1.6: Polygons and Quadrilaterals
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Angles in Polygons Investigation 6-1 Investigation 6-2 |
Properties of Parallelograms Investigation 6-3 |
Quiz 1 Start Proving Quadrilaterals are Parallelograms |
Finish Proving Quadrilaterals are Parallelograms |
Rectangles, Rhombuses, and Squares Investigation 6-4 Investigation 6-5 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Quiz 2 Start Trapezoids and Kites |
Finish Trapezoids and Kites Investigation 6-6 |
Quiz 3 Start Review of Chapter 6 |
Finish Review of Chapter 6 | Chapter 6 Test |
Angles in Polygons
Goal
Students will use the Triangle Sum Theorem to derive the Polygonal Sum Theorem by dividing a convex polygon into triangles. Students will also be reintroduced to the Exterior Angle Sum Theorem, but now it will be applied to any polygon.
Teaching Strategies
Using the Know What? at the beginning of this lesson, discuss where someone might see polygons in nature and the real world. Determine if any of these polygons are regular polygons or not. Students might need a review of the definition of a regular polygon.
Investigation 6-1 is intended to be a student-driven activity while the teacher monitors and leads or answers questions. Students should know what a quadrilateral, pentagon, and hexagon are from Chapter 1; however they may need a little review. Diagonals were also addressed in Chapter 1. Make sure that students only draw the diagonals in step 2 from one vertex so that none of the triangles overlap. In step 3 it might be helpful to write out the last column as a list, including the triangle; \begin{align*}180^\circ, 360^\circ, 540^\circ, 720^\circ\end{align*}. Then, ask students what the next number in the pattern should be. Continue this for a few more terms and then generalize into the Polygon Sum Formula.
Students might think that the Polygon Sum Formula and the Equiangular Polygon Formula are two different formulas for them to memorize. This is not the case. Tell students that they need to memorize the Polygon Sum Formula and then the Equiangular Polygon Formula simply divides the Polygon Sum Formula by the number of angles in the polygon. Stress to students that the Equiangular Polygon Formula can be used on equiangular polygons as well as regular polygons.
To introduce exterior angles for polygons, draw a triangle with its exterior angles. Students should remember that each set of exterior angles of a triangle add up to \begin{align*}360^\circ\end{align*}. Then, show students the exterior angles for a square. Each exterior angle is \begin{align*}90^\circ\end{align*}, so their sum would be \begin{align*}360^\circ\end{align*} as well. Now, have students complete Investigation 6-2 in pairs.
The first question in the review questions is a table with angle sums and individual angles in a regular n-gon. Complete this table at the end of the lesson so that students see the relationship between all the angles and their sums. If you would like, add a final column labeled “Each exterior angle in a regular n-gon.” Students can either use linear pairs with column 4 or divide \begin{align*}360^\circ\end{align*} by the number of sides.
Properties of Parallelograms
Goal
The purpose of this lesson is to familiarize students with properties special to parallelograms.
Notation Note
The notation for any quadrilateral or is the list of vertices, usually clockwise, such as \begin{align*}ABCD\end{align*} (quadrilateral) or \begin{align*}LMNOP\end{align*} (pentagon). Students can start at any vertex they would like. The only requirement is that the vertices are listed such that they are next to each other in the picture. Quadrilateral \begin{align*}ABCD\end{align*} could be:
Notice that the first vertex listed and the last vertex listed are next to each other.
Teaching Strategies
Investigation 6-3 enables students to discover the properties of parallelograms on their own. Encourage students to label the vertices of the parallelogram (\begin{align*}ABCD\end{align*}, for example) and then they can write equality/congruence statements for the angles and sides. Once they have all the equality/congruence statements written (step 3), have students write down all their conclusions and see if they can generate the parallelogram theorems on their own.
