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Clarifications
Clarification 1: Postulates, relationships and theorems include opposite sides are congruent, consecutive angles are supplementary, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.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
Terms from the K-12 Glossary
- Angle
- Parallelogram
- Quadrilateral
- Rectangle
- Rhombus
- Square
- Supplementary Angles
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In elementary grades, students identified and classified quadrilaterals, including parallelograms. In grade 7, students solved problems involving the area of parallelograms. In Geometry, students establish relationships between sides, angles and diagonals of parallelograms (including the special cases of parallelograms: rectangles, rhombi and squares), prove the theorems related to these relationships and use them to solve mathematical and real-world problems. In later courses, parallelograms play an important role in work with vectors.- Relationships, postulates and theorems in this benchmark should focus on, but are not limited to, the ones stated in Clarification 1. 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)
- 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.
- Instruction includes discussing that the definition of a parallelogram only states that the opposite sides are parallel; anything else (e.g., opposite sides are congruent or opposite angles are congruent) is a property and needs to be proven. Students should understand that all parallelograms are trapezoids based on the definition within the K-12 Glossary.
- Instruction includes the connection to triangle congruence and the relationship between angles formed by a transversal through parallel lines when completing proofs about parallelograms.
- When the properties of parallelograms are introduced and proven, it is important to discuss the definitions of rectangles, rhombi and squares. Precision and accuracy are important when discussing the definitions of the special parallelograms. (MTR.3.1, MTR.4.1)
- For example, some possible discussion questions include:
- What makes a parallelogram a rectangle?
- What is the unique feature of a rhombus?
- Is a square always a rectangle?
- Is a square sometimes a rhombus?
- Is a rectangle always a square? Is a rhombus always a square?
- What are the properties of a parallelogram observed in a rhombus?
- For example, some possible discussion questions include:
- Instruction includes proving that the diagonals of a parallelogram bisect each other and that the diagonals of a parallelogram are congruent if and only if the parallelogram is a rectangle. Clarify that just having congruent diagonals will not be enough to identify a quadrilateral as a rectangle as this is also a property of isosceles trapezoids. The quadrilateral has to be proven a parallelogram to use this property to classify it as a rectangle.
- Instruction includes the understanding that properties of parallelograms apply to all parallelograms, including squares, rhombi and rectangles.
Common Misconceptions or Errors
- Students may think of squares, parallelograms, rectangles and rhombi as being exclusive to each other. A square is a rectangle and a rhombus.
- Students may think that parallelograms are not trapezoids. The K-12 Mathematics Glossary defines trapezoids as quadrilaterals with at least one pair of parallel sides. Therefore, all parallelograms are trapezoids.
Instructional Tasks
Instructional Task 1 (MTR.3.1)- Given parallelogram ABCD, prove that angle A and angle B are supplementary.
- Given quadrilateral JKLM with JK ≅ LM ≅ MJ.
- Part A. Draw the diagonal connecting points M and K. Determine and prove that two triangles are congruent.
- Part B. Using the congruent triangles from Part A, what is true about segments MJ and LK?
- Part C. Prove that quadrilateral JKLM is a parallelogram.
Instructional Items
Instructional Item 1- Given parallelogram WXYZ, where WX = 2 + 15, XY = + 27 and YZ = 4 − 21, determine the length of ZW, in inches.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
MFAS Formative Assessments
Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measures of all four angles describing any theorems used.
Students are given parallelogram ABCD along with midpoint E of diagonal AC and are asked to determine the relationship between the lengths AE + ED and BE + EC.
Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measure of an angle opposite one of the given angles.
Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.
Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.
Students are asked to prove that a rectangle is a parallelogram.
Students are asked to prove that the diagonals of a rectangle are congruent.
Students are asked to prove that opposite angles of a parallelogram are congruent.
Students are asked to prove that the diagonals of a parallelogram bisect each other.
Students are asked to prove that opposite sides of a parallelogram are congruent.
Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.
Student Resources
Perspectives Video: Teaching Idea
Students will learn to use basic geometric construction tools to copy a segment and an angle, then apply these skills repeatedly to construct a rhombus. The definition of a parallelogram is verified.
Type: Perspectives Video: Teaching Idea
Problem-Solving Tasks
This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles.
Type: Problem-Solving Task
This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.
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
This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles.
Type: Problem-Solving Task
This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.
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
