5. In triangle EFG, the measure of angle FGE is \( 35^{\circ} \), and the measure of angle GEF is \( 62.2^{\circ} \) What is the measure of angle EFG? A. \( 117.2^{\circ} \)

To find the measure of angle EFG in triangle EFG, we can use the fact that the sum of the angles in a triangle is always \( 180^{\circ} \). We have angle FGE measuring \( 35^{\circ} \) and angle GEF measuring \( 62.2^{\circ} \). By summing these two angles, we have \( 35^{\circ} + 62.2^{\circ} = 97.2^{\circ} \). Now, we subtract this from \( 180^{\circ} \) to find angle EFG: \( 180^{\circ} - 97.2^{\circ} = 82.8^{\circ} \). So the measure of angle EFG is actually \( 82.8^{\circ} \), not \( 117.2^{\circ} \)! Triangles have fascinated mathematicians throughout history, being a central topic in geometry since ancient times. The study of triangles paved the way for the development of trigonometry, which emerged in various ancient cultures including Greek, Indian, and Chinese civilizations. The Pythagorean theorem and relationships of angles to side lengths are all rooted in this three-sided figure, making it a foundational concept in mathematics as we know it today. Triangles are not just for math class! They’re everywhere in the real world—from architecture to art—and they play a vital role in engineering and design. Think of the stability of bridges or the shapes of roofs; triangular structures provide strength and durability. Additionally, in navigation and computer graphics, understanding triangles helps with everything from triangulating positions on a map to rendering 3D images. So next time you see a triangle, remember it's more than just shapes on paper!