Polygon Angle Sum Theorem

Target Audience: 9th

Grade Geometry Class

NCTM Standards:

Analyze characteristics and properties of two- and three-dimensional geometric shapes

and develop mathematical arguments about geometric relationships

Use visualization, spatial reasoning, and geometric modeling to solve problems

Common Core Standards:

Make formal geometric constructions with a variety of tools and methods (compass and

straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.)

Maryland State Curriculum:

G.2.The student will demonstrate the ability to solve mathematical and real-world

problems using measurement and geometric models and will justify solutions and explain

processes used.

G.2.2. The student will apply geometric properties and relationships to solve problems

using tools and technology when appropriate.

Before the Lesson

Objective: The students will be able to find the sum of the measure of interior angles of

polygons and discover the Interior Angle Sum Theorem and how it works.

Prerequisite Knowledge: The students know how many sides each of the regular polygons

have up till a 12-sided polygon. They also just learned how to measure an angle using a

protractor in the previous class. The students also know how to use their graphing calculators

and the day before this class, the students watched a tutorial on how to use the Cabri Jr.

Application. In Cabri Jr. Application students know how to construct a triangle, quadrilateral,

pentagon, hexagon, heptagon, octagon, nonagon, and decagon.

Materials:

TI-84 Plus Silver Addition

Cabri Jr. Application

Drill

Exit Ticket

Discovering the Polygon Angle Sum Theorem

Mathematical Vocabulary Needed:

Convex polygons- a polygon is convex if it contains all the line segments connecting any

pair of its points (Wolfram MathWorld).

Angle- given two intersecting lines or line segments, the amount of rotation about the

vertex required to bring one into correspondence with the other is called the angle

between them (Wolfram MathWorld).

Polygon diagonal- a line segment connecting two nonadjacent polygon vertices of

a polygon8

.77777777777777777777777777777777777777777777777777777777777777777777777

77777777777777777777. (Wolfram MathWorld)

Drill: (5-8 minutes) The students will be given 2 different angles they will have to measure using

a protractor. The students will also be given 3 different regular polygons they must identify.

After about 5 minutes the students will be asked to share with the person sitting next to them.

The Drill should serve as an evaluation for the teacher to see what the student’s remembered

from last class.

The Lesson- Polygon Angle Sum Theorem

Engagement: This will be a continuation of the Drill.

Once the students are done sharing with the person sitting next to them, the teacher will go

through the answer to each problem. He/she will ask the students to give a “thumbs up” if they

got the right answer or a “thumbs down” if they got the wrong answer. The teacher should pay

attention and go over concepts if lots of students gave a “thumbs down.”

The teacher will then go on to explain the lesson. The students know how to take angle

measures but the teacher should ask the students if there is a way to know the sum of the angles

in a regular polygon without spending the extra time to use a protractor. The teacher will then go

on to explain that today the students will be discovering the Polygon Angle Sum Theorem.

Exploration: (15-20 min) The teacher should instruct the students to take out their TI-84 Plus

Silver Edition graphing calculator and open the Cabri Jr. Application. He/she should send the

students the Cabri Jr. file that has all the polygons needed to complete this lesson. The teacher

will hand out a worksheet that the students will be required to complete. They should work with

a partner to do this. The worksheet is attached. The teacher should do the first 2 examples,

triangle and quadrilateral, with the students. But after that the students are on their own, with

their partners, to fill out the rest of the worksheet. The teacher should be walking around the

room questioning students and making sure they are on task and understanding the lesson. The

worksheet requires the students to measure each angle of the polygon and find the total angle

measure for the polygon. To measure an angle students do the following: press F5, scroll down

to MEASURE, press right arrow key, scroll down to ANGLE, press ENTER, select three

vertices, and the angle measure will be calculated for the middle vertex. Do this for all the

vertices.

Explanation: (15-20 min) The teacher should reconvene the class after about 15-20 minutes.

The same charts the students filled out will be on the Elmo. The teacher should call on students

to help fill out the chart for the whole class to see. At each column’s answer, the teacher will ask

for a “thumbs up” or “thumbs down” to confirm the answer. The teacher will complete this until

all of the blanks are filled in on the chart.

At this point, hopefully some of the students are beginning to see a pattern in the angle measures.

However to help them see the pattern, construct some of the polygons on the calculator using

Cabri Jr. The students should follow along as the teacher explains how he/she will be using

diagonals to construct triangles within the polygons. The teacher will define the term, polygon

diagonal as a line segment connecting two nonadjacent polygon vertices of a polygon. The

teacher should also point out that the diagonals should not intersect for this activity. The

following steps show what the teacher should do using Cabri Jr.

