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Clarifications
Clarification 1: Postulates, relationships and theorems include vertical angles are congruent; when a transversal crosses parallel lines, the consecutive angles are supplementary and alternate (interior and exterior) angles and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.
Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
- MA.912.GR.3.2
- MA.912.GR.3.3
- MA.912.GR.5.1
- MA.912.GR.5.2
- MA.912.GR.6.1
- MA.912.GR.6.2
- MA.912.T.1.1
- MA.912.LT.4.3
- MA.912.LT.4.8
- MA.912.LT.4.10
Terms from the K-12 Glossary
- Angle
- Congruent
- Corresponding Angles
- Supplementary Angles
- Transversal
- Vertical
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 3, students described and identified line segments, rays, perpendicular lines and parallel lines. In grade 4, students classified and solved problems involving acute, right, obtuse, straight and obtuse angle measures. In grade 8, students solved problems involving supplementary, complementary, adjacent and vertical angles. In Geometry, students prove relationships and theorems and solve problems involving lines and angle measure. In later courses, students will study lines and angles in two and three dimensions using vectors and they will use radians to measure angles.- While focus of this benchmark are those postulates, relationships and theorems listed in Clarification 1, instruction includes other definitions, postulates, relationships or theorems such as the midpoint of a segment, angle bisector, segment bisector, perpendicular bisector, the angle addition postulate and the segment addition postulate. Additionally, some postulates and theorems have a converse (i.e., if conclusion, then hypothesis) that can be included.
- Instruction includes the connection to the Logic and Discrete Theory benchmarks when developing proofs. Additionally, with the construction of proofs, instruction reinforces the Properties of Operations, Equality and Inequality. (MTR.5.1)
- For example, when proving that vertical angles are congruent, students must be able to understand and use the Substitution Property of Equality and the Subtraction Property of Equality.
- Instruction utilizes different ways students can organize their reasoning by constructing various proofs when proving geometric statements. It is important to explain the terms statements and reasons, their roles in a geometric proof, and how they must correspond to each other. Regardless of the style, a geometric proof is a carefully written argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the statement you are trying to prove. (MTR.2.1)
- For examples of different types of proofs, please see MA.912.LT.4.8.
- Instruction includes the connection to compass and straight edge constructions and how the validity of the construction is justified by a proof. (MTR.5.1)
- Students should develop an understanding for the difference between a postulate, which is assumed true without a proof, and a theorem, which is a true statement that can be proven. Additionally, students should understand why relationships and theorems can be proven and postulates cannot.
- Instruction includes the use of hatch marks, hash marks, arc marks or tick marks, a form of mathematical notation, to represent segments of equal length or angles of equal measure in diagrams and images.
- Students should understand the difference between congruent and equal. If two segments are congruent (i.e., PQ ≅ MN), then they have equivalent lengths (i.e., PQ = MN) and the converse is true. If two angles are congruent (i.e., ∠ABC ≅ ∠PQR), then they have equivalent angle measure (i.e., ∠ABC = ∠PQR) and the converse is true.
- Instruction includes the use of hands-on manipulatives and geometric software for students to explore relationships, postulates and theorems.
- For example, folding paper (e.g., patty paper) can be used to explore what happens with angle pairs when two parallel lines are cut by a transversal. Students can discuss the possible pairs of corresponding angles, alternate interior angles, alternate exterior angles and consecutive (same-side interior and same-side exterior) angles. (MTR.2.1, MTR.4.1)
- Problem types include mathematical and real-world context where students evaluate the value of a variable that will make two lines parallel; utilize two sets of parallel lines or more than two parallel lines; or write and solve equations to determine an unknown segment length or angle measure.
- Instruction for some relationships or postulates may be necessary in order to prove theorems.
- For example, to prove that consecutive angles are supplementary when a given transversal crosses parallel lines, students will need to know the postulate that states corresponding angles are congruent.
- For example, to prove or use the perpendicular bisector theorem, students will need to have knowledge of the definition of a perpendicular bisector of a segment and midpoint.
Common Misconceptions or Errors
- Students may misuse the terms corresponding, alternate interior and alternate exterior as synonyms of congruent. To help address this conception, students should develop the understanding that these angles are congruent if and only if two parallel lines are cut by a transversal. Similarly, same-side angles are supplementary if and only if the two lines being cut by a transversal are parallel.
Instructional Tasks
Instructional Task 1 (MTR.7.1)- Runways are identified by their orientation relative to Magnetic North as viewed by an approaching aircraft. Runway directions are always rounded to the nearest ten degrees and the zero in the “ones” column is never depicted (i.e., 170 degrees would be viewed as “17” and 20 degrees would be seen as “2”). The same runway has two names which are dependent on the direction of approach.
- Use the aerial of Northeast Florida Regional Airport in St. Augustine, FL to answer the
questions below.
- Part A. Flying to the runway from point F, the runway is Runway 13. This means the heading is 130° off magnetic north. Draw a line through point R that goes to magnetic north, what is true about that line and line NF
- Part B. On the line drawn in Part A, draw and label a point, A.
- Part C. Measure angle FRA.
- Part D. How does your answer from Part C support you answer from Part A?
- Part E. What is the name of the runway when approaching from point R
Instructional Task 2 (MTR.2.1, MTR.4.1, MTR.5.1)
- Part A. Given AB, use a compass and straightedge to construct line such that line and AB form 90° and the point of intersection, M, is the midpoint of AB
- Part B. Suppose that point P lies on line as shown below. What conjecture can be made about point P? Which endpoint of AB is closest to point P? Use a compass to test your conjecture.
- Part C. What if a point, Q, was added to line Which endpoint of AB is closest to point Q How does this compare with your conjecture in Part B?
- Part D. How can the construction from Part A and the conjectures from Part B and Part C be used to prove that AP = BP given that line is the perpendicular bisector of AB and point P lies on line ?
Instructional Items
Instructional Item 1- What value of will make M the midpoint of PQ if PM = 3 − 1 and PQ = 5 + 3?
Instructional Item 2
- Two lines intersect at point P. If the measures of a pair of vertical angles are (2 − 7)° and ( + 13)°, determine and the measures of the other two angles?
Instructional Item 3
- Based on the figure below, complete a proof to prove that ∠1 ≅ ∠2 given that || and || .
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes.
Students are asked to construct a line parallel to a given line through a given point.
Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.
Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers.
Students are asked to find the measures of angles formed by two parallel lines and a transversal.
Students are asked to find the measures of angles formed by two parallel lines and two transversals.
Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles.
In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent.
Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.
Students are asked to describe and justify the relationship between same side interior angles.
Original Student Tutorials Mathematics - Grades 9-12
Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.
NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.
Student Resources
Original Student Tutorial
Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.
NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.
Type: Original Student Tutorial
Problem-Solving Tasks
Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.
Type: Problem-Solving Task
This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.
Type: Problem-Solving Task
This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.
Type: Problem-Solving Task
This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.
Type: Problem-Solving Task
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
Type: Problem-Solving Task
This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.
Type: Problem-Solving Task
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.
Type: Problem-Solving Task
This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.
Type: Problem-Solving Task
This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.
Type: Problem-Solving Task
This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.
Type: Problem-Solving Task
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
Type: Problem-Solving Task
This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.
Type: Problem-Solving Task
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
