9.3.3E Properties of Polygons
Use properties of polygonsincluding quadrilaterals and regular polygonsto define them, classify them, solve problems and logically justify results.
For example: Recognize that a rectangle is a special case of a trapezoid.
Another example: Give a concise and clear definition of a kite.
Overview
Standard 9.3.3 Essential Understandings
The study of geometric figures might be considered one of the "main topics" in geometry, along with, among others, proportional thinking, reasoning and sense making. The focus of this standard is on geometric figures, with the others listed above also being involved to a great extent.
In this standard, students will work with a transversal that intersects two lines. They will study angles formed by the three lines, and determine whether or not the two intersected lines are parallel. They will see many types of triangles, including "special" triangles such as isosceles, equilateral, right, 306090 and 454590 triangles, focusing on the properties of each of these figures.
Students will study similarity and understand that congruence is a special case of similarity. They will work with scale factors in "reallife" applications of similar figures. They will look at quadrilaterals and put them into a hierarchy, with "special" quadrilaterals being "nested" within others. Also, in this standard, students will study circles and their properties, with special focus on their applications.
Benchmark Group E  Properties of Polygons
9.3.3
9.3.3.7 Use properties of polygonsincluding quadrilaterals and regular polygonsto define them, classify them, solve problems and logically justify results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 Classify quadrilaterals into a "hierarchy."
 Show knowledge that not all quadrilaterals fit into one of the categories of "special" quadrilaterals.
Work from previous grades that supports this new learning includes:
 Display familiarity with polygons, such as triangles, squares, rectangles, etc..
NCTM Standards
Geometry
 Analyze characteristics and properties of two and threedimensional geometric shapes and develop mathematical arguments about geometric relationships:
 Analyze properties and determine attributes of two and threedimensional objects;
 Explore relationships (including congruence and similarity) among classes of two and threedimensional geometric objects, make and test conjectures about them, and solve problems involving them;
 Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
 Use trigonometric relationships to determine lengths and angle measures.
 Use visualization, spatial reasoning, and geometric modeling to solve problems:
 Use geometric models to gain insights into, and answer questions in, other areas of mathematics;
 Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSM)
 HS.GCO (Congruence) Prove geometric theorems.
 HS.GCO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
 HS.GCO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
 HS.GCO.11. Prove theorems about parallelograms. 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
 HS.GSRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
 HS.GSRT.7. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
 HS.GSRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
 HS.GSRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
 HS.GC (Circles) Understand and apply theorems about circles.
 HS.GC.1. Prove that all circles are similar.
 HS.GC.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
 HS.GC.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
 HS.FTF (Trigonometric Functions) Extend the domain of trigonometric functions using the unit circle.
 HS.FTF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
Misconceptions
 Students sometimes have a hard time with the "nesting" feature of quadrilaterals, i.e., a square is nested inside parallelograms, which is nested inside quadrilaterals, which is nested inside polygons, etc.
 Students sometimes interchange properties where they shouldn't. For example, a parallelogram has diagonals that bisect each other, and some students will extend that to other types of quadrilaterals.
Vignette
In the Classroom
In this vignette, students examine quadrilaterals and their characteristics.
Note: This activity is best done in a computer lab situation. If no lab is available, the activity can be done in "demo" mode with interactive geometry software, or by having the students work at their desks with patty paper or rulers and protractors.
The files included in this activity titled "Quads lab" are Geometer's Sketchpad documents, and can be downloaded at the end of the Resources section of this Framework. If you have no access to Geometer's Sketchpad, then you can create similar documents in GeoGebra, which is a free interactive geometry program.
Insert link to download file '9.3.3 Download files', which contains sketches from Geometer's Sketchpad.
Teacher: You will be working at computers, two students per computer.
There is a folder on your desktop titled "Quads lab." When you open that folder, there will be examples of each of these quadrilaterals. Open them one at a time.
Note that when you move the individual figure on your screen, it will stay as an example of that figure. For example, when you move the "Parallelogram" figure around, it will always stay a parallelogram.
Make whatever measurements you need to make, and move the figure around as much as you need to, in order to formulate a conjecture about whether the quadrilateral MUST HAVE the property you're investigating. If it must have that property, then put a check mark or an "x" in the box. If it does not have that property, then leave the box blank.
One of you should start by doing the work on the computer while the other records the conjectures. When we're halfway done with our time in the lab, I'll have you switch duties.
Property  Parallelogram  Rhombus  Rectangle  Square  Kite  Trapezoid  Isosceles Trapezoid 
Opposite sides are parallel 