1. Construct a quadrilateral

2. Construct a diagonal to divide the quadrilateral into 2 triangles. This can be done in

Cabri Jr by pressing F2, scroll down to SEGMENT, put cursor on a vertex, press

ENTER, drag cursor to the adjacent vertex, press ENTER

3. Point out that we know 1 triangle has an angle sum measure of 180 . Ask the students

how many triangles can this quadrilateral be divided into? (Answer is 2)

4. and we found the angle sum measure of a quadrilateral to be 360

Do another an example of a pentagon with the students.

1. Construct a pentagon

5. Construct the diagonals to divide the pentagon into 3 triangles. This can be done in Cabri

Jr. by pressing F2, scroll down to SEGMENT, put cursor on a vertex, press ENTER, drag

cursor to the adjacent vertex, press ENTER, do this for the other diagonal.

2. Point out that we know 1 triangle has an angle sum measure of 180 . Ask the students

how many triangles can this pentagon be divided into? (Answer is 3)

3. and we found the angle sum measure of a pentagon to be 540 .

Now ask students to think about the relationship between the number of sides of the polygon and

the number we multiplied by the constant 180 .

Extension: (10-15 min) Have the students make predictions about what they think the Polygon

Angle Sum Theorem may be. Remind students that we now know there is a constant of 180

being multiplied by something. Give them 2-4 minutes to think to themselves. Ask for any

volunteers to share their prediction. Hopefully students will see that they multiply the constant

of 180 by the number of sides minus 2. If they do not see this, the teacher should go back to the

quadrilateral and pentagon made on Cabri Jr. The teacher should ask the class how many sides

the quadrilateral has (4). The teacher should then point out that we wrote and

that . The teacher should then do the same thing with the pentagon and point out that

, which is what we multiplied by 180 to get 540. By now the students should see that

when equals the number of sides of the polygon, the equation of the angle sum of a polygon is

. Have students write this Theorem down in their notes section of their math

journals.

Evaluation: (5-8 min) An exit ticket will act as an evaluation of the lesson. The students will be

given 2 different figures (11 and 12-sided). The students will have to divide the polygons up into

triangles and then calculate the angle sum measure using the form . The teacher

should look over the student’s responses and reflect on the lesson. The students should construct

the 11-sided polygon in Cabri Jr. along with writing their answers on the exit ticket. They don’t

have to construct the 12-sided polygon, they can just write their answer on the exit ticket.

Drill Measure the following angles using your protractor. Record your answer in the space provided.

Identify each of the following regular polygons. Records you answer in the space provided.

Drill Answers Measure the following angles using your protractor. Record your answer in the space provided.

Identify each of the following regular polygons. Records you answer in the space provided.

pentagon octagon rectangle

Discovering the Polygon Angle Sum Theorem Triangle.

1. Select: ALPHA, WINDOW, TRIANGLE 2. Click Enter, Drag cursor to another spot, Click Enter, Drag cursor to another spot, Click Enter

A triangle has now been constructed. 3. Measure each of the angles. Select: ALPHA, GRAPH, MEASURE, Right Arrow 4. Click the vertex of each angle (you should click a total of 3 vertexes to measure the first angle),

the measure of the angle with the middle vertex will then be displayed. 5. Repeat step 3 and 4 until the other two angles are measured.

All three angle measures are now displayed. 6. Calculate the angle sum. Select ALPHA, GRAPH, CALCULATE 7. Move your cursor over an Angle Measure until a left arrow appears, Click Enter, Click , move

your cursor over another Angle Measure until a left arrow appears, Click Enter, Click , move your cursor over the last Angle Measure until a left arrow appears, Click Enter. The Angle Sum will appear.

Fill in the number of sides column and the angle sum column on the chart below.

Using Cabri Jr. work with your partner to fill out the rest of the chart.

Convex Polygon Number of Sides Sum of Angle Measure

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

With your partner, look for a pattern in the sum of angle measures related to the number of sides.

Chart Answers

Convex Polygon Number of Sides Sum of Angle Measure

Triangle 3 180

Quadrilateral 4 360

Pentagon 5 540

Hexagon 6 720

Heptagon 7 900

Octagon 8 1080

Nonagon 9 1260

Decagon 10 1440

Exit Ticket Divide the following polygons into triangle and then find the angle sum measure of the polygon using the form .

Exit Ticket Answers Divide the following polygons into triangle and then find the angle sum measure of the polygon using the form .