1. If two angles of a triangle are 75° and \( \frac{\pi^c}{6} \), then what is the measure of the third angle?

(a) 75° (b) 60° (c) 65° (d) 70°

2. What will be the measure of the third angle of a triangle in radians if the other two angles are \(65^\circ 56' 44''\) and \(64^\circ 3' 16''\)?

3. Give the definition of a radian. The angles of a triangle are in the ratio 2:5:3; what is the radian measure of the smallest angle of the triangle?

4. If two angles of a triangle measure 35°57′4″ and 39°2′56″, then calculate the radian measure of the third angle.

5. If the measures of two angles of a triangle are 39°2′56″ and 35°57′4″, determine the circular (degree) measure of the third angle.

6. > If two angles of a triangle are 35°5'74" and 39°2'56", find the radian measure of the third angle. Let me know if you'd like me to solve it for you too.

7. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?

8. Two unequal arcs of a circle subtend two angles at the center in the ratio 5:3, and the sexagesimal (degree) measure of the second angle is 45°. What is the radian measure of the first angle?

9. 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?

10. If the ratio of three consecutive angles of a cyclic quadrilateral is 1 : 2 : 3, what are the measures of the first and third angles?

11. In a right-angled triangle, the difference between the two acute angles is \(\frac{2\pi}{5}\). Express the measures of these two angles in both radians and degrees.

12. In a right-angled triangle, the difference between the two acute angles is 30°. Express the measures of those two angles in both radians and degrees.

13. In a triangle, one angle is 60° and another angle is \(\frac{\pi}{6}\) radians. What is the measure of the third angle in degrees?

14. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?

15. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?

16. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

17. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

18. The angles of a triangle are in the ratio \(2:5:3\); calculate and write the circular (radian) measure of the smallest angle.

19. If the three angles of a triangle are in the ratio 2:3:4, then calculate and write the radian measure of the largest angle.

20. If two angles of a triangle measure \(65^\circ 56' 55''\) and \(64^\circ 3' 5''\), find the circular (radian) measure of the third angle.

21. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

22. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?

(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

23. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

24. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

25. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.

26. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.

27. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

28. 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.

29. If three angles of a quadrilateral are \(\frac{π}{3}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the measure of the fourth angle in both sexagesimal (degree) and circular (radian) units.

30. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.