1. When one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.

2. When one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the opposite interior angle.

3. 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.

4. 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.

5. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

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

7. Prove that the quadrilateral formed by the intersection of the angle bisectors of the four angles of any quadrilateral is a cyclic quadrilateral.

8. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

9. Let us prove that the quadrilateral formed by the intersection of the internal angle bisectors of the four angles of a quadrilateral is a cyclic quadrilateral.

10. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

11. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.

12. 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.

13. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

14. ABCD is a cyclic quadrilateral. Side AB is extended to point X. If \(\angle XBC = 82^\circ\) and \(\angle ADB = 47^\circ\), find the value of \(\angle BAC\).

15. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

16. If one angle of a cyclic quadrilateral is 75°, what is the measure of its opposite angle?

17. Prove that the line segment joining the midpoints of two sides of a triangle is equal to half of the third side.

18. 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.

19. A 5-meter-high house is located on one side of a park. From both the roof and the base of the house, the base and the top of a palm tree on the opposite side of the park are seen at angles of depression and elevation of 30° and 60°, respectively. What is the height of the palm tree? What is the distance between the house and the palm tree?

20. Prove that if the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on a circle (i.e., the quadrilateral is cyclic).

21. Perfectly explained. The triangle formed is a right-angled isosceles triangle, where the angle of elevation is 45°, making the opposite and adjacent sides equal. So, the height of the pillar equals the distance from its base: 20 meters.

(a) 1:2:3 (b) 1:5:7 (c) 1:7:9 (d) None of the above

22. ABCD is a cyclic quadrilateral. The sides AD and AB are extended to E and F, respectively. If \(\angle\) CBF = 120°, then find the measure of \(\angle\) CDE.

23. ABCD is a cyclic quadrilateral. The bisectors of \(\angle\)DAB and \(\angle\)BCD intersect the circle at points X and Y. Prove that XY is the diameter of the circle.

24. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

25. Prove that if a perpendicular is drawn from the right-angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of the perpendicular are similar to each other.

26. Prove that the area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the other two sides.

27. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

28. In the cyclic quadrilateral ABCD, the side \( AB \) is extended to point \( X \), forming \( \angle XBC = 82^\circ \) and \( \angle ADB = 47^\circ \). Find the measure of \( \angle BAC \).

29. The central angle formed by an arc of a circle is twice any inscribed angle formed by the same arc—prove it.

30. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.