Angles：練習題

In the figure, \( PQ \) is a straight line. If point \( O \) lies on \( PQ \) and ray \( OR \) is drawn such that \( \angle POR = 4x^{\circ} \) and \( \angle QOR = 2x^{\circ} \), find the value of \( x \).

In the figure, four rays meet at a point \(O\). The angles formed are \(\angle AOB = x + 20^{\circ}\), \(\angle BOC = 2x - 10^{\circ}\), \(\angle COD = x + 30^{\circ}\), and \(\angle DOA = 2x + 20^{\circ}\). Find the value of \(x\).

In the figure, \( AB \) is a straight line. Three rays \( OC \), \( OD \), and \( OE \) are drawn from point \( O \) such that \( ∠AOC = x^{\circ} \), \( ∠COD = y^{\circ} \), \( ∠DOE = z^{\circ} \), and \( ∠EOB = w^{\circ} \). Given that \( x:y:z = 2:3:4 \) and \( w = y + 10 \), find the value of \( z \).

In the figure, \( AOB \) is a straight line. If \( ∠AOC = 72^{\circ} \) and \( ∠COD = 58^{\circ} \), find the size of \( ∠DOB \).

In the figure, four angles are formed at point \( O \). They are \( ∠AOB = 3x^{\circ} \), \( ∠BOC = 120^{\circ} \), \( ∠COD = (2x + 10)^{\circ} \), and \( ∠DOA = 80^{\circ} \). Find the value of \( x \).

In the figure, AOB is a straight line. If \(\angle AOC = 55^{\circ}\) and OD is a ray such that \(\angle COD = 90^{\circ}\), find the size of \(\angle DOB\).

In the figure, AB and CD intersect at O. If \(\angle AOC + \angle BOD = 130^{\circ}\), find the size of the obtuse angle \(\angle AOD\).

In the figure, straight lines AB, CD, and EF intersect at point O. If \(\angle AOC = 2x^{\circ}\), \(\angle COE = 3x^{\circ}\), and \(\angle EOB = 80^{\circ} - x^{\circ}\), find the value of \(x\).

In the figure, AOB is a straight line. Two rays OC and OD are drawn from point O such that \(\angle AOC = 2x^{\circ}\), \(\angle COD = 3x^{\circ}\), and \(\angle DOB = 4x^{\circ}\).
(a) Find the value of \(x\).
(b) Find the size of \(\angle AOD\).

In the figure, the ratio of three angles around a point O, namely ∠AOB, ∠BOC, and ∠COA, is 2 : 3 : 5.

(a) Find the sizes of ∠AOB, ∠BOC, and ∠COA.
(b) If BOD is a straight line, find the size of ∠AOD.