1)) Two of the angles in a triangle measure \( 62^{\circ} \) and \( 40^{\circ} \). What is the measure of the third angle?

To find the measure of the third angle in a triangle, you can use the fact that the sum of all three angles in a triangle always equals \( 180^{\circ} \). In this case, you have the angles measuring \( 62^{\circ} \) and \( 40^{\circ} \). So, you can calculate the third angle by subtracting the sum of the first two angles from \( 180^{\circ} \): \( 180^{\circ} - (62^{\circ} + 40^{\circ}) = 180^{\circ} - 102^{\circ} = 78^{\circ} \). So, the measure of the third angle is \( 78^{\circ} \)! Understanding triangles and their angle relationships can be crucial in various fields, like architecture and engineering, where precision is essential. Knowing the triangle angle sum property helps in designing stable structures, ensuring that all angles work harmoniously to bear loads efficiently. Remember, when working with angles in triangles, it’s also easy to mix things up! One common mistake is to forget that the angles must sum to \( 180^{\circ} \). Always double-check your calculations or even draw a quick sketch to visualize the triangle, which can help avoid errors and solidify your understanding!