# Properties of Quadrilaterals

Properties of Quadrilaterals will be observed in this project, you will use the Focus Question on the following slide to develop the Teach aspect for this Learn-Use-Teach sequence of this module.

Your Teach moment will involve creating your own proof about quadrilaterals using the information about congruent triangles that you learned in module 4.

Upload your completed Teach project at the bottom of the last slide of this project.

Check your calendars for the date and time of the Collaboration Session set up by your instructor for when you will make your individual or group presentation to the rest of the class.

For further instructions on how to complete this portion of the module, contact your instructor or consult your Learning Guide.

Focus Question: How are congruent triangles used to prove some of the properties of special quadrilaterals?

Follow these directions to answer the Focus Question.

Read

Review information on quadrilaterals and congruent triangles.

Research

Find the definitions of some special quadrilaterals. Use interactive websites or geometry software to conjecture additional properties for these quadrilaterals. Review an online proof that uses both the definitions and congruent triangles.

Respond

Begin by writing precise definitions for several special quadrilaterals. Organize the properties that these quadrilaterals appear to possess. Next, consider several theorems and prove some of the properties of quadrilaterals. Finally, apply your understanding to answer the Focus Question.

What makes one quadrilateral different from another?

A quadrilateral is a four-sided polygon. Since it is a polygon, all the sides of a quadrilateral are coplanar. The intersection of any two sides is called a vertex. The two segments joining opposite vertices are diagonals.

There are several specific quadrilaterals with special shapes and characteristics. Among them are the trapezoid, parallelogram, rectangle, rhombus, and square. Although they have common characteristics, these quadrilaterals have distinct differences. Is a square a rectangle? Is a rhombus a square? These questions can be confusing. Unless you are familiar with the precise definitions and the theorems associated with these special quadrilaterals, you can easily make incorrect deductions when you hear or use their names.

What does CPCTC stand for and how is it used?

CPCTC represents the sentence “Corresponding Parts of Congruent Triangles Are Congruent.” It is often abbreviated as CPCTC or as “corr parts of ≅ Δs are ≅” in proofs. It is a restatement of the definition of congruent triangles, which is “Two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.”

For instance, if△ABC≅△XYZ, then the definition of congruent triangles or CPCTC would indicate that each of the following statements is true.

Within a proof and after the statement of two congruent triangles, the CPCTC statement is used as the reason for a corresponding pair of angles or sides to be listed as congruent. Notice how it is used in the following proof.

Research

Site 1

AlgebraLab Glossary : http://www.algebralab.org/Glossary/glossary.aspx

• Find the definitions of commonly used geometric terms.

• Use the Read page and this site to write the definitions requested in problem 1 on the Respond sheet.

Site 2

Explore a Paralelogram: http://www.regentsprep.org/Regents/math/geometry/GP9/JavaParallel.htm%20

• Scroll down to the Java Sketchpad applet. When you manipulate vertex A or B of the parallelogram, the measurements of the angles and sides can be observed.

• Move the vertices and note the congruent angles and sides and their relationships.

• In Respond question 3, complete the parallelogram column of the table.

Site 3

Explore a Rectangle: http://www.regentsprep.org/Regents/math/geometry/GP9/JavaRectangle.htm%20

• Scroll down to the Java Sketchpad applet. When you manipulate vertex A, B, or D of the rectangle, the measurements of the angles and sides can be observed.

• Move the vertices and note the congruent angles and sides and their relationships.

• In Respond question 3, complete the rectangle column of the table.

Site 4

Explore a Rhombus: http://www.regentsprep.org/Regents/math/geometry/GP9/JavaRhombus.htm%20

• Scroll down to the Java Sketchpad applet. When you manipulate vertex A, B, or D of the rhombus, the measurements of the angles and sides can be observed.

• Move the vertices and note the congruent angles and sides and their relationships.

• In Respond question 3, complete the rhombus column of the table.

