Find the values of $x$and $y$

Arrowheads denotes line segment $\overline{AB}$is parallel to $\overline{CD}$

Double arrowheads denotes line segment $\overline{BC}$is parallel torole="math" localid="1637750713301" $\overline{DE}$

Line segments $\overline{BC}\text{and}\overline{XY}$are transversal to the parallel lines $\overline{AB}$and$\overline{CD}$

Line segments $\overline{CD}\text{and}\overline{XY}$are transversal to the parallel lines $\overline{BC}$and$\overline{DE}$

$\angle ABC$and $\angle BCD$forms a pair of alternate interior angles

$\angle BCD$and $\angle CDE$forms another pair of alternate interior angles

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior angles are congruent.

$\begin{array}{c}\angle BCD\cong \angle ABC\end{array}$

$m\angle BCD=m\angle ABC$

$m\angle BCD=55$

$\angle CDE\cong \angle BCD$

$m\angle CDE=m\angle BCD$

$x=55$

$\angle ACB$and $\angle CED$forms a pair of corresponding angles

$\angle BAC$and $\angle DCE$forms a pair of corresponding angles

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

$\angle ACB\cong \angle CED$

$m\angle ACB=m\angle CED$

$m\angle ACB=50$

$\angle DCE\cong \angle BAC$

$m\angle DCE=m\angle BAC$

$m\angle DCE=y$

Since, $\angle ACB$, $\angle BCD$and $\angle DCE$forms the straight line $\overline{AE}$ so these angles are supplementary angles. That is,

$m\angle ACB+m\angle BCD+m\angle DCE=180$

$50+55+y=180$

$105+y=180$

$y=180$

$y=75$

Thus $y=75$