See complete Problem 4
Triangle, Quadrilateral, Angles, Congruence. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Sunday, May 18, 2008
Elearn Geometry Problem 4
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Trazamos el segmento BD
ReplyDeleteEl triangulo BDC es isosceles => angulo(DBC)=angulo(BDC)=(180-2A)/2= 90-A
Tomamos AD=DC=BC=1
En el triangulo BDC ; 1/sen(90-A)=BD/sen(2A) => BD=2 sen(A)
En el tringulo ABD ; 1/sen(ABD)=2sen(A) / sen (A) => sen(ABD)= 1/2 => angulo (ABD) =30
Angulo(x)=angulo(DBC)+angulo(ABD)=90-A + 30= 120 - A
It is possible to provide a proof using only elementary geometry!
ReplyDeleteLet point E be a reflection of point D over AB.
ReplyDeleteThen AEB and ADB triangles are congruent
Angle EAB = angle DAB = alpha => angle EAD = 2 alpha => triangles AED and CBD are congruent (AE = AD = DC = BC) => ED = DB
But BE = DB (because AEB and ADB are congruent) => EBD is an equilateral triangle.
Angle EBD = 60 => angle ABD = 30 (since ABD = ABE)
Angle DBC = 90 - alpha (since DCB is an isosceles triangle)
x = angle ABD + angle DBC = 30 + (90 - alpha) = 120 - alpha.
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ReplyDeletehttp://mate-facil.co.cc/35/
ReplyDeleteBy TiNo
Let the bisector of <DCB intersect AB at X. Now since <XCA = <XAD = a, AD = CD and XD is common, by SAS, AXD is congruent to XCD. Hence <DXC = <DXA. Since X is on the perpendicular bisector of BD it also follows that <DXC = <BXC. Therefore since <DXC = <DXA = <BXC and all three angles are supplementary, <BXC = 60. Hence <XBD = 30. Now, <XBC = <XBD + <DBC = 30 + 90 - a = 120 - a.
ReplyDeletehttp://ahmetelmas.files.wordpress.com/2010/05/cozumlu-ornekler.pdf
ReplyDeleteVideo solution (Spanish version) by Eder Contreras and Cristian Baeza at Geometry problem 4. Thanks Eder.
ReplyDeletesolution is uploaded to the following link
ReplyDeletehttps://docs.google.com/open?id=0B6XXCq92fLJJajJOTFRfNTlSTDZGdUlVZEJfSml4UQ
Proof:
ReplyDeleteJoin AC
Through D draw a line perpendicular to AC and meet AC at M and meet AB at E.
∵ AD = DC
∴ EM is the perpendicular bisector of AC
∴ AE = EC and ∠AEM = ∠CEM
∴ ∠EAD = ∠ECD = α
∵ DC = CB
∠DCE = ∠BCE = α
Δ DCE ≡ Δ BCE
∠CEM = ∠CEB
∵ ∠AEM = ∠CEM = ∠CEB = 60o
∵ In ΔBCE, x + α = 120o
Join BD.
ReplyDeleteLet E be the circumcentre of ∆ABD.
∠DEB = 2∠DAB = 2α.
∆s DEB, DCB are congruent (Each isosceles, common base BD,
equal vertical angles, each 2α)
Follows EA = ED = DC = DA, ∆EAD is equilateral, and ∠AED = 60°
So ∠ABD = (1/2)∠AED = 30°
Observe that ∠DBC = 90° - α
Hence x = 30° + (90° - α) = 120° - α
hallo my dear brothers and sisters ...... here is a simple solution for this problem
ReplyDeletelets start
join A,C. join B,D.
draw an angle bisector of angle DCB.it intersects DB at the point F and
also intersects AB , at the point E.
now consider triangle DBC
CD=CB (given)
angle DCF=angle FCB=alpha
angle FDC = angle FBC = (180- 2 alpha)/2=90-alpha
so angle DFC = angle BFC = 90
so angle EFD = angle EFB = 90
now consider two triangle DFC and BFC
cf is common
angle DCF=angle FCB=alpha
angle DCF=angle FCB=alpha
triangle CDF is congruent with triangle CBF
so FD = FB
now consider two triangle DEF and EFB
EF is common
FD = FB
angle EFD = angle EFB = 90
two triangle DEF and EFB are congruent with each other
so angle BEF = angle FED............... (1)
consider triangle ADC
AD = DC
this is an isosceles triangle
so angle DAC = angle DCA
consider triangle AEC
angle EAC = angle EAD + angle DAC
angle ECA = angle ECD + angle DCA
angle EAD = angle ECD = alpha
so angle EAC = angle ECA
so triangle AEC is isosceles triangle
so AE = EC
now consider two triangle AED and CED
AE=CE
ED is common
AD=DC
triangle AED and CED are congruent
so angle AED = angle FED ..........(2)
from (1) and (2)
angle BEF = angle FED = angle AED = 180/3 =60
so angle EBF = 90 - 60 = 30
ANGLE ABC= angle EBF + angle FBC
= 30 +90 - alpha = 120 - alpha
Draw an angle bisector of angle BCD and let it meet at the point E on the side AB
ReplyDeleteThen connect E and D.
