Lesson Description
This lesson consolidates Unit 3 concepts. Students review angle relationships formed by parallel lines and transversals, slope relationships, and analytic geometry methods for determining line relationships.
Mixed Review
- Two parallel lines are cut by a transversal. If one corresponding angle measures 65°, find all acute angles formed.
- If alternate interior angles are congruent, what can you conclude about the lines?
- Find the slope of a line passing through (2, -1) and (6, 7).
- Determine whether lines with slopes \( \frac{3}{4} \) and \( -\frac{4}{3} \) are parallel, perpendicular, or neither.
Analytic Geometry Review
- Write the equation of a line parallel to \( y = -2x + 1 \) passing through (3,5).
- Write the equation of a line perpendicular to \( y = \frac{1}{3}x - 4 \) passing through (0,2).
- Determine if the lines through points A(1,2), B(5,6) and C(2,7), D(6,3) are parallel or perpendicular.
- Explain why vertical lines cannot be expressed in slope-intercept form.
Application Problems
- A road rises 4 feet for every 20 feet traveled horizontally. What is its slope?
- A wheelchair ramp must meet safety guidelines with a maximum slope of \( \frac{1}{12} \). Is a ramp rising 3 feet over 30 feet compliant?
- Architectural beams are designed to intersect at right angles. How can slope calculations confirm perpendicular alignment?
Challenge Problems
- Given triangle vertices A(1,2), B(5,2), and C(3,6), determine whether the triangle contains a right angle.
- Prove algebraically that if two lines are perpendicular, the product of their slopes equals -1 (when defined).
- Create a real-world scenario involving parallel and perpendicular lines and model it with equations.