Students might wonder if there is a difference between consecutive angles and same side interior angles. Discuss this with your students. One could argue they are the same. Another could argue they are different because consecutive refers to two angles that are next to each other in a polygon. Same side interior angles refer to two angles that are formed by parallel lines and one transversal.
Encourage students to make as many connections as possible. For example, students have learned parallel lines are equidistant from each other. Make this connection to a parallelogram. If students drawn in the diagonals of a parallelogram; review that alternate interior angles are congruent.
In Example 3, we place the parallelogram in a coordinate plane. In this chapter we will show that certain quadrilaterals are parallelograms (and rhombuses, squares, and rectangles) when they are in the coordinate plane. In this section we introduce one method. Because the diagonals of a parallelogram bisect each other, their point of intersection should be the midpoint of each. Therefore, the midpoint of each diagonal will be the same point.
Proving Quadrilaterals are Parallelograms
Goal
Students will use triangle congruence postulates and theorems to prove quadrilaterals are parallelograms. Students will also determine if a quadrilateral is a parallelogram when placed in the coordinate plane.
Teaching Strategies
There are two main ways to prove that a quadrilateral is a parallelogram: formal proof and using the coordinate plane. If students are given a formal proof, they must prove that the two halves of a parallelogram (split by one of the diagonals) are congruent and then use of the converses used in this lesson. They will also have to use CPCTC somehow.
If students are doing a problem with a quadrilateral in the coordinate plane, then they must use the distance formula, slope formula, or midpoint formula. The distance formula would lend itself to the Opposite Sides Theorem Converse, and finding the slopes of all four sides is the definition of a parallelogram and the midpoint formula is the Parallelogram Diagonals Theorem Converse. Discuss all these options with students before allowing them to start class work or homework.
As a way to introduce the proof of the Opposite Sides Theorem Converse and the proof of Theorem 6-10 (Example 1), you can copy these theorems and then cut up the statements and reasons and put them in an envelope. Given an envelope to either pairs or groups of students and have them match up the statements with the corresponding reasons and put them in the correct order. On one side of the envelope, put the Given and Prove as well as the picture. This technique can be done for any proof.
Rectangles, Rhombuses and Squares
Goal
This lesson introduces rectangles, rhombuses, and squares. These are more specific types of parallelograms.
Teaching Strategies
Make sure students understand that everything that falls within a rhombus possess the same characteristics and properties of a parallelogram. A rectangle also has all the properties of a parallelogram as well as its properties. A square has all the properties of a rhombus, rectangle and parallelogram. Squares do not have any of its own unique properties.
Investigation 6-4 shows us that the diagonals of a rectangle are congruent. So, if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Investigation 6-5 explores the properties of the diagonals of a rhombus. Here, the diagonals are perpendicular and they bisect each angle in the rhombus. Note that students do not need to show both to be a rhombus. However, if they decide to use Theorem 6-16, they do need to show that the diagonals bisect all four angles of the rhombus. Therefore, it is much easier for students to use Theorem 6-15 (showing that the diagonals are perpendicular) to show a parallelogram is a rhombus.
Because there are no theorems regarding squares, make sure you go over Example 4 thoroughly. This is the only place we discuss all the properties of squares. Stress that a square holds all the properties of a parallelogram, rectangle, and rhombus. Ask students if they think it has any of its own properties. A square is a specific version of a rhombus, so the angles are bisected. Because a square is also a rectangle, the angles are all \begin{align*}90^\circ\end{align*}, so the bisected angles are all \begin{align*}45^\circ\end{align*}. In addition to the properties listed in Example 4, show students other ways to write some of these properties. For example, a rectangle has four congruent angles. Students can also write this as the sides are all perpendicular to each other.
At this point, students might have the different types of parallelograms confused. There are a lot of different properties and students might have them all mixed up in their heads. One way to help is to draw a hierarchy diagram with QUADRILATERALS at the top and then two arrows. One points to PARALLELOGRAMS and the other points to OTHER TYPES. Then, ask students if they can determine how rectangles, squares, and rhombuses fit into this diagram. The final diagram should look like:
Students can add to this diagram in the next lesson.
