11. The measure of one of the equal angles in an isosceles triangle is twice the measure of the remaining angle. Determine the exact radian measures of the three angles in the triangle.
Answer
Final Answer: The exact radian measures of the three angles in the triangle are \(\boxed{\frac{2\pi}{5}}\), \(\boxed{\frac{2\pi}{5}}\), and \(\boxed{\frac{\pi}{5}}\).
Step 1 ：Let's denote the measure of the equal angles in an isosceles triangle as \(x\) and the measure of the remaining angle as \(y\).
Step 2 ：According to the problem, \(x = 2y\).
Step 3 ：Also, the sum of the angles in a triangle is equal to \(\pi\) radians. Therefore, we have the equation \(2x + y = \pi\).
Step 4 ：Substituting \(x = 2y\) into this equation, we can solve for \(y\) and then find \(x\).
Step 5 ：The solution to the system of equations is \(x = \frac{2\pi}{5}\) and \(y = \frac{\pi}{5}\).
Step 6 ：This means that the measure of the equal angles in the triangle is \(\frac{2\pi}{5}\) radians and the measure of the remaining angle is \(\frac{\pi}{5}\) radians.
Step 7 ：Final Answer: The exact radian measures of the three angles in the triangle are \(\boxed{\frac{2\pi}{5}}\), \(\boxed{\frac{2\pi}{5}}\), and \(\boxed{\frac{\pi}{5}}\).