Site 5

Explore an Isosceles Trapezoid: http://www.regentsprep.org/Regents/math/geometry/GP9/JavaTrapezoid.htm%20

• Scroll down to the Java Sketchpad applet. When you manipulate point A, B, or D of the trapezoid, the measurements of the angles and sides can be observed.
Note: This site incorrectly references the figure as a rhombus one time.

• Move the vertices and note the congruent angles and sides and their relationships.

• In Respond question 3, complete the isosceles trapezoid column of the table.

Site 6

Math Warehouse: CPCTC: http://www.mathwarehouse.com/geometry/congruent_triangles/congruent-parts-CPCTC.php

• Scroll down to the third proof involving the diagonals of a rhombus.

• Review what is given, what you must prove, and the accompanying figure.

• Click Next to begin the proof. Try to anticipate the correct statement or reason before clicking Next after each step.

• Notice how this proof uses congruent triangles and Corresponding Parts of Congruent Triangles Are Congruent (CPCTC) in two steps.

• Study the entire proof and answer the questions in problem 8.

Respond: Questions 1 and 2

Follow the directions below to complete questions 1–2.

1) Use Site 1 and refer to the Read screens to write complete definitions for each of the following. Include a drawing, marking appropriate angles and/or sides as definition warrants.

2) Compare these definitions to those used in your textbook or by your class. Adjust as necessary. I will do that

Upload your work:

Respond: Questions 3 and 4

3) Copy the table. For each special quadrilateral listed, place an X to the right of the property if it is true according to the following criteria.

A) First use only the definition for each figure you wrote in problem 1.

B) Use Sites 2–5 to investigate the properties of a parallelogram, rectangle, rhombus, and isosceles trapezoid. During your investigation, place an X by any additional property that appears to be true. You will complete the column for a square in question 4 below.

◦ Begin by looking for patterns and relationships between angles and sides.

◦ Next, study the relationship between the two diagonals and between the diagonals and angles.

4) At Site 4, drag point B to create a square. Observe the measurements for the sides and angles. Complete the column for the properties of a square.
- What other types of quadrilaterals can be used to describe a square?

– What do you notice about the properties of a square in relation to these other quadrilaterals?

Upload your completed table:

Respond: Question 5

5) Are there other properties that are not listed in the table that appear to be true for these quadrilaterals? If so, list them below.

Respond: Questions 6

6) Draw a Venn diagram that shows the relationship between parallelograms, rectangles, rhombi, squares, trapezoids, and isosceles trapezoids.

Upload your diagram:

Respond: Question 7

7) At Site 6, investigate a proof that uses one of these special quadrilaterals. Note how congruent triangles and Corresponding Parts of Congruent Triangles (CPCTC) are used in this two-column format proof.

A) What definition is used as a reason?

B) Which pair of triangles are proven congruent?

Respond: Question 8

8) Many of the properties from the previous table are stated in the theorems we generally associate with quadrilaterals. Select two of the following theorems to prove. Select a figure to use from those given. Then state the given and prove in terms of the figure and the theorem. Use congruent triangles and CPCTC to prove the theorems. Use a traditional two-column format for your proof unless otherwise instructed by your teacher.
Note: Any theorem you prove in this list can be used in the proof of any of the other theorems.

A. The opposite sides of a parallelogram are congruent.

B. The opposite angles of a parallelogram are congruent.

C. The diagonals of a parallelogram bisect each other.

D. The diagonals of a rectangle are congruent.

E. The diagonals of a rhombus are perpendicular.

Upload your proofs. Be sure your proofs are clearly labeled.

Conclusion

9) How are congruent triangles used to prove some of the properties of special quadrilaterals? Why is it important to know which properties are stated in the definition and which properties usually appear as theorems?

Create Your Own Proof

Create your own proof that utilizes congruent triangles and CPCTC.

Requirements for your proof:

• Include the given, prove, and image.

• Use at least six steps.
(Note: the given counts as one step only.)

• Use congruent triangles in order to prove something additional.

• Use various theorems, postulates, and definitions that you have learned throughout the course.

• Include the solution to the proof.

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