The steps after Example 5 are very helpful for students to determine if a parallelogram is a rectangle, rhombus or square. Here is an additional example, using these steps.
Additional Example: The vertices of \begin{align*}WXYZ\end{align*} are below. Determine the type of quadrilateral it is.
Solution: Let’s follow the steps.
1. The quadrilateral is graphed.
2. Do the diagonals bisect each other?
\begin{align*}midpoint_{WY} & = \left ( \frac{-8+2}{2}, \frac{3-7}{2} \right ) = (-3,-2)\\ midpoint_{XZ} & = \left ( \frac{0-6}{2}, \frac{1-5}{2} \right ) = (-3,-2)\end{align*}
The diagonals bisect each other. This is a parallelogram.
3. Are the diagonals equal?
\begin{align*}WY & = \sqrt{(-8-2)^2 + (3-(-7))^2} = \sqrt{10^2 + 10^2} = \sqrt{200}\\ XZ & = \sqrt{(0-(-6))^2 +(1-(-5))^2} = \sqrt{6^2 + 6^2} = \sqrt{72}\end{align*}
The diagonals are not equal. This is not a rectangle.
4. At this step, we know this figure is a rhombus. It cannot be a square because the diagonals are not equal, from step 3. So, to prove that it is a rhombus, we can either show that all the sides are equal or that the diagonals are perpendicular. It is easier to find the slopes of the diagonals than do the distance formula four times.
\begin{align*}m_{WY} = \frac{3-(-7)}{-8 - 2} = \frac{10}{-10} = -1 && m_{XZ} = \frac{1-(-5)}{0-(-6)} = \frac{6}{6} = 1\end{align*}
The diagonals are perpendicular, so this reaffirms our earlier conclusion that \begin{align*}WXYZ\end{align*} is a rhombus.
Trapezoids and Kites
Goal
This lesson introduces students to the special properties of kites, trapezoids, and isosceles trapezoids.
Teaching Strategies
After going over the definition of a trapezoid, discuss the difference between a trapezoid and a parallelogram. Students need to realize that a trapezoid has exactly one pair of parallel sides. It is not like the definition of an isosceles triangle (at least two congruent angles). Therefore, a parallelogram is not a trapezoid.
For a trapezoid to be isosceles, the non-parallel sides must be congruent. Describe an isosceles trapezoid as cutting off the top of an isosceles triangle. Therefore, Theorem 6-17 is a form of the Base Angles Theorem for isosceles triangles. Isosceles trapezoids also have congruent diagonals, like a rectangle. Ask students if isosceles trapezoids are rectangles. This is a great example of two different quadrilaterals that have the same property. Of course, rectangles are not isosceles trapezoids because rectangles have four congruent angles and two sets of parallel sides.
Just like triangles, a trapezoid also has a midsegment. Trapezoids only have one midsegment because it connects the non-parallel sides. For this reason, the midsegment is also parallel with the parallel sides. Stress that the length of the midsegment is the average of the lengths of the parallel sides. You could also say that the midsegment is halfway between the parallel sides, so its length is halfway between the lengths of the parallel sides.
Kites are very similar to rhombuses, but a rhombus is not a kite. The definition of a kite says “a quadrilateral with two sets of adjacent congruent sides.” All sides are congruent in a rhombus. Because a kite has two sets of congruent adjacent sides, it has some properties of rhombuses. Go over the similarities and differences between kites and rhombuses. At the end of this lesson, complete the hierarchy diagram that was started in the previous lesson.
Review Game
Have your students create flashcards with quadrilateral names on one side and important information or properties on the reverse. Have various types of quadrilaterals, both abstract and real world, ready to show students. Once students believe they have classified the quadrilateral, they are to hold up the appropriate name. Have students do this in pairs and keep score. You can give the winners bonus points on the test, candy, or a homework pass.
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