Find the size of each lettered angle in the following figure:

Here, AB || CD and BC || AD

So, ABCD is a || gm

y = ∠ABC (As opposite angles of || gm are equal)

y = 2∠ABD (As diagonals bisect vertex angle so BD bisects ∠ABC)

y = 2 x 53° = 106°

Also, y + ∠DAB = 180° (DC || AB, sum of co-int ∠s = 180°)

∠DAB = 180° – 106° = 74°.

Thus, x = $\dfrac{1}{2}$∠DAB [As AC bisects ∠DAB]

⇒ x = $\dfrac{1}{2}$ x 74° = 37°

and ∠DAC = x = 37°

Also, z = ∠DAC = 37° [Alternate angles are equal]

Hence, x = 37°, y = 106° and z = 37°.