1. **State the problem:** In triangle $\triangle ACD$, sides $AC$ and $AD$ are congruent, so $\triangle ACD$ is isosceles with $AC \cong AD$. Given: - $m\angle A = 3x - 4$ - $m\angle C = 5x + 1$ - $m\angle D = 7x - 27$ Find $x$ and the measure of each angle. 2. **Use properties of isosceles triangles:** - Since $AC \cong AD$, the angles opposite these sides are congruent. - Therefore, $m\angle C = m\angle D$. 3. **Set up the equation:** $$5x + 1 = 7x - 27$$ 4. **Solve for $x$:** $$5x + 1 = 7x - 27$$ $$1 + 27 = 7x - 5x$$ $$28 = 2x$$ $$x = 14$$ 5. **Find each angle measure by substituting $x=14$:** - $m\angle A = 3(14) - 4 = 42 - 4 = 38$ degrees - $m\angle C = 5(14) + 1 = 70 + 1 = 71$ degrees - $m\angle D = 7(14) - 27 = 98 - 27 = 71$ degrees 6. **Check sum of angles:** $$m\angle A + m\angle C + m\angle D = 38 + 71 + 71 = 180$$ degrees, which is correct for a triangle. **Final answers:** - $x = 14$ - $m\angle A = 38$ degrees - $m\angle C = 71$ degrees - $m\angle D = 71$ degrees