[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]What is the mathematical process for finding the angle between two lines in 2D space?[/td][br][td]Why is it important to understand the concept of angle between two lines in planar geometry?[/td][br][td]How can the study of angles between lines enhance our understanding of spatial relationships in professional fields like architecture?[/td][br][/tr][br][tr][br][td]How does the slope of a line relate to the angle it forms with another line?[/td][br][td]What does the angle between two lines tell us about the relationship between those lines?[/td][br][td]Should computational tools replace traditional methods for teaching geometry concepts like the angle between two lines?[/td][br][/tr][br][tr][br][td]Under what conditions will two lines be perpendicular or parallel to each other?[/td][br][td]How can the principles of angles between two lines be applied to solve real-world problems?[/td][br][td]Could reliance on technology to solve geometric problems hinder students' understanding of fundamental mathematical concepts?[/td][br][/tr][br][/table][br]

Mini-Investigation: Exploring Angles Between Two Lines in 2D[br][br]Objective:[br]Use the applet to understand how to calculate the angle between two lines in a two-dimensional space.[br][br]Questions:[br]1. What is the angle between Line 1 (y = 2.000x + 3.000) and Line 2 (y = -4.000x - 2.000) as calculated by the applet?[br]2. How does changing the slope of Line 1 affect the angle between the two lines? Experiment by adjusting the slope and recording the new angles.[br]3. If Line 1 is parallel to the x-axis, what would be the angle between Line 1 and Line 2? Test your prediction using the applet.[br]4. What happens to the angle if Line 1 and Line 2 are perpendicular? Can you find the slopes that make this true?[br]5. How can we use the applet to demonstrate that the sum of angles in a linear pair is 180 degrees?[br]6. Suppose Line 1 represents the ground and Line 2 represents a ramp. How would changing the angle affect the steepness of the ramp? [br]7. Challenge: Can you create two lines that are neither parallel nor perpendicular but still have a specific angle between them, say 45 degrees? Use the applet to find the slopes of these lines.[br]8. Use the applet to explore the relationship between the slopes of two lines and the angle between them. What do you notice about the slopes of lines that form acute vs. obtuse angles?[br][br]Extension Activity:[br]Investigate the effect of the y-intercept on the angle between two lines. Keep the slope constant and only change the y-intercept of one line. How does this affect the angle? Use the applet to support your findings.[br]