Geometric Reasoning: Find Angle ABC in Rhombus ABDF

[Natasha1]

Homework Statement

As shown in the diagram (attached), ABDF is a rhombus, ACE is an equilateral triangle, and AB  AC . Find ABC through geometric reasoning (a scale diagram will gain no credit).

2. See drawing (picture attached)

The Attempt at a Solution

If I make angle ABC = x then BCA is also = x (as triangle ABC is isosceles) so I got x + x + 180 - 2x = 180

angles ACE = CEA = EAC = 60 degrees (Equilateral triangle)

Must I use alternate angles to help me solve this problem...

 

[haruspex]

You need to find some way to use that ABDF is a rhombus, not just a parallelogram.

 

[Natasha1]

if BCA = x then CAF = x (alternate angles)

CAE = 60 but what is EAF = ?

 

[haruspex]

if BCA = x then CAF = x (alternate angles)

CAE = 60 but what is EAF = ?

What has that to do with my hint? What is the difference between a rhombus and a general parallelogram?

 

[Natasha1]

sides are equal

 

[Natasha1]

how does that help me?

 

[Natasha1]

Is BC a 1/3 of BD?

 

[haruspex]

how does that help me?

Maybe more isosceles triangles?

 

[Natasha1]

So we have ABC = BDF = DFA = x

 

[haruspex]

So we have ABC = BDF = DFA = x

No, BDF is not equal to the other two.
You need to use that it is a rhombus, not just a parallelogram. Find two sides that are equal in the rhombus but might not be in a parallelogram.

 

[Natasha1]

So ABC = DFA = x

in each triangle we get x + x + 180 - 2x = 180 (but then this is obvious)

 

[haruspex]

So ABC = DFA = x

in each triangle we get x + x + 180 - 2x = 180 (but then this is obvious)

I repeat:

Find two sides that are equal in the rhombus but might not be in a parallelogram.

 

[Natasha1]

In quadrilateral CEFA we have 60+x+x+180-2x+60+60=360 but then 360 = 360 leads nowhere

 

[Natasha1]

all 4 sides are equal in a rhombus

 

[haruspex]

all 4 sides are equal in a rhombus

Yes, but some are also equal in a parallelogram. I repeat

Find two sides that are equal in the rhombus but might not be in a parallelogram.

 

[Natasha1]

i give up

 

[haruspex]

i give up

As you said, all four sides are equal in a rhombus. That is not true of a parallelogram. In a parallelogram, which sides must be equal? Which sides need not be equal? Answer in terms of 'adjacent' and 'opposite'.

 

[Natasha1]

Opposite

 

[haruspex]

Opposite

Which question is that the answer to? I asked which must be equal and which need not be equal. The answers are different, obviously.

 

[Natasha1]

As you said, all four sides are equal in a rhombus. That is not true of a parallelogram. In a parallelogram, which sides must be equal? Which sides need not be equal? Answer in terms of 'adjacent' and 'opposite'.

In the rhombus ABDF, AB = BD = DF = FA
In a parallelogram opposite = equal, adjacent = not equal

 

[Natasha1]

is x = 30 degrees? (a guess)

 

[Natasha1]

How can knowing that triangles ABC and AFE are isosceles and similar help me in finding angle ABC?

 

[haruspex]

As you said, all four sides are equal in a rhombus. That is not true of a parallelogram. In a parallelogram, which sides must be equal? Which sides need not be equal? Answer in terms of 'adjacent' and 'opposite'.

In the rhombus ABDF, AB = BD = DF = FA
In a parallelogram opposite = equal, adjacent = not equal

Right, so pick some pair of adjacent sides.. AB and BD say. You know these are equal. What isosceles triangle does that give you?

 

[Natasha1]

similar isosceles triangles

 

[haruspex]

similar isosceles triangles

I mean with reference to the points in the diagram. Choose two adjacent sides of the rhombus and say which you have chosen. You know they are equal length, so they form two sides of an isosceles triangle. Which triangle? What angles are therefore equal?