Since In Tri ACD , angle ACD = angle CAD
angle ECD = angle EAD = alpha
CE = AE ( Isosceles tri AEC )
Since In Tri BCE and Tri ADE
BC = AD ( given )
angle BCE = angle EAD = alpha
CE = AE (proved)
Tri BCE congruent to Tri ADE. This implies x = angle CBE = angle ADE
Since in Tri CDE and in Tri ADE
CE = AE ( proved )
CD = AD ( given )
angle ECD = angle EAD
Tri CED congruent Tri ADE , angle CDE = angle ADE = x
Since Tri BCE , Tri CDE , Tri ADE are congruent to each other
angle BEC = angle CED = angle AED
and BEC + CED + AED =180 degree
3(AED) = 180 degree
AED = 60 degree
IN Tri CEA , angle CEA = 2(AED) angle = 120 degree. Since it Is isosceles
angle ECA = angle EAC = 60 degree
IN Tri CDA , angle DCA = angle DAC = 60 - alpha , and angle CDA = 120 + 2(alpha)
AND FINALLY
ANGLE CDA = 360 - (ANGLE EDC + ANGLE EDA)
ANGLE CDA = 360 - (ANGLE EDC + ANGLE EDC) ( EDC = EDA)
ANGLE CDA = 360 - ANGLE 2(EDC)
ANGLE CDA = 360 - 2X (EDC = X)
120 + 2(ALPHA) = 360 - 2X (ANGLE CDA = 120 + 2(ALPHA))
2X - 360 = - 2(ALPHA) - 120
2X = 360 - 120 - 2(ALPHA)
= 240 - 2(ALPHA)
= 2( 120 - ALPHA)
THEREFORE X = 120 - ALPHA
Let the bisector of < DCB meet AB at E and BD at F. Draw altitude DG of Tr. ABD.
ReplyDeleteTr. s BCF & DCF are congruent and these are also congruent with Tr. ADG. So DG = DF = FB hence in right Tr. BGD BD = 2DG which means that < ABD = 30 and since < DBC = 90-@ the result follows
Sumith Peiris
Moratuwa
Sri Lanka
https://www.youtube.com/watch?v=2XwhJStz4bA
ReplyDeleteConstruct point E such that E lies on AB and EC is an angle bisector of <DCB. Hence, <DCE=<BCE.
ReplyDeleteConstruct line ED.
Triangle DCE is congruent to BCE. Hence <EDC=x
Since <EAD=<ECD and <DAC=<DCA,<EAC=<ECA
Hence, EA=EC
Hence triangle EAD and ECD are congruent.
Hence <EDA=x
By angle sum of quadrilateral 3x+3a=360°
So x+a=120°
x=120°-a
Proved
Grab and pen and a piece of paper, and illustrate accordingly.
ReplyDeleteAdd point E to the figure and connect it to line C. Do so in such a way that line CE is the angle bisector of angle BCD. Next, connect point D to point E. That is all for the illustrations.
We can see that:
> Triangles CDE and CBE are congruent because
~ They are share line CE
~ Line CD = line BC
~ Angle DCE = angle BCE
> Triangles CDE and ADE are congruent because
~ Due to two equal opposite angles and equal adjacent lengths, quadrilateral ADCE is a concave kite.
~ They share side DE
~ This is a special case where SSA works, but only because they make up a kite.
> Since triangles ADE, CDE and CBE are congruent, angle AED = angle CED = angle CEB.
> Angle AED = angle CED = angle CEB = 180 / 3 = 60
Now, we solve the problem:
Just focus on triangle CBE. The three angles in a triangle add up to 180.
x + a + 60 = 180
x + a = 120
x = 120 - a
Let angle bisector of m(BCD) meet AB at E.
ReplyDeleteConnect ED and since given AD=DC=BC=>ED is perpendicular to AC, AEC is an isosceles and BCDE is a kite => AE=EC and EB=ED
Observe that the triangles DAE, DCE and BCE are congruent (SAS)
Hence m(AED)=m(DEC)=m(CEB)=180/3=60 =>m(DBE)=30
Now its easy to see x=30+90-alpha=120-alpha
For easy typing, I use z instead of "alpha" for the following
ReplyDeleteangle CDB=90-z
Consider triangle CDB
sin2z/BD=sin(90-z)/CD
CD=BDcosz/sin2z
CD=BD/2sinz
Consider triangle BAD
angle ABD=x-angle CBD=x-90+z
sinz/BD=sin(x-90+z)/AD
AD=BDsin(x+z-90)/sinz
AD=-BDcos(x+z)/sinz
CD=AD
BD/2sinz=-BDcos(x+z)/sinz
1/2=-cos(x+z)
x+z=120
x=120-z