G-GPE.2.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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Formative Assessments
Formative Assessments
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Students are asked to prove that two parallel lines have equal slopes.
Proving Slope Criterion for Parallel Lines – One
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Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.
Proving Slope Criterion for Perpendicular Lines – 2
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Students are asked to identify the slope of a line parallel to a given line and write an equation for the line given a point.
Writing Equations for Parallel Lines
Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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Assessment Questions
Assessment Questions
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Videos of Expert Instruction
Videos of Expert Instruction
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This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.
Perpendicular Lines 2
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This video shows how to determine which lines are parallel from a set of three different equations.
Parallel Lines 2
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Hook/Performance Tasks
Hook/Performance Tasks
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This fact provides an elegant example of a theorem in geometry that is somewhat tricky to prove using classical methods, but relatively easy to prove using coordinates.
A Midpoint Miracle
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This task has students apply their knowledge of parallel lines to solve a geometric problem on areas of triangles.
Equal Area Triangles on the Same Base II
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This task is an relatively easy application of the formula for the area of a triangle and use of parallel lines.
Equal Area Triangles on the Same Base I
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Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of △ABC?
Unit Squares and Triangles
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The goal of this task is to examine when two lines in the plane are perpendicular in terms of their slopes.
When are Two Lines Perpendicular?
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In the solution, an algebraic description is provided as well as an indication, touching on the third bullet above, about how this algebra can be visualized.
Parallel Lines in the Coordinate Plane
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The goal of this task is to use ideas about linear functions in order to determine when certain angles are right angles.
Triangles Inscribed in a Circle
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The goal of this task is to use similar triangles to establish the slope criterion for perpendicular lines. Students need to be familiar with scaling and with the side-side-side congruence criterion.
Slope Criterion for Perpendicular Lines