The residents of a downtown neighborhood designed a triangular-shaped park as part of a city beautification program. The park is bound by streets on all sides. The second angle of the triangle is \( 4^{\circ} \) more than the first. The third angle is \( 4^{\circ} \) less than four times the first. If the sum of the interior angles of a triangle measure \( 180^{\circ} \), determine the measures of the three angles. The measure of the first angle is \( x=\square^{\circ} \).

To find the angles of the triangular-shaped park, let's label the first angle as \( x \). According to the problem, the second angle can be expressed as \( x + 4^\circ \), and the third angle can be expressed as \( 4x - 4^\circ \). We know that the sum of all interior angles in a triangle is \( 180^\circ \). Thus, we can set up the following equation: \[ x + (x + 4^\circ) + (4x - 4^\circ) = 180^\circ \] Simplifying this, we have: \[ x + x + 4^\circ + 4x - 4^\circ = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ \] Now that we have the first angle, we can determine the measures of the other two angles. The second angle is \( x + 4^\circ = 34^\circ \), and the third angle is \( 4x - 4^\circ = 116^\circ \). So, the measures of the three angles are: - First angle: \( 30^\circ \) - Second angle: \( 34^\circ \) - Third angle: \( 116^\circ \) Now, let's explore some history behind triangles and their angles! Did you know that the study of triangles dates back to ancient civilizations like the Egyptians and Babylonians? They used geometric principles for everything from land surveying to building pyramids. The Greeks, particularly Euclid, further developed geometric theories, establishing the foundation of what we now call Euclidean geometry, which famously includes the sum of a triangle's interior angles being \( 180^\circ\). Triangles are not just beautiful shapes; they also have cool real-world applications! From engineering to architecture, triangular shapes are often used because of their structural integrity. Triangles distribute weight evenly, making them ideal for constructing bridges and roofs. If you’ve ever seen a truss in a building or a bridge, you’ve seen triangles in action!