# MAFS.912.G-CO.3.11

Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
General Information
Subject Area: Mathematics
Domain-Subdomain: Geometry: Congruence
Cluster: Level 3: Strategic Thinking & Complex Reasoning
Cluster: Prove geometric theorems. (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 of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

• Assessment Limits :
Items may require the student to be familiar with similarities and
differences between types of parallelograms (e.g., squares and
rectangles).

Items may require the student to identify a specific parallelogram.

Items may assess theorems and their converses for opposite sides of
a parallelogram, opposite angles of a parallelogram, diagonals of a
parallelogram, and consecutive angles of a parallelogram.

Items may assess theorems and their converses for rectangles and
rhombuses.

Items may include narrative proofs, flow-chart proofs, two-column
proofs, or informal proofs.

In items that require the student to justify, the student should not be
required to recall from memory the formal name of a theorem.

• Calculator :

Neutral

• Clarification :
Students will prove theorems about parallelograms.

Students will use properties of parallelograms to solve problems.

• Stimulus Attributes :
Items may be set in real-world or mathematical context.
• Response Attributes :
Items may require the student to classify a quadrilateral as a
parallelogram based on given properties or measures.

Items may require the student to prove that a quadrilateral is a
parallelogram

Sample Test Items (1)
• Test Item #: Sample Item 1
• Question: Drag the correct statement from the statements column and the correct reasons from the reasons column to the table to complete line 3 of the proof.

• Difficulty: N/A
• Type: DDHT: Drag-and-Drop Hot Text

## 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)
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)
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-CO.3.AP.11a: Measure the angles and sides of parallelograms to establish facts about parallelograms.

## Related Resources

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

## Assessments

Sample 4 - High School Geometry State Interim Assessment:

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

Type: Assessment

Sample 2 - High School Geometry State Interim Assessment:

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

Type: Assessment

## Formative Assessments

Comparing Lengths in a Parallelogram:

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.

Type: Formative Assessment

Finding Angle C:

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.

Type: Formative Assessment

Frame It Up:

Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.

Type: Formative Assessment

Two Congruent Triangles:

Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.

Type: Formative Assessment

Angles of a Parallelogram:

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.

Type: Formative Assessment

Proving Congruent Diagonals:

Students are asked to prove that the diagonals of a rectangle are congruent.

Type: Formative Assessment

Proving a Rectangle Is a Parallelogram:

Students are asked to prove that a rectangle is a parallelogram.

Type: Formative Assessment

Proving Parallelogram Angle Congruence:

Students are asked to prove that opposite angles of a parallelogram are congruent.

Type: Formative Assessment

Proving Parallelogram Diagonals Bisect:

Students are asked to prove that the diagonals of a parallelogram bisect each other.

Type: Formative Assessment

Proving Parallelogram Side Congruence:

Students are asked to prove that opposite sides of a parallelogram are congruent.

Type: Formative Assessment

## Lesson Plans

To Be or Not to Be a Parallelogram:

A lesson where students apply parallelogram properties and theorems to solve real world problems. The teacher models a problem solving strategy, which involves drawing a picture, highlighting important information, estimating and/or writing equation, and solving problem (P.I.E.S.).

Type: Lesson Plan

Diagonally Half of Me!:

This lesson is an exploration activity assisting students prove that diagonals of parallelograms bisect each other. It allows them to compare some quadrilaterals with parallelograms in order to make conjectures about the diagonals of parallelograms.

Type: Lesson Plan

Proving Parallelograms Algebraically :

This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram.

Type: Lesson Plan

Evaluating Statements About Length and Area:

This lesson unit is intended to help you assess how well students can understand the concepts of length and area, use the concept of area in proving why two areas are or are not equal and construct their own examples and counterexamples to help justify or refute conjectures.

Type: Lesson Plan

Midpoints of the Side of a Parallelogram:

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.

## MFAS Formative Assessments

Angles of a Parallelogram:

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.

Comparing Lengths in a Parallelogram:

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.

Finding Angle C:

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.

Frame It Up:

Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.

Proving a Rectangle Is a Parallelogram:

Students are asked to prove that a rectangle is a parallelogram.

Proving Congruent Diagonals:

Students are asked to prove that the diagonals of a rectangle are congruent.

Proving Parallelogram Angle Congruence:

Students are asked to prove that opposite angles of a parallelogram are congruent.

Proving Parallelogram Diagonals Bisect:

Students are asked to prove that the diagonals of a parallelogram bisect each other.

Proving Parallelogram Side Congruence:

Students are asked to prove that opposite sides of a parallelogram are congruent.

Two Congruent Triangles:

Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.

## Student Resources

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

Midpoints of the Side of a Parallelogram:

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.