . !3 .
*Id
Hs
>FU
UN
ul- *F N ', utu
W * {) Ir/' o,
Proofs
l.
E
,qC; nA nC
=
=
Prove: a,4RE =^ARC
Given:
Statement
Reasons
, Tv=- ghu LR ZKC
2 [A A-r.
,
:
AAKE
A A(C-
"
t /otvU',;. "T?"l lerrv<,
3. 5::
2. Given; B is the midpoint of AE and
Prove: yABC
=aEBD
Statement
rbr:6r/pt {Ya G
z
3
rAsc
:4
OsE
4 AAhL !AEbD
t.6iw^'
i. [,{ 6n'dpo'nt
3 Venhtco 'L5
4. 5A5
3. Given: Pf m; ITPR= IPRA
=
Prove: {RP =yAPR
FT lAA
,
mYm
Arnp
LTPRY- TPRA
r A APR
6tv^"-* q*
e_+le { l\
545
4. Given:
fd
=
S,q;
,qf
is an angle bisector
4LYL3
Statement
1fn=S,A? L bi:rclar
2 R"=ffi
s
Zl5AT
o /t1r
ArAt
5,i Le* L 5"ASAT
t (t}'e'n:
i, {e lle*w
4 ef L br:ccfor
{,
5A5
CPLTC
5. Given: ,qnlt nC; An nC
=
Prove: aABD =aCDB
D L-'
Statement
Reasons
1.
z.
J.
4.
Gtw*
t
At+ l'4+ Ls
f& {letii"<--
5.A3
C:
Problems
t.
AB= BD
.
.qn = Uf
:-:
i{\\hLrlr4'" (\LI
if
fn(6ltcrn'.> Dr )
a1
L hr *etnr Atr
J- hrSctrrr:; &irm
:
Name all segments that
are: a. altitudes, b. medians, c. angle bisectors d. perpendicular
bisectors
\
2.
J.
Two sides of
triangle measure 5 ft and 13 ft. Find all possible measures of the third side.
a
1t
^L
tt
ALNG is isosceles triangle with vertex L
w1.ANS =35o
m,/1 - J -t
,73 = AO
4
SP is angle bisector of IQSU
Pf
is perpendicular bisectorof trS
PR ir purpundicular to QS
IQ,
ir
XS= ll
mZpSR =
-an
rtlrr-
UT= l"l
PS= lu
4..l
J
I
t'
y',
PX= U
w
14
o
,zz =45
,2a= 50
Multiple Choice Practice
D:
Ee
Multiple Choice Grven LX
LZ = LO,
find, the value of
=.r l/
and
$.
r.
Muttiple Choice Whrch
posrulate or
rem can be used r
rheo_
"qiffiF#;;iJffnT"ni."*:ru
@
SSS
@ SAS
(pe)
(O38
(A)
@zo
@) ssn
G) AAA
c)95
L
6, lfuttipte Choice W_hich starement
correctly
congruence of tl_," ,.r"ngl",
,n
$:,r:rtb"rlhe
. ,b,XercrSe
5?
@
d.
Multiple Choice Use the diagram rn
Exercrsej to find mLZ
@ lg'
@38.
@ 70.
€"e
@ 95"
LKML= ApeM
@* AKIM = [.peM ,
(e:@AD
CD
AKLA4: LpMe
AKUL = Lpeta
@
7^ Muttipte
3.
MuLtiple Choice If I,ABC
= LXyZ,whrch
of the followrng statements below
rs not true,/
@ LB=Ly
(-\
f -e*-.'""-'..'.-"..-'\
r-D^
W@
-
LRAC
@ TE=Xy
,.
ae=YZ
= Lyxz
;:ltfr'j:: i-!*"*'
proving the tnangles
;;,
=pet
;
;;:;
.'Ji:
Verhcal Angles ,Itreorem
9
P-*!=".
@
C)
rf Multiple Choice LTse
the dragram below
Which congruence rs correct to prove
AXYZ = A]KL?
Choice
congruenr,"
5
",.-^*^'P
You grve ro prove
KL
iprnp oI Cong^.n..
