Side DC of a parallelogram ABCD is produced to a point E such that \(\angle BCE = 70^\circ\). If \(\angle DAB = x\), \(\angle ADB = 40^\circ\), \(\angle DBC = y\) and \(\angle BDC = z\), then value of \(x+y+z\) is equal to:
180°
Since DCE is a straight line, \(\angle BCD\) and \(\angle BCE\) are supplementary, so \(\angle BCD = 180^\circ - 70^\circ = 110^\circ\).
In parallelogram ABCD, opposite angles are equal, so \(\angle DAB = \angle BCD\), which gives \(x = 110^\circ\).
Consecutive angles of a parallelogram are supplementary, so \(\angle ADC = 180^\circ - \angle DAB = 180^\circ - 110^\circ = 70^\circ\).
Since diagonal BD splits \(\angle ADC\) into \(\angle ADB\) and \(\angle BDC\), \(40^\circ + z = 70^\circ\), so \(z = 30^\circ\).
In triangle BDC, the angles sum to \(180^\circ\): \(y + z + \angle BCD = 180^\circ\), that is \(y + 30^\circ + 110^\circ = 180^\circ\), so \(y = 40^\circ\).
Adding all three, \(x + y + z = 110^\circ + 40^\circ + 30^\circ = 180^\circ\).
Hence, the value of x+y+z is 180 degrees.
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