1. Problem 8: Rohit has 6 wooden sticks of equal length and wants to join them to form a regular polygon. Find the internal angle between the sticks. Formula: The internal angle of a regular polygon with $n$ sides is given by $$\text{Internal angle} = \frac{(n-2) \times 180^\circ}{n}$$ Since $n=6$ (hexagon), $$\text{Internal angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ$$ 2. Problem 9: ABCD is a rectangle with diagonals intersecting at O. Given $OD = 4x + 9$ and $OA = 5x + 6$, find $x$. In a rectangle, diagonals bisect each other, so $OD = OA$. Set equal: $$4x + 9 = 5x + 6$$ $$9 - 6 = 5x - 4x$$ $$3 = x$$ 3. Problem 10: WEST is a rhombus with diagonals $WS = 24$ cm and $TE = 10$ cm. Find the length of side $WE$. In a rhombus, diagonals bisect each other at right angles. Half diagonals: $$\frac{WS}{2} = 12, \quad \frac{TE}{2} = 5$$ Using Pythagoras theorem for triangle $WEO$: $$WE = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}$$ 4. Problem 11: Two adjacent angles of a parallelogram are in ratio 2:1. Find the difference between these angles. Let angles be $2x$ and $x$. Sum of adjacent angles in parallelogram is $180^\circ$: $$2x + x = 180^\circ$$ $$3x = 180^\circ$$ $$x = 60^\circ$$ Difference: $$2x - x = x = 60^\circ$$ 5. Problem 12: Find the length of diagonals of a rectangle with length 24 cm and breadth 7 cm. Diagonal length formula: $$d = \sqrt{l^2 + b^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \text{ cm}$$ Both diagonals are equal, so both are 25 cm. 6. Problem 13: Find $x$ in trapezoid with angles 78°, 39°, 115°, and $x$. Sum of interior angles in quadrilateral is $360^\circ$: $$78 + 39 + 115 + x = 360$$ $$232 + x = 360$$ $$x = 360 - 232 = 128^\circ$$ 7. Problem 14: Find number of diagonals in a regular 9-sided polygon. Formula: $$\text{Number of diagonals} = \frac{n(n-3)}{2}$$ For $n=9$: $$\frac{9 \times 6}{2} = \frac{54}{2} = 27$$ 8. Problem 15: In rectangle ABCD, diagonals intersect at O. Given $\angle CDB = 54^\circ$, find $\angle OAB$. In rectangle, diagonals bisect each other and $\angle OAB = \frac{1}{2} \angle CDB$ (since triangle formed is isosceles right triangle properties). So, $$\angle OAB = \frac{54^\circ}{2} = 27^\circ$$ 9. Problem 16: Find each interior angle of a regular decagon. Formula: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ For $n=10$: $$\frac{(10-2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = 144^\circ$$ Final answers: 8. 120° 9. 3 10. 13 cm 11. 60° 12. 25 cm 13. 128° 14. 27 15. 27° 16. 144°