Question

Find the value of x, y, and z in the rhombus below.
108°
(-x+1)°
(-2y+4)°
(3z+9)°

Answer 1

Label the four consecutive angles A1 = 108°, A2 = (−x+1)°, A3 = (−2y+4)°, A4 = (3z+9)°.

In a rhombus opposite angles are equal and adjacent angles are supplementary.

1) Opposite equality: A1 = A3 → 108 = −2y + 4
−2y = 104 → y = −52.

2) Adjacent supplementary: A1 + A2 = 180 → 108 + (−x+1) = 180
−x + 109 = 180 → −x = 71 → x = −71.

3) Opposite equality: A2 = A4 → −x+1 = 3z+9.
Substitute x = −71: 71 + 1 = 3z + 9 → 72 = 3z + 9 → 3z = 63 → z = 21.

So x = −71, y = −52, z = 21. (Checks: angles are 108°, 72°, 108°, 72°.)