Properties of Polygons
Directions: Write your answers in your math journal.
Label this exercise Properties of Polygons.
6. Which of these statements are true and which are false?
a. A square is a rectangle.
b. A rectangle is a square.
c. A trapezoid is a parallelogram.
d. A parallelogram is a trapezoid.
e. A square is a quadrilateral.
f. A quadrilateral is a square.
g. An equilateral triangle is isosceles.
h. A scalene triangle is equilateral.
7. List and identify as many of the different kinds of triangles and quadrilaterals as you can in the following figure.
8. A polygon divides a plane into what three regions?
9. How does the number of sides in a polygon compare with the number of its vertices?
10. Find the measures of the interior and exterior angles of each of these regular polygons:
a. Heptagon b. Octagon c. Nonagon d. Decagon
11. Find the values of all the angles in this sketch if AEF is a straight angle and
x A
B C
D
E 3
D x 8, C 2x 5, B 2x 10, and A x 13.
∠ = + ∠ = + ∠ = + ∠ = +
12. If the figure is a regular hexagon, find the measure of x∠ . (Hint: What do the angle markings mean?)
Explorations
1. To find the sum of the angles of a quadrilateral:
a. Make a sketch of a rectangle. Draw a line from one vertex to another non-adjacent vertex. This is called a diagonal of the rectangle. It divides the rectangle into two ___________. Make a conjecture concerning the sum of the angles of a rectangle.
b. Make sketches of the following quadrilaterals: a square, a trapezoid, a rhombus, and a general parallelogram. Draw a diagonal in each figure. How many triangles are formed in each figure? Make a conjecture concerning the sum of the angles in any quadrilateral.
2. To construct a special angle:
a. Mark a point on your paper and draw a large circle with your compass. Draw a diameter using your original point. Pick a point on the circle and connect it to the endpoints of the diameter. Now mark a point on the circle on the other side of the diameter and connect it to the ends of the diameter. Cut out the two triangles and compare the size of the angles with their vertex on the circle.
b. Try the experiment again using a circle of a different size. Compare the angles you get with the angles from the first circle. What did you find?
c. How large do you think these angles are? How can you tell for sure?
d. How does this knowledge help you?
x
3. a. Complete the following chart for the regular polygons with the given number of sides.
Number of sides of regular polygon
Number of triangles formed by diagonals from a single
vertex
Sum of the measure of interior angles
c. If sketches of regular polygons are placed side by side with ever increasing number of sides, the polygons begin to look more and more like _______________.
4. A quadrilateral has two diagonals if diagonals are drawn from every vertex and not just a single vertex.
a. Sketch a pentagon, hexagon, and an octagon.
b. Draw in all diagonals and count them.
c. Make a conjecture about the number of diagonals that can be drawn altogether in an n-gon where n is greater than 3.
Proofs/Justifications/Constructions
1. Is this figure a quadrilateral? Justify your answer.
2. Can a right triangle be obtuse? Justify your answer.
3. Mark two points on your paper (not too close together) and label them points A and B.
Using only your compass and your straight edge (the straight edge is the piece of equipment in your supplies that looks like a ruler, but has no markings on it), construct a right angle at point B. Explain why you think your construction is correct.
4. Although it includes a reflex angle, show that the sum of the measures of the interior angles of the quadrilateral is 360 . Justify your answer. 0
5. The sum of the angles in a regular polygon is 1800 . How many sides does the polygon 0 have? What is the measure of one of its interior angles? Justify your answer.
6. Suppose a polygon has n sides with n>3. Make a conjecture about the sum of the measures of its angles in terms of the variable n. Justify your answer.
7. Make a conjecture about the measure of each interior angle of a regular n-sided polygon with n>3. Justify your answer.
8. In Exploration #2 on page 11 you made a conjecture as to the number of diagonals that can be drawn in an n-sided polygon if n>3. You may have written your conjecture in this form: An n-sided polygon has
(
3)
2 n n−
diagonals. Justify this answer. (Hint: How many diagonals are there from each vertex? Why? How many vertices does an n-sided figure have? Why is it necessary to divide by 2?)
1. Figure b represents the circle. Figure a represents the interior of a circle.
2. A circle is not a polygon since it is not made up of line segments.
3. The least number of sides a polygon can have is three. If you decrease the size of one side of a triangle until it no longer exists, you end up with two line segments instead of a polygon.