Opposite sides are congruent 







All four sides are congruent 







All angles are right angles 







Opposite angles are congruent 







Diagonals are congruent 







Diagonals bisect each other 







Diagonals are perpendicular 







Diagonals bisect opposite angles 







Students work on the sheet. When they're finished, the sheet will have the following checks:
Property  Parallelogram  Rhombus  Rectangle  Square  Kite  Trapezoid  Isosceles Trapezoid 
Opposite sides are parallel  X  X  X  X 
 One pair  One pair 
Opposite sides are congruent  X  X  X  X 

 One pair 
All four sides are congruent 
 X 
 X 



All angles are right angles 

 X  X 



Opposite angles are congruent  X  X  X  X  One pair 


Diagonals are congruent 

 X  X 

 X 
Diagonals bisect each other  X  X  X  X  One of them 


Diagonals are perpendicular 
 X 
 X  X 


Diagonals bisect opposite angles 
 X 
 X  One pair 


Note: In this activity, a trapezoid is defined as "a quadrilateral with exactly one pair of parallel sides." In some geometry courses, a trapezoid is defined as "a quadrilateral with at least one pair of parallel sides." The definition used will not cause a change in the checks in the boxes above.
Resources
Teacher Notes
 Create a Venntype diagram to show the "nesting" of quadrilaterals.
 Emphasize the properties as students are filling in the chart.
Diagonals to Quadrilaterals
Instead of considering the diagonals within a quadrilateral, this lesson provides a unique opportunity: Students start with the diagonals and deduce the type of quadrilateral that surrounds them. Using an applet, students explore certain characteristics of diagonals and the quadrilaterals that are associated with them.
Polygon Capture
In this lesson, students classify polygons according to more than one property at a time. In the context of a game, students move from a simple description of shapes to an analysis of how properties are related.
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: NCTM, 2010.
New Vocabulary
trapezoid: a quadrilateral with exactly one pair of parallel sides.
isosceles trapezoid: a trapezoid with congruent legs.
kite: a quadrilateral with two pair of distinct congruent consecutive sides.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks
 What other instructional strategies can be used to engage students with properties of triangles?
 How can manipulatives be used to help students visualize these abstract concepts?
 How can the instruction be scaffolded for students?
 What additional scaffolding should be provided to ELL students?
 Do the tasks that have been designed connect to underlying concepts or focus on memorization?
 How can it be determined if students have reached this learning goal?
 How can the lesson be differentiated?
Assessment
Note to teachers: This performancebased assessment is best done in a computer lab setting. If one is not available, then the activity could be done in "teacher demo" mode, or using worksheets with different quadrilaterals printed on them.
Midpoint quadrilaterals lab
Name ______________________________ Period _______
In this lab, you will be working with quadrilaterals and the figures formed when you connect the midpoints of the sides.
1. On your screen, construct a convex quadrilateral. Next, construct the midpoints of each of the sides. Then, connect the midpoints with segments, forming a new quadrilateral. We will call this the "midpoint quadrilateral."
To help avoid confusion, change the color of the segments of the midpoint quadrilateral.
Draw a sketch of your figure in the space below.
2. As you move vertices and/or segments of your original quadrilateral, make a conjecture about what appears to be true about the midpoint quadrilateral. What measurements can you make to support your conjecture? What theorems, postulates, definitions or previous knowledge can you use to justify your conjecture?
Call your teacher over to discuss your conjecture before moving on to the next step.
3. Move a vertex of your original figure so that it is a nonconvex polygon. Is your conjecture from number 2 above still true? Why?
4. Move a vertex of your original figure so that it isn't a polygon, but rather two triangles with a common vertex. Is your conjecture still true? Why?
5. By now, it is clear that the midpoint quadrilateral will always be a parallelogram.
Is there a way to manipulate the original quadrilateral so that the midpoint quadrilateral is a rectangle? If so, how? What type of figure must your original quadrilateral be, in order to make this happen? Draw a sketch below.
