1. The circular measure of each interior angle of a regular hexagon is –
(a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)
2. The circular measure of each interior angle of a regular hexagon is _____.
(a) \(\cfrac{\pi^c}{4}\) (b) \(\cfrac{\pi^c}{6}\) (c) \(\cfrac{\pi^c}{3}\) (d) \(\cfrac{2\pi^c}{3}\)
3. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –
(a) 60° (b) 45° (c) 30° (d) 90°
4. The central angle of each interior angle of a regular hexagon:
5. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.
6. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –
(a) 30° (b) 60° (c) 90° (d) 120°
7. If the circular (radian) measure of an angle is \( \frac{7\pi}{12} \), what is its value in the sexagesimal (degree) system?
(a) 90° (b) 105° (c) 135° (d) 160°
8. In an equilateral triangle ABC, the base BC is extended to a point E such that CE = BC. A is joined to E to form triangle ACE. Find the circular (radian) measures of the angles of triangle ACE.
9. If the sum of two angles is 135° and their difference is \(\frac{π}{12}\), write the angles in sexagesimal and circular measure.
10. A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?
11. The measure of all angles in a circular segment is ______.
12. If each side of an equilateral triangle is 4 cm, then the measurement of the triangle's height is —
(a) \(4\sqrt{3}\) cm (b) \(16\sqrt{3}\) cm (c) \(8\sqrt{3}\) cm (d) \(2\sqrt{3}\) cm
13. If one side of a rhombus is 6 cm and one of its angles measures 60°, then the area of the rhombus-shaped field is –
(a) \(9\sqrt{3}\) square cm (b) \(18\sqrt{3}\) square cm (c) \(36\sqrt{3}\) square cm (d) \(6\sqrt{3}\)square cm
14. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?
(a) 105° (b) 90° (c) 150° (d) 75°
15. Draw an isosceles triangle with a base length of 5.2 cm and each of the equal sides measuring 7 cm. Then, construct the circumcircle of the triangle and measure the circumradius. (Only mark the construction steps).
16. ABCD is a cyclic trapezium where AD and BC are parallel sides. If \(\angle\)ABC = 75°, then the measure of \(\angle\)BCD is –
(a) 30° (b) \(\frac{75°}{2}\) (c) 45° (d) 75°
17. In a circle centered at \(O\), \(AB\) is a diameter. \(ABCD\) is a cyclic quadrilateral. If \(\angle ADC = 120^\circ\), then the measure of \(\angle BAC\) is –?
(a) 60° (b) 40° (c) 50° (d) 30°
18. If the difference between the measures of the two acute angles in a right-angled triangle is 10°, determine their circular measures.
19. The radius of a circle is 28 cm. What is the circular measure of the central angle subtended by an arc of length 5.5 cm in that circle?
20. The sum of two angles is 135°, and their difference is \(\cfrac{\pi}{12}\). Determine the sexagesimal and circular measures of both angles.
21. The circular measure of the complementary angle of \(\frac{3\pi}{8}\) is _____.
22. If the sum of two angles is 135° and their difference is \(\frac{\pi^c}{12}\), find their sexagesimal and circular measures.
23. If the sum of two angles is 135° and their difference is \( \cfrac{\pi}{12} \), write the sexagesimal and circular measures of the two angles.
24. If the sum of two angles is 135° and their difference is \(\cfrac{π}{12}\), write their sexagesimal and circular measures.
25. If the sum of two angles is 135° and their difference is \(\cfrac{π}{12}\), write their sexagesimal and circular measures.
26. The English translation is: **"If the sum of two angles is 135° and their difference is \(\frac{\pi}{12}\), determine the angles in both sexagesimal and circular measure."**
27. An equilateral triangle where each side is 6 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]
28. An isosceles triangle where the base is 7.8 cm and each of the equal sides is 6.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).
29. A rotating ray turns counterclockwise from a certain position, completing two full revolutions and then an additional angle of \(30^\circ\). Calculate and write the angle in trigonometric measurement in both sexagesimal (degree) and circular (radian) systems.
30. In triangle \( \triangle ABC \), AC = BC and side BC is extended up to point D. If \( \angle ACD = 144^\circ \), then find the radian measure of each angle of triangle ABC.
