Geometrical reasoning：练习题

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

In a triangle, the angles are given by \((x + 10)^{\circ}\), \((2x - 5)^{\circ}\), and \((x + 15)^{\circ}\). By setting up a linear equation, calculate the value of \(x\) and provide the geometrical reasoning for your calculation.

A pentagon has interior angles of size \(x^{\circ}\), \((x+20)^{\circ}\), \((x+40)^{\circ}\), \((x+60)^{\circ}\), and \((x+80)^{\circ}\). Calculate the size of the largest interior angle in this pentagon.

Two angles lie on a straight line. If the size of one angle is \(65^{\circ}\), what is the size of the other angle?

Calculate the size of each exterior angle of a regular octagon.

In the diagram, \(ABCD\) is a cyclic quadrilateral. The angle \(ABC = (2x + 15)^{\circ}\) and the angle \(ADC = (x + 30)^{\circ}\).
Find the value of \(x\).

In a circle with centre \(O\), a chord \(AB\) of length \(16\text{ cm}\) is at a perpendicular distance of \(6\text{ cm}\) from \(O\).
Calculate the area of the circle, giving your answer in terms of \(\pi\).

A regular polygon has an exterior angle of \(24^{\circ}\).
Work out the sum of the interior angles of this polygon.

In the diagram, \(ABCD\) is a cyclic quadrilateral. The points \(A\), \(B\), \(C\), and \(D\) lie on a circle with centre \(O\).
Angle \(ADC = 104^{\circ}\).
(a) Calculate the size of the obtuse angle \(ABC\). Give a reason for your answer.
(b) Find the size of the reflex angle \(AOC\). Give a reason for your answer.
(c) If \(OA = OB\), calculate the size of angle \(OAB\).

A regular polygon has \(n\) sides. Each interior angle of the regular polygon is \(157.5^{\circ}\).
(a) Calculate the number of sides, \(n\), of the polygon.
(b) A second polygon is a regular hexagon. Show that the sum of the interior angles of the first polygon is exactly \(5.25\) times the sum of the interior angles of the hexagon.
(c) A third polygon is an irregular pentagon with four interior angles of \(100^{\circ}\), \(110^{\circ}\), \(120^{\circ}\), and \(115^{\circ}\). Calculate the size of the fifth interior angle.