Two co-interior angles between parallel lines have sizes in a ratio \( 4: 5 \). ind the angles. (4)

To find the sizes of the angles, let’s denote the angles as \( 4x \) and \( 5x \) based on their ratio. Since the angles are co-interior angles (which means they are on the same side of the transversal), they add up to \( 180^\circ \). So, we can set up the equation: \[ 4x + 5x = 180^\circ \] Combining like terms gives us \( 9x = 180^\circ \). Solving for \( x \) yields \( x = 20^\circ \). Thus, the angles are: \[ 4x = 80^\circ \quad \text{and} \quad 5x = 100^\circ \] Therefore, the two co-interior angles are \( 80^\circ \) and \( 100^\circ \). Co-interior angles create a fun opportunity to explore transversal lines! Think of them as buddies hanging out on parallel lines—when they’re together (and you measure their angles), they always form a perfect 180-degree angle combo, making their friendship strong and united! It's geometry's way of keeping friends together! In practical terms, understanding this concept is crucial, particularly in architecture and design. For instance, when constructing buildings or designing furniture, you want to ensure that angles created between parallel lines adhere to these principles. This knowledge helps prevent potential structural issues and ensures aesthetic appeal! So next time you spot parallel lines, remember their angle friends!