4. Let x = the measure of one of the angles in the equiangular triangle. Since the other angles have the same measure, each angle is 60 . 0
7.
8.
But the question asks for the measure of y.
9. A pentagon can be divided into 3 triangles by drawing diagonals from a single vertex and since there are 5 vertices in a pentagon, the measure of an interior angle of a regular pentagon is 3 180
(
0)
05 =108 . Since an exterior angle is supplementary
0− 0 = 0
Explorations
#1. You should have discovered that the sum of all three angles of a triangle is 180°.
#2.
#3. 1. The measure of the sum of the interior angles of a hexagon is
(
0)
04 180 =720 .
2. The measure of the sum of the interior angles of an octagon is
(
0)
06 180 =1080 .
3. The measure of the sum of the interior angles of a decagon is
(
0)
08 180 =1440 . 4.
No. of sides in polygon No. of diagonals from a vertex Sum of interior angles
4 2 2 180
(
0)
=36005 3 3 180
(
0)
=54006 4 4 180
(
0)
=72008 6 6 180
(
0)
=10800( )
Number of sides on polygon Number of triangles
6 4
Answers to Practice Problems
Connections and Modeling
1. The polygons are equilateral triangles, rhombi, and squares.
2. Classified by the measure of their angles Classified by the measure of their sides Equiangular triangles Scalene triangles Acute triangles Equilateral triangles Right triangles Isosceles triangles
5. Let x = the measure of the smallest angle. Then the measure of the second angle is 3x and the measure of the third angle is x+200.
The angles are 32 , 0 96 , and 0 52 . 0
6. a. True b. False c. False d. False e. True f. False g. True h. False
7. There are 6 triangles – 2 right triangles, 4 other ones (appear to be isosceles). There are 9 quadrilaterals – 6 rectangles, 3 trapezoids (2 appear to be isosceles).
0 0 0
8. A polygon divides a plane into these three regions: the interior region, the exterior region, and the polygon itself.
9. The number of vertices in a polygon is the same as the number of sides it has.
10. a. An interior angle of a regular heptagon is5 180
(
0)
00 0 The marked angles in the triangle are equal.
If y = the measure of one of these angles, then
Furthermore since x+ =y 1200, x=1200− =y 1200−300 =900
Explorations 1. a. Triangles
The sum of the measures of the angles of a rectangle is 360 . 0
b. Each figure is divided into two triangles. The sum of the measures of the angles of a quadrilateral is 360 . 0
2. a. The angles have the same measure.
b. The angles have the same measure.
c. The angles are all 90 (right angles). You can compare them to the corner angle on a 0 piece of paper, use a protractor to measure them, or put two together and see that they form a straight angle.
d. This provides a way to construct a right angle.
3. a.
60 58 58 180
(
0)
=104400 104400 1740b. There is a limit. The values are approaching 180 , but they cannot reach that value 0 because an interior angle must be less than a straight angle.
c. The polygons will begin to look more like a circle.
4. a.
Justifications / Constructions
1. Yes, the figure is a quadrilateral since it has four sides each of which are line segments.
2. No, a right triangle can’t also be obtuse. An obtuse angle is greater than 90 . If the right 0 angle and the obtuse angle are added together, the sum is already greater than 180 , leaving 0 nothing for the measure of the third angle.
3. Make a circle with center at point A and passing through point B.
Draw a diameter through point A and connect point B with the ends of the diameter.
4. The figure can be divided into two triangles . Therefore the sum of the measures of its interior angles is 2 180
(
0)
=3600.5. If n = the number of sides in the polygon, the sum of its angles is
The polygon has 12 sides. Since the polygon is regular, each interior angle is
0
1800 0
12 =150 6. Conjecture: The measure of the sum of the interior angles of an n-sides polygon is
( )
1800 n−2 . This is true since the n-sided polygon can be divided into n−2 triangles, each of which is 180 . 0
7. Conjecture: The measure of each interior angle of a regular n-sided polygon with n>3 is
( )
1800 n 2 n
− . The sum of all the interior angles of the polygon must be divided by the
number of equal angles, n.
8. The number of diagonals that can be drawn from a single vertex is n−3. There are n vertices so it would at first thought seem that there would be n n
(
−3)
diagonals altogether.However, that would be counting each diagonal twice. For example, the diagonal from vertex A to vertex B would be counted as a vertex from A and as a vertex from B, but there
A
NOTES