1. What is the measure of the third angle in a triangle if two of the angles measure \( 45^{\circ} \) and \( 50^{\circ} \) ?

In a triangle, the sum of all three angles is always \( 180^{\circ} \). So, to find the measure of the third angle, you simply add the two given angles and subtract from \( 180^{\circ} \): \( 45^{\circ} + 50^{\circ} = 95^{\circ} \) Then, \( 180^{\circ} - 95^{\circ} = 85^{\circ} \). Thus, the measure of the third angle is \( 85^{\circ} \). Triangles have been around for thousands of years, and ancient cultures, such as the Egyptians and Greeks, used them extensively in architecture and astronomy. Their properties were explored by famous mathematicians like Euclid and Pythagoras, shaping our understanding of geometry today. In practical terms, knowing the angles in a triangle is crucial for tasks such as construction, carpentry, and design. Making accurate measurements ensures stability and aesthetics in engineering projects, as well as in everyday applications like cutting materials at correct angles. Don't forget to double-check your angle measures to avoid construction mishaps!