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1) x=58, y=109

2) x=72, y=36

3) x=60, y=48

Step-by-step explanation:

1) 58 and x are so-called "F-angles" (a.k.a "corresponding angles), so x=58°

The angle on the other side of the 58 is 180-58 = 122 because they are supplementary. We need this one to calculate y.

The sum of angles in a quadrilateral is 360°, so 71 + 58 + 122 + y = 360, so y = 109

2) There are two isosceles triangles. Due to the symmetry, the bottom left angle is also 72, leaving 180-72-72 = 36 for y. The top triangle is the same triangle rotated, so x = 72.

3) Due to the parallel lines, the left angle of the top triangle is also 72, making the supplementary angle below it 180-72=108. The sum in the quadrilateral must be 360, so 72+108+2x+x=360, so 3x=180, x=60.

The sum in the outer triangle must be 180, so 72+y+x = 180, leaving y = 180-72-60 = 48

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

25.45584

care to share were the trapezoid shown is ?