# Homework Help: Geometric reasoning

1. Mar 24, 2016

### Natasha1

1. The problem statement, all variables and given/known data
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)

3. 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...

2. Mar 24, 2016

### haruspex

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

3. Mar 24, 2016

### Natasha1

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

CAE = 60 but what is EAF = ?

4. Mar 24, 2016

### haruspex

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

5. Mar 24, 2016

### Natasha1

sides are equal

6. Mar 24, 2016

### Natasha1

how does that help me?

7. Mar 24, 2016

### Natasha1

Is BC a 1/3 of BD?

8. Mar 24, 2016

### haruspex

Maybe more isosceles triangles?

9. Mar 24, 2016

### Natasha1

So we have ABC = BDF = DFA = x

10. Mar 24, 2016

### haruspex

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.

11. Mar 24, 2016

### Natasha1

So ABC = DFA = x

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

12. Mar 24, 2016

### haruspex

I repeat:

13. Mar 24, 2016

### Natasha1

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

14. Mar 24, 2016

### Natasha1

all 4 sides are equal in a rhombus

15. Mar 24, 2016

### haruspex

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

16. Mar 24, 2016

### Natasha1

i give up

17. Mar 24, 2016

### 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'.

18. Mar 24, 2016

### Natasha1

Opposite

19. Mar 24, 2016

### haruspex

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

20. Mar 24, 2016

### 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