ASA
Definitron of mrdpoinr
8, Muttiple Choice
Whrch posrulate or rheorem
can be used ro prove rhat
LnfC ii?cz
=
X
D\
A\
z--t\
--a
@ Ly = LK by SAS Congruence postulare
-p*g:g*u!{s*gqlerq-"s.rio',:lgg((e-)
ZY = LK br SA: Congruence t;,r$
C-. B orC
@
(g@
SsS
@ SAS
tFl
@
none of the above
ASA
Mut4ple Choice Whar is the thrrd congru_
ence needed to prove rhat AABD
= ACBD by
AAS?
Q.
83. &ruhpte Chobe
@ x:
<d;;=:qBD
l;\ r=::--
lO.
and
)
120,
@ ":60,
y:60
O x: 30,
LtL
@ LDBA = LCDB
@ BorC
r
y:6O
(A) ,qB=fC
/1L/:
Solve for
y=120
A
Multiple Choice What rs the thrrd congmence needed to prove that LMNe
= Ap1/O
by ASA?
@ _e=tp
aA)
A/^r/^\ -i
11 *'tt.vv
-
14. futultiple Choice
Solve for x and v
@ x: J0,), : 55
tE, -l = 55 rr : ttr'r
M
/
-P/VO
(D /M= Lo
/.K'--;- -:;\
watr)
.g
lE"
r.]v\ _,ya1
==--j-.-:z*.._-**j-y
vn
@ x=70,y:145
@ x:55, y:145
O
t\v
Multiple Choice Grven rhat LX LD, and
=
DE = Xl{r, what rs the third congruence
need_
ed to prove rhat AXI4IF
ADEb
bv
ASA?
=
Gl Ly= LC
O Lw= 29
L]
b.
Multiple choice
@ LY: LE
(A) x: 72.5,y
@ x: j5.,:
<D -w=LE )
"-.
none of the above
\-\z_/
-%---------
(D)
/-E-)
\-*Y
{
lQ"
Mulnple Choice Grven rhat LG
= LE and
LI : LD, what rs the third congruence
need_
ed ro prove that AG111
= AEF bv AAS?
@
/H:
r-n\
\12-l
Hl:
@
tD
FD
)
@
Hr =ED
ED =ej
What are the values of x
and y?
J-)
j
,:35,|:
tln
55
x: 110,):
J5
,6, Multiple Choice In the dragram, ffi rs the
perpendicular brseclor ol ,qB Strat is the
value o1 r?
none of the above
z
16
(ffi>
10
1,. Multiple Choiie In rhe dragram. e
KM
and7P =* PE Frnd LM.
--*\
,.T; 8 ',
{\-----7
@10
G)
72
@13
(E-)
1)
20, Muttiple choice Lrst
the
longest to shortest
in order from
tAl 7r Ei r.-.
@
) Lr
, cLI
EG,GF,EF
Fr--..\,!4, Lf
, LG, GF
ffi,
w
\E I:U.tk,GF
/f
M-
-
t8. Multiple Choice Use the diagram below to
frnd FG.
srdes
-
4u"y"Wr: Choice In the diagram, W
are rnrdsegmenh
of LXYZ Find t{T.
andW-D
@10
@15
@,4
12x+8
--_\
@3
.f
@44
-_\
M
G)
6
6l
j
L2- Multiple
Choice Vlhrch posfulare or theorem
can be used to prove that
XenC = ABADT
@ sss
14. Mutnpte Choice Use rhe
dragram below
Which addrttonal congrllence _rs correct to
prove A,4BC = LDEF?
FE
@
@
LB = LE by SAS Congruence postulate
FE by SSS Congruence postulate
-BC =
O tA = -D by SAS Congruence postujate
@ AorB
@ SAS
--6, --:;:
--Y--:j.W
(D AAS
@ none of the above
A
73. nfuttiple Choice The tnangle below
ciassified
@
u.u,.
as l
rro*il
@ acute scalene
/-6-'T_ -:-*;-_
ru
\Llr obtuse SCa.lene
@ nght scalene
can be
Q4" rw"uiple choice A tnangle has rwo srdes
rhal
have lengths of g cm ard 14 cm.
Which length
below could nol.represent the length of tbe third
srde
I
@ 7cm
--*\
22cm
@) LLeM
I
45. Muttipte
@ 13cm O
)
1
choice
(E, I)
18cm
Cm
What is the value of x?
@3
2x+8
&&. wutaple Choice
Whieh. tyee of segment need nob
cotrta.in a vertex of the
triangle?
altibude
b"
angte bi,sector
c.
median