+0

+2
1
253
4
+281

In acute-angled triangle ABC, let D be the foot of the altitude from A, and E be the midpoint of BC. Let F be the midpoint of AC. Suppose 6 BAE = 40◦ . If 6 DAE = 6 DF E, what is the magnitude of 6 ADF in degrees?

kes1968  Jun 6, 2017
Sort:

#1
+520
+1

Are these spurious 6s supposed to be angle symbols, viz., should it read:

If ∠DAE = ∠DFE, what is ....

‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

edited by Badinage  Jun 6, 2017
edited by Badinage  Jun 6, 2017
#2
+6900
+2

Because D and F are the mid-points of BC and AC respectively, DF || AB. Therefore, angle AEF should equal 40 deg.

Let G be the intersection point of AD and EF.

Angle DAE + 40 deg = angle AGF

Angle DFE + angle ADF = angle AGF

Because angle DAE = angle DFE,

angle ADF should equal 40 deg :D

MaxWong  Jun 6, 2017
#3
+79894
+2

Max.....E is the mid-point of AB, not D....so.....  CE/EB = CF/FA ....

Which would mean that EF ll AB.....instead of  DF ll AB .....

Then angle FED  = angle ABE   and angle GDE = angle ADB

So by  AA congruency

Triangle  DGE  is similar to triangle DAB

Thus....angle DGE  =  angle DAB

And angle DAB =   angle DAE + angle EAB

So.....this means that

Angle DGE  =  angle DAE + angle EAB

But, by the exterior angle theorem,  angle DGE  also  = angle DFE + angle GDF

And angle DAE  = angle DFE

So.....this means that

angle DAE  + angle EAB   =  angle DFE + angle GDF

Subtracting like angles on either side of the equation, we have that

angle EAB  =  angle GDF

But angle EAB  = 40°....so  angle GDF  has the same measure

But  angle GDF  =  angle ADF

And Max's assertion that angle ADF = 40°  is correct, after all  !!!

**Thanks, Max, for some of the hints you gave.....you actually did 90% of the proof....I just "cheated" off you a little.....LOL!!!!!!

Here is a "quasi" pic :

CPhill  Jun 6, 2017
edited by CPhill  Jun 6, 2017
edited by CPhill  Jun 6, 2017
#4
+5576
+1

Here is CPhill's picture:

hectictar  Jun 6, 2017
edited by hectictar  Jun 6, 2017

### 13 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details