## Sunday, November 20, 2011

### Problem 691: Isosceles Triangle, 80-80-20 Degrees, Angles, Congruence, Mind Map, Polya

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 691.

1. This is far easy one!
AC=CB and AE perpendicular to DC, then ACED is a deltoid so <CDE=30.
x=50-30=20 and we are done here (:

2. http://img191.imageshack.us/img191/4097/problem691.png
Since triangle ABC is isosceles so (BAC)=(BCA)=80
And (DAE)=40 and (DCE)=30
Triangle DAE congruent to Tri. CAE ( case SAS)
So DE=EC and ( CDE)=(ECD)=30
In triangle DAF, x=180-80-80=20
Peter Tran

3. Sí pero yo creo que quisiste decir que AC = AD o no?

5. <ADC is an exterior angle, then tr ADC is isosceles, so we have AD = AC, cause this triangles CAE and DAE are congruents by SAS.
Then ADF is 80 80 20 , therefore x=20

Greetings ^^

6. AE is easily seen to be the perpendicular bisector of DC hence Tr. DEC is isoceles and < CDE = 30 and so x= 20

Can anyone show that Tr. BEF is isoceles?

Sumith Peiris
Moratuwa
Sri Lanka

1. Using symmetry, <CEF and <DEB are 60 each. Since <BDE is 100 and <CEF is 100 by difference (180- 60-20). Therefore, TR DEB is congruent to Tr ECF (ASA). Since corr sides of cong Tr are equal, EF = EB making TR BEF isoceles.

7. ....and that BD = CF?

8. ...also that AE is perpendicular to BF?