Math Geometry?

Question

ABCD is a parallelogarm. The angles at A and D have been trisected. Find the sum of the measure of x and y.

Here is a pic

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Answer 1

The answer is 180 degrees. The sum of the angles in a four sided polygon is always 360 degrees. Because it is a parallelogram, you know that adjacent angles must add to 180. Looking at the picture, angle A + angle D = 180 degrees. The sum of the angles in a triangle must add up to 180 degrees.
Start with triangle Y. The angles involved are 1/3A, 1/3D and Y (because A and D were trisected.)
180 = Y + 1/3A + 1/3D
180 = Y + 1/3(A+D)
180 = Y + 1/3(180) (because A+D = 180)
Y = 180 - 60 = 120 degrees.
WIth triangle X, the angles involved are 2/3A and 2/3D.
So 180 = X + 2/3A + 2/3D
and following the same math as before, X = 60 degrees.
60 + 120 = 180 degrees

Answer 2

Call the white dot a and the black dot b. Since the consecutive angles of a parallelogram are supplementary (add up to 180), the three angle a's plus the three angle b's add to up 180. This gives the first equation:

3a + 3b = 180 (equation 1)

The sum of the angles in a triangle is 180, so let's look at the triangle containing angle x and the one containing angle y:

a + b + y = 180 (equation 2) and

2a + 2b +x = 180 (equation 3) time to play with the equations.

Add equations 2 and 3:

3a + 3b + x + y = 360 now subtract equation 1:

3a + 3b + x + y = 360
- (3a + 3b = 180)
____________________

x + y = 180

QED

Answer 3

in a parallelogram, the sum of a bigger angle and a smaller one is 180
hence, A + D = 180 ---> (1)

now D and A are trisected, for the triangle AYD :
(D/3) + (A/3) + Y = 180

Multiply the Eqn. by 3

D + A + 3Y = 540
from (1)... D + A = 180
therefore,
3Y = 540 - 180

Y = 120

For the triangle AXD :

(2D/3) + (2A/3) + X = 180

multiply the Eqn by 3/2

D + A + (3X/2) = 270

substituting from (1);

180 + (3X/2) = 270

X = (2/3) (270-180)

X = 60

Answer 4

Rather than repeat what everyone else said, let me point out an interesting thing: There is no reason for point X to lie on the side BC. The two lines making angle x could meet below or above the top side, with the same result. Same goes for angle y.

Answer 5

It will be 180 degrees
Reason-x will be 180-2/3of 180 and y will be 180-1/3 of 180 and together they will add up to 180 because