Geometrical Elements, Angles and Distance：練習問題

In the figure, three straight lines intersect at a single point. If two of the vertically opposite angles are given as \( (2x + 15)^\circ \) and \( (3x - 10)^\circ \), find the value of \( x \).

In the figure, line \( AB \) is parallel to line \( CD \). A transversal line \( EF \) intersects \( AB \) at point \( G \) and \( CD \) at point \( H \). If the alternate interior angle \( \angle AGH = (4y - 20)^\circ \) and the corresponding angle to \( \angle AGH \) at point \( H \), which is \( \angle GHD \), is \( (2y + 40)^\circ \), find the value of \( y \).

In the figure, line \( L_1 \) is parallel to line \( L_2 \). A zigzag line connects them, forming angles of \( 130^\circ \) and \( 110^\circ \) with the parallel lines respectively (measured from the same interior direction). Find the interior angle \( x \) formed at the vertex of the zigzag.

Five angles lie on a straight line at a single point. If the sizes of these angles are in the ratio \( 1:2:3:4:5 \), what is the difference between the largest angle and the smallest angle?

In the figure, \( AB \parallel CD \). Point \( E \) lies between the two parallel lines such that \( \angle BAE = 35^\circ \) and \( \angle DCE = 45^\circ \). A reflex angle \( \angle AEC \) is formed on the outside of the V-shape. What is the size of the reflex angle \( \angle AEC \)?

In the figure, lines \( AB \) and \( CD \) intersect at point \( O \). Given that \( \angle AOC = (3x + 10)^\circ \) and \( \angle BOD = (5x - 20)^\circ \), find the value of \( x \).