1. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

2. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.

3. The distance between two pillars is 150 meters. One pillar's height is three times the other. The angles of elevation to their tops from the midpoint of the line joining their bases are complementary. Find the height of the smaller pillar.

4. The sum of two angles is \(\cfrac{13π}{12}\). If one of the angles is \(58°\), find the other angle.

(a) 117° (b) 127° (c) 137° (d) 147°

5. 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.

6. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.

7. 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?

8. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

9. The distance between two pillars is 150 meters. One is three times as tall as the other. From the midpoint of the line connecting the bases of the two pillars, the angles of elevation to their tops are complementary. What is the height of the shorter pillar?

10. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

11. If the angle of elevation of the top of a chimney from a point on level ground at its base is 60°, and from another point on the same straight line 24 meters further away from the base, the angle of elevation is 30°, then calculate and write the height of the chimney. [Take the approximate value of \(\sqrt{3} = 1.732\) and find the result correct to three decimal places.]

12. There is a bridge perpendicular to the bank of a river. From one end of the bridge on the bank, if a person walks a certain distance along the riverbank, the other end of the bridge is seen at an angle of 45°. Walking 400 meters farther along the bank, the same end of the bridge is then seen at an angle of 30°. Find the length of the bridge.

13. From the roof of a 15-meter-high house situated at one end of a park, the foot and the top of a brick kiln chimney located at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Find the height of the chimney and the distance between the brick kiln and the house.

14. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, find the first and third angles in radians.

15. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument and the angle of depression to its base are complementary angles. If the height of the monument is four times the height of the building, find the measures of the elevation and depression angles.

16. Here's the full translation of your math problem into English: > The height of one building forms a right angle with the window of another building situated directly across the street. From that window, the angle of elevation to the top of the first building is 30°. > If the width of the street is \(4\sqrt{3}\) meters, find: > (1) the height of the first building, and > (2) how high the window of the second building is above the street level.

17. 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.

(a) 60° (b) 45° (c) 30° (d) 90°

18. 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.

19. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

20. A person standing on a railway overbridge 5√3 meters high first observes the engine of a train at a depression angle of 30° on one side of the bridge. After 2 seconds, he sees the engine at a depression angle of 45° on the other side of the bridge. What is the speed of the train?

21. If each side of a rhombus is 5 cm and one of its diagonals is 4 cm, find the length of the other diagonal.

22. 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?

23. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)

24. The distance between two places is 200 km. The time taken to travel from one place to the other by a motor car is 2 hours more than the time taken by a jeep. The speed of the jeep is 5 km/h more than that of the motor car. Find the speed of the motor car.

25. 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?

26. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

27. 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.

28. 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.

29. There is a palm tree on the bank of a river. Directly opposite it, on the other side of the river, a pole is fixed into the ground. If someone walks 7\(\sqrt{3}\) meters along the riverbank from the pole toward the tree, the base of the tree appears to form a 60° angle with the riverbank at that point. Determine the width of the river.

30. In a right-angled triangle, if one of the acute angles is 30°, determine the measure of the other acute angle in sexagesimal system.