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Introduction :
By using the Geometers Sketchpad software, students will create and manipulate
triangles, measure the interior angles, and find the sum of the interior angles in a triangle.
The students then are required to repeat the process for quadrilateral, pentagon and
hexagon. Students will discover a relationship between the number of sides of a polygon
and the numbers of triangles that can be create inside the polygons. Students will also
discover a relationship between the number of triangles in the polygons and the sum of the
interior angles of the polygons. In the end, students find the formula for the sum of the
interior angles of polygon and use the formula to solve a problem.
Topic : Polygons II (Mathematics Form 3 KBSM)
Learning Objective : Understand and use the knowledge of exterior and interior
angles of polygons.
Learning Outcome : At the end of the lessons, students will be able to
i. Determine the sum of the interior angles of polygons.
ii. Find the size of an exterior angle when the interior of a
polygon is given and vice versa.
iii. Determine the sum of the exterior angles of polygons.
Directions :
i. Form a group of two or three students. Each group will share a computer.
ii. Follow the steps given below and use the diagrams a reference.
iii. Do all the activity that has been listed.
iv. Complete the worksheet attached.
STEPS:
1. Start the Geometers Sketchpad software. Youll see a blank screen with icons at the
left side of it.
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2. Click on the segment tool and go anywhere in the sketching region (white space).
Click and drag the arrow to the desired place and let go. The segment should be
highlighted in pink.
3. Next, click anywhere on the sketchpad and drag the arrow to one of the endpoints
of the segment.
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4. Then, click at one of the endpoints of the segments and drag the arrow to the other
endpoints so that it will looks like a triangle.
5. Click the selection arrow tool. Then click anywhere on the sketchpad so that the
segments that were highlighted in pink will disappear.
6. Move the cursor to one of the vertices until it turns horizontal and click on it. Then
in clockwise or anticlockwise, choose the other two vertices.
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7. Click Measure at the top of the screen and then select Angle. There will be
automatic label on each vertices and an angle indicated to the second vertices.
8. Click on the white space so that there are no items that highlighted in pink. Repeat
steps 6-8 for the other two vertices until you get two other angles.
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9. Click on the white space. Then click Measure before selecting Calculate. A
calculator will appear on the screen. To calculate the sum of the interior angles of
a triangle, click on the first angle and then click the plus sign on the calculator.
Repeat for the other two angles and clickOK. Then, please answer question 1 on
the worksheet. Complete the third columns on the table of questions 3 in the
worksheet.
10.Click on the white space. Then click at one of the vertices and move it anywhere
so that the shape of the triangle is change. Repeat the process for a few times.
Then, make a conclusion about the sum of interior angles of any triangles and
write it in the worksheet for question 2.
11.To explore the exterior angles of the triangle, you must extend the segments of
the triangle to form rays.
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12.First, choose any vertex. Let say, you choose vertex B. You must choose to draw
either ray or . Let say you want to draw ray . Click on point A, and thenpoint B. From the construct menu, select Ray.
13.Click anywhere in the Blank space to deselect ray . Repeat step 12 for the othertwo vertices.
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14.Select the Point tool and click at each ray so that there will be a new point on eachray.
15.Click Selection tool, and choose all three new points. From the Display menu,select Label point...
16.There will be a pop-up window ofLabel Multiple Object. Change the first labelto P and clickOk.
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17.So, you have drawn an exterior angle of PCA, QAB and CBR. Note that oneinterior angle is adjacent to each exterior angle.
18.To find the measures of all the exterior angles, select point P, C and A in thatorder. The three point are highlighted. From the Measure menu, select Angle.
Repeat the process for the other two exterior angles.
19.Identify the pair of interior angle and its exterior angle at each vertex. Completethe table for the question 9 in the worksheet.
20.To calculate the sum of interior angle and exterior angle at each vertex, fromMeasure menu, select Calculate. Click the interior angle and click the plus (+)
sign. Then click the corresponding exterior angle before press Enter.
21.By using the same step as above, calculate the sum of the exterior angles of atriangle and complete the table for question 11.
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ACTIVITY
The activity is for the students to investigate the relationship between the sum of interior
angles for triangles and the sum of interior angles of polygons (quadrilateral, pentagon,
hexagon, heptagon, octagon, etc).
Quadrilateral
1. ClickFile and select New Sketch. A new blank sketchpad will open.
2. Follow steps 2-9 but this time draw a quadrilateral where it has 4 segment or 4 sides.
3. Then, choose any vertex and put the cursor over it till it turns horizontal before click
on it. Draw as many diagonals as you can from the chosen vertex. Fill out the row for
quadrilateral in the table of questions 3 in the worksheet.
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4. To explore the exterior angles of the quadrilateral, you must extend the segments of
the quadrilateral to form rays.
