Angles：练习题

Two lines intersect to form four angles. If one angle is \(x^\circ\) and the angle adjacent to it on a straight line is \((3x - 20)^\circ\), find the value of \(x\).

In the diagram, the lines AB and CD are parallel. A transversal line intersects AB at point X and CD at point Y. If a pair of co-interior angles are given by \((2x + 10)^\circ\) and \((3x - 15)^\circ\), find the value of \(x\).

In the diagram, line \(AB\) is parallel to line \(CD\). Point \(E\) is located between the two lines such that \(\angle BAE = 130^\circ\) and \(\angle DCE = 120^\circ\). Calculate the size of the reflex angle \(AEC\).

Two angles are supplementary. If one of the angles measures \( 115^\circ \), find the size of the other angle.

In a quadrilateral \( ABCD \), the interior angles are represented by the expressions \( 3x^∘ \), \( (x + 30)^∘ \), \( (2x + 10)^∘ \), and \( 80^∘ \). Find the value of \( x \).

Three of the interior angles of a quadrilateral are \(85^\circ\), \(110^\circ\), and \(75^\circ\). Calculate the size of the fourth interior angle.

In triangle \(PQR\), side \(PQ = PR\). A line \(XY\) is drawn parallel to \(QR\) which intersects \(PQ\) at \(X\) and \(PR\) at \(Y\). Given that \(\angle QPR = 44^\circ\), calculate the size of \(\angle PXY\).

In the diagram, a chord AB of a circle with centre O subtends an angle of 140° at the centre. Point D lies on the circumference such that ACBD is a cyclic quadrilateral. Calculate the size of the angle ADB.

In the diagram, lines PQ and RS are parallel. A transversal line cuts them such that a pair of co-interior angles are \( (x + 45)^\circ \) and \( (2x - 15)^\circ \).
(a) Set up an equation to find \( x \).
(b) Determine the value of \( x \).
(c) Calculate the size of the larger of the two angles.

For a regular polygon with 9 sides (a nonagon):
(a) Calculate the size of each exterior angle.
(b) Calculate the size of each interior angle.
(c) Find the sum of all the interior angles of the nonagon.