MAFS.912.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).

Clarifications

Geometry - Fluency Recommendations

Fluency with the use of coordinates to establish geometric results, calculate length and angle, and use geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields.
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Geometry: Expressing Geometric Properties with Equations
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Use coordinates to prove simple geometric theorems algebraically. (Geometry - Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

  • Assessment Limits :
    Lines may include horizontal and vertical lines.

    Items may not ask the student to provide only the slope of a parallel
    or perpendicular line.

  • Calculator :

    Neutral

  • Clarification :
    Students will prove the slope criteria for parallel lines.

    Students will prove the slope criteria for perpendicular lines.

    Students will find equations of lines using the slope criteria for
    parallel and perpendicular lines.

  • Stimulus Attributes :
    Items may be set in a real-world or mathematical context.
  • Response Attributes :
    Items may require the student to be familiar with slope-intercept
    form of a line, standard form of a line, and point-slope form of a line.
Sample Test Items (1)
  • Test Item #: Sample Item 1
  • Question:

    The equation for line A is shown.

    y=begin mathsize 12px style negative 2 over 3 x minus 4 end style

    Line A and line B are perpendicular, and the point (-2,1_ lies on line B.

    Write an equation for line B.

  • Difficulty: N/A
  • Type: EE: Equation Editor

Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1207310: Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912065: Access Geometry (Specifically in versions: 2015 - 2022 (current), 2022 and beyond)

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.G-GPE.2.AP.5a: Using slope, prove lines are parallel or perpendicular.
MAFS.912.G-GPE.2.AP.5b: Find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Assessments

Sample 1 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 4 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 3 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 2 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Formative Assessments

Proving Slope Criterion for Perpendicular Lines - 2:

Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.

Type: Formative Assessment

Proving Slope Criterion for Perpendicular Lines - 1:

Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.

Type: Formative Assessment

Proving Slope Criterion for Parallel Lines - Two:

Students are asked to prove that two lines with equal slopes are parallel.

Type: Formative Assessment

Proving Slope Criterion for Parallel Lines - One:

Students are asked to prove that two parallel lines have equal slopes.

Type: Formative Assessment

Writing Equations for Parallel Lines:

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.

Type: Formative Assessment

Writing Equations for Perpendicular Lines:

Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given a point.

Type: Formative Assessment

Finding Equations of Parallel and Perpendicular Lines:

This lesson is intended to help you assess how well students are able to understand the relationship between the slopes of parallel and perpendicular lines and, in particular, to help identify students who find it difficult to:

  • Find, from their equations, lines that are parallel and perpendicular.
  • Identify and use intercepts.
It also aims to encourage discussion on some common misconceptions about equations of lines.

Type: Formative Assessment

Lesson Plans

Graphing Equations on the Cartesian Plane: Slope:

The lesson teaches students about an important characteristic of lines: their slope. Slope can be determined either in graphical or algebraic form. Slope can also be described as positive, negative, zero, or undefined. Students get an explanation of when and how these different types of slope occur. Finally, students learn how slope relates to parallel and perpendicular lines. When two lines are parallel, they have the same slope and when they are perpendicular their slopes are negative reciprocals of one another. Prerequisite knowledge: Students must know how to graph points on the Cartesian plane. They must be familiar with the x- and y- axes on the plane in both the positive and negative directions.

Type: Lesson Plan

"When will we ever meet?":

Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine whether lines are parallel, perpendicular, or neither by looking at the graph and the equation.

Type: Lesson Plan

Forget Waldo - Where is 'the orthocenter'?:

Starting with a set of three points, students will practice finding equations of lines and the lines that are perpendicular to them. The students will repeat this process three times - using different colors for differentiating one line from the next. The big finale brings all the work together and the students realize this activity leads to finding the orthocenter of a triangle.

Type: Lesson Plan

Investigating Lines With Our Minds!:

Students will investigate lines to discover the relationships of the slopes of parallel and perpendicular lines using X-Y Coordinate Pegboards.

Type: Lesson Plan

Problem-Solving Tasks

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Type: Problem-Solving Task

Tutorials

Parallel Lines:

Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.

Type: Tutorial

Perpendicular Lines:

Perpendicular lines have slopes which are negative reciprocals of each other, but why?

Type: Tutorial

Video/Audio/Animations

Parallel Lines 2:

This video shows how to determine which lines are parallel from a set of three different equations.

Type: Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

Perpendicular Lines 2:

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.

Type: Video/Audio/Animation

Worksheet

Midpoints of the Sides of a Quadrilateral:

The students will construct a quadrilateral on graph paper, determine the midpoints of each of the four sides, then connect the midpoints of adjacent sides. The question then is the following: what are the properties of the resulting quadrilateral? Students need to justify their conclusions.

Type: Worksheet

MFAS Formative Assessments

Proving Slope Criterion for Parallel Lines - One:

Students are asked to prove that two parallel lines have equal slopes.

Proving Slope Criterion for Parallel Lines - Two:

Students are asked to prove that two lines with equal slopes are parallel.

Proving Slope Criterion for Perpendicular Lines - 1:

Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.

Proving Slope Criterion for Perpendicular Lines - 2:

Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.

Writing Equations for Parallel Lines:

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 Perpendicular Lines:

Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given a point.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Problem-Solving Tasks

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Type: Problem-Solving Task

Tutorials

Parallel Lines:

Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.

Type: Tutorial

Perpendicular Lines:

Perpendicular lines have slopes which are negative reciprocals of each other, but why?

Type: Tutorial

Video/Audio/Animations

Parallel Lines 2:

This video shows how to determine which lines are parallel from a set of three different equations.

Type: Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

Perpendicular Lines 2:

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.

Type: Video/Audio/Animation

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Tasks

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Type: Problem-Solving Task