5. Then repeat Steps 12 20 to find the sum of pair of interior-exterior angle of the
quadrilateral. Complete the table for the question 9 in the worksheet.
6. By using the same step as above, calculate the sum of the exterior angles of aquadrilateral and complete the table for question 11.
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Pentagon
1. Next is to find the sum of interior angles of a pentagon. Click File and select New
Sketch. A new blank sketchpad will open.
2. Follow steps 2-9 but this time draw a pentagon where it has 5 segment or 5 sides.
3. Then, choose any vertex and put the cursor over it till it turns horizontal before click
on it. Draw as many diagonals as you can from the chosen vertex. Fill out the row for
pentagon in the table of questions 3 in the worksheet.
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4. To explore the exterior angles of the pentagon, you must extend the segments of the
pentagon to form rays.
5. Then repeat Steps 12 20 to find the sum of pair of interior-exterior angle of the
pentagon. Complete the table for the question 9 in the worksheet.
6. By using the same step as above, calculate the sum of the exterior angles of a triangleand complete the table for question 11.
Hexagon
1. Next is to find the sum of interior angles of a hexagon. Click File and select New
Sketch. A new blank sketchpad will open.
2. Follow steps 2-9 but this time draw a hexagon where it has 6 segment or 6 sides.
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3. Then, choose any vertex and put the cursor over it till it turns horizontal before click
on it. Draw as many diagonals as you can from the chosen vertex. Fill out the row for
hexagon in the table of questions 3 in the worksheet.
4. To explore the exterior angles of the pentagon, you must extend the segments of the
pentagon to form rays.
5. Then repeat Steps 12 20 to find the sum of pair of interior-exterior angle of the
pentagon. Complete the table for the question 9 in the worksheet.
6. By using the same step as above, calculate the sum of the exterior angles of a triangleand complete the table for question 11.
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OBSERVATIONS
1. What is the sum of interior angles of a triangle? ______
2. What is the sum of interior angles ofany triangles? _____
Is the sum of interior angles for a triangle is differing from other triangles? Yes / No.
3. Fill in the tables :
Name of
Polygon
Number of
Sides
(n)
Sum of interior
angles
Number of
Triangles made
by diagonal(s)
Sum of interior
angles according
to triangles
Triangle 3 1
Quadrilateral 4 2
Pentagon 5 3
Hexagon 6 4
4. Is the sum of interior angles of using the software for each polygons are equal with the
ones that you calculate according to the triangles? Yes / No
5. Can you deduce, or find the sum of interior angles of heptagon and octagon withoutusing the Geometers Sketchpad?
Heptagon :
Octagon :
6. Please explain how you found the answer for Question 4.
We just follow the sequence of number of triangles made by diagonals in heptagon
and octagon from the table above and times it with
7. Can you find the relationship or a function between the numbers of sides of thepolygons with the number of triangles made by diagonals?
Number of triangles made by diagonals = n2.
8. So, can you make a conclusion or a rule to find the sum of interior angles for any
polygons that has n number of sides.
The sum of interior angles = ( )
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9. Complete the table below for each pair of interiorexterior angle of a triangle.
Polygon Interior Angles Exterior AnglesThe sum of interior
and exterior angles
Triangle
mABC = mRBC =
mBCA = mPCA = mCAB = mQAB =
Quadrilateral
mABC = mQBC =
mBCD = mRCD =
mCDA = mSDA =
mDAB = mPAB =
Pentagon
Hexagon
10.Write down any relationship that you see with the exterior angles and the remote
interior angles.
The sum of a pair interiorexterior angles of a polygon is
Interior angle + Exterior angle =
11.Complete the table below for the sum of exterior angles of a polygon.
Polygon Triangle Quadrilateral Pentagon Hexagon
Sum of
exterior angles
12.Can you make a conclusion from the table above?
The sum of exterior angles of any polygon =
Notes: Words that are highlighted in the red are suppose to be written by the students.
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CONCLUSION
From the activity, students can find the sum of the interior angles of triangles,
quadrilateral, pentagon and hexagon. Then, the students can investigate, describe and
reason about the sum of interior angles of other polygons like heptagon and octagon.
Lastly, the students can make and test conjectures about the properties of polygons and
relationship with the number triangles made by diagonals in the polygons before making a
conclusions on how to determine the sum of the interior angles in a polygons.
EXERCISES
1. Determine the number of sides in a polygon if the sum of the interior angles is 5400.
2. A Canadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an
undecagon.
a) Define a regular polygon with 11 sides.
b) Determine the sum of the interior angles of the loonie.
3. What is the fifth interior angle of the pentagon below?
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4.
In the diagram above, ABE and BCF are straight line. Find the values of x and y.
5.
Find the value of x in the diagram.