If \( 3 x^{\circ}, 4 x^{\circ} \), and \( 5 x^{\circ} \) are the angles of a triangle, find these angle: If the angles of a triangle are in the ratio 2:3:5, find them.

To find the angles \( 3x^\circ, 4x^\circ, \) and \( 5x^\circ \), we start with the fact that the sum of the angles in a triangle is \( 180^\circ \). So, we can set up the equation: \[ 3x + 4x + 5x = 180 \] Combining like terms gives us \( 12x = 180 \). Solving for \( x \) yields \( x = 15 \). Now, substituting this value back gives us the angles: - \( 3x = 45^\circ \) - \( 4x = 60^\circ \) - \( 5x = 75^\circ \) So the angles are \( 45^\circ, 60^\circ, \) and \( 75^\circ \). Now, for the second part, if the angles are in the ratio \( 2:3:5 \), let these angles be \( 2y, 3y, \) and \( 5y \). Again, since the sum of the angles in a triangle is \( 180^\circ \): \[ 2y + 3y + 5y = 180 \] This simplifies to \( 10y = 180 \), allowing us to solve for \( y \) to find \( y = 18 \). Substituting back, the angles are: - \( 2y = 36^\circ \) - \( 3y = 54^\circ \) - \( 5y = 90^\circ \) Thus, the angles corresponding to the ratio \( 2:3:5 \) are \( 36^\circ, 54^\circ, \) and \( 90^\circ \). Did you know that the concept of angles in triangles goes back to ancient civilizations? The Egyptians and Greeks studied geometry and used it for building structures and understanding the stars. They laid the groundwork for what we know today as trigonometry! Triangles are not only a mathematical concept but also a crucial element in architecture and engineering. From the Eiffel Tower to modern skyscrapers, triangles provide stability and strength in structures. Understanding triangles and their properties helps in designing buildings that stand the test of time and nature!