SOLUTION: Each side of Traingle ABC is produced. AB extends till D, AC extends till F and BC extends till E. If AB=AC, BD=BE and AF=DF, find angle ABC.

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Question 1163536: Each side of Traingle ABC is produced. AB extends till D, AC extends till F and BC extends till E. If AB=AC, BD=BE and AF=DF, find angle ABC.
Found 3 solutions by Edwin McCravy, greenestamps, MathTherapy:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Just being given that those three triangles are isosceles is not enough
information to determine a unique answer.  You must be given something that
you left out.  Otherwise there are infinitely many answers.  What did you
leave out?  Tell me in the space below and I'll get back to you by email.

Edwin
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


A sketch (NOT to scale)....

AB=AC, so triangle ABC is isosceles;
BD=BE, so triangle BDE is isosceles;
AF=DF, so triangle AFD is isosceles.



Let x be the measure of angle ABC.

Then x is the measure of angle ACB (isosceles triangle ABC).

Then the measure of angle ECF is also x (vertical angle).

Angle BAC is 180-2x (angle sum of triangle ABC).

Angle ADF is 180-2x (isosceles triangle AFD).

Angle BED is 180-2x (isosceles triangle DBE).

Angle CFE is x (angle sum of triangle CEF).

Angle AFD is 180-x (supplement to angle CFE).

Angle DBC is 180-x (supplement to angle ABC).

Now here is the sketch (still not to scale):



Now look at quadrilateral DBCF. The angle sum is

FDB + DBC + BCF + CFD = (180-2x)+(180-x)+(180-x)+(180-x) = 720-5x

But the angle sum of any quadrilateral is 360 degrees:





ANSWER: The measure of angle ABC is 72 degrees.

Note that, with a measure of 72 degrees for angle ABC, the problem description defines part of the figure of a regular 5-pointed star.

Viewing the given not-to-scale sketch as part of a picture of a regular 5-pointed star makes it easy to see that all of the conditions of the problem as stated are satisfied.

-------------------------------------------

Comment to tutor @MathTherapy....

Nice, elegant solution, achieved by defining x to be the measure of angle BDE....
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Each side of Traingle ABC is produced. AB extends till D, AC extends till F and BC extends till E. If AB=AC, BD=BE and AF=DF, find angle ABC.
Borrowing Tutor @GREENESTAMPS' Diagram
.
AB = AC, so ∡ABC = ∡ACB
BD = BE, so ∡BDE = ∡BED
AF = DF, so ∡ADF = ∡FAD

Let ∡BDE be x
∡s BDE and ADF are the same
Then ∡BDE, ∡BED, and ∡CAB are each, x, as well

Since an exterior angle of a triangle is equal to the sum of its interior opposite angles, we can say that: ∡ABC = x + x, or 2x
In addition, ∡ACB also = 2x (AB = AC)
From ΔABC, its 3 angles are: x, 2x, and 2x. so we get: x + 2x + 2x = 180
5x = 180


Therefore, ∡ABC = 2x = 2(36) =

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