In convex quadrilateral ABCD, angle A is congruent to angle C, AB=CD=180, and AD is not equal to BC. The perimeter of ABCD is 640. Find cos A.

Hmm...I drew the diagram, and labeled some things, I drew an altitude from D to AB and did some other things, but no success was obtained. Any help? Thanks

joint BD, and use the cosine law twice, once for each triangle

we know AB = DC = 180
let angle A be Ø
let AD = a and BC = b

BD^2 = a^2 + 180^2 - 2(180)a cosØ
BD^2 = b^2 + 180^2 - 2(180)bcosØ

a^2 + 180^2 - 360a cosØ = b^2 + 180^2 - 360b cosØ
a^2 - b^2 = 360a cosØ - 360b cosØ
(a+b)(a-b) = 360cosØ (a-b)
a+b = 360cosØ
cosØ = (a+b)/360

but perimeter = 640
a+b+180+180=640
a+b = 280

then cosØ = 280/360= 7/9

ohhhhh that makes sense i should have thought of that thanks!

To find the value of cos A, we need to use the given information and properties of a convex quadrilateral.

First, let's analyze the given information:
1. Angle A is congruent to angle C. This tells us that angle A + angle C = 180 degrees (as their sum must be supplementary).
2. AB = CD = 180. This means that sides AB and CD have the same length.

Now, let's proceed with solving the problem:
1. Since AB=CD, the perimeter of the quadrilateral ABCD can be expressed as follows: AB + BC + CD + DA = 640.
Using the given information, we can write this as: 180 + BC + 180 + DA = 640.
Simplifying, we get: BC + DA = 280.

2. Let's consider the given property that AD is not equal to BC. This means that the lengths of sides AD and BC are different.

3. Now, draw the perpendicular from D to AB, and let's call the point of intersection E. As a result, we have two right triangles, ADE and CDE. Let's focus on triangle ADE.

4. In triangle ADE, we have the following sides:
- AD (unknown length)
- AE (180, as AB=180)
- DE (unknown length)

5. Using the Pythagorean theorem, we can express DE in terms of AD. It states that the square of the hypotenuse (DE) is equal to the sum of the squares of the other two sides (AD and AE):
DE^2 = AD^2 + AE^2
DE^2 = AD^2 + 180^2
DE^2 = AD^2 + 32400

6. Now, let's consider triangle CDE. Since angle A is congruent to angle C, this means that triangle CDE is similar to triangle ADE.
This similarity gives us the following ratio of corresponding sides:
DE / CD = AD / AE
DE / 180 = AD / 180
DE = AD

7. Substituting DE = AD in the equation from step 5, we can solve for AD:
DE^2 = AD^2 + 32400
AD^2 = AD^2 + 32400
0 = 32400

8. As we obtained an erroneous equation, it means the initial premises are not satisfied, and there is no valid solution for this problem.

Hence, it seems there may be some error in the problem statement or the problem is invalid.