6. Is there a way to manipulate the original quadrilateral so that the midpoint quadrilateral is a rhombus? If so, how? What type of figure must your original quadrilateral be, in order to make this happen? Draw a sketch below.
7. Is there a way to manipulate the original quadrilateral so that the midpoint quadrilateral is a square? If so, how? What type of figure must your original quadrilateral be, in order to make this happen? Draw a sketch below.
DOK Level: 4
Assessment hints/solutions:
1. My "original figure" is GEOM, in blue, and my "midpoint quadrilateral" is QUAD, in red.
2. I think it's a parallelogram. I drew in the diagonals, segment EM and segment GO, and turned them green, to show them more easily.
Because of the Midsegment Theorem, segment AD and segment QU are each half the length of segment EM, so they must be congruent to each other. Also, segment UA and segment QD are each half the length of segment GO, so they're congruent to each other. Since both pairs of opposite sides are congruent, QUAD must be a parallelogram.
Note: Many students will measure the sides (or angles) of the midpoint quadrilateral and find that both pair of opposite sides (or angles) have the same measure when they move the figure around. While this is certainly a convincing argument, especially as the figure is moved around on the screen, it wouldn't be considered a "proof." The level of rigor required in this case is left to the individual teacher.
Remind students to call you over to discuss their conjectures before moving on to the next step.
3.
It is still a parallelogram, for the same reason it was a parallelogram when GEOM was a convex polygon.
4.
It is still a parallelogram, for the same reason it was a parallelogram when GEOM was a convex polygon.
5.
In this case, the midpoint quadrilateral is a rectangle. I did a lot of measuring and couldn't find anything in particular about the original figure, but then I measured angle EIO and found that to be a 90degree angle, so the diagonals are perpendicular. This makes sense, due to the Midsegment Theorem.
6.
In this case, the midpoint quadrilateral is a rhombus. At first, it looked like my original figure was a trapezoid, but I measured the slopes of segment GM and segment EO, and it turns out they aren't equal, so I know those segments aren't parallel.
Then I looked at the diagonals. Angle EIO isn't a right angle, so the diagonals aren't perpendicular, but I measured their lengths, and they're the same, so the diagonals are congruent. This makes sense, too, because the lengths of the sides of the midpoint quadrilateral are half the length of the sides of the diagonals which are parallel to them, so in order for them all to be congruent, the diagonals have to be congruent.
7. I looked at the two previous questions, just above, and figured that since a square is a rectangle and a rhombus, that I should just combine the two requirements for those, so that the diagonals must be perpendicular and congruent.
Then I figured that if the diagonals were perpendicular and congruent, the original figure must be a square, because the diagonals of a square are perpendicular and congruent.
That turned out to not be the case. I have an example of a figure where the midpoint quadrilateral is a square, but the original figure is not.
Note to teachers: This is easier to see if you start with a sketch in which two segments (which will end up being the diagonals of the original figure) are perpendicular and congruent, but which DO NOT intersect at their midpoints. Then, make your "original figure" from the endpoints of those diagonals, and the midpoint quadrilateral of that figure will be a square.
Differentiation
Be sure the chart (with properties of special quadrilaterals) is correctly filled in and handy during the Assessment.
Keep the definitions of the different quadrilaterals handy, so students can refer to them during the activity.
Have students determine what are necessary and sufficient conditions for the midpoint quadrilateral to be a special quadrilateral. Determine the type of midpoint quadrilateral formed in each of the special quadrilaterals.
Parents/Admin
Students are: (descriptive list)  Teachers are: (descriptive list) 
using computer software to determine properties of quadrilaterals.  observing and helping when necessary. 
discussing results with a partner and/or another pair/group.  asking guiding questions, especially of students who are struggling. 
Parent Resources
 Sophia geometry lessons
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 Khan Academy geometry lessons
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 Teacher Tube, YouTube
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 Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.