Download (2) The student erred because the included the measures of angles

(2) (2) The student erred because the included the measures of angles F, G, K,
and N which are not angles of the polygon. Since these angles form a
circle, the student can get the correct answer of 540 by subtracting 360
from the answer that they got. Another approach would be to divide the
pentagon into 3 triangles instead of 5 in a similar way to their answer of
part 4. I would give the student 5 points for their work.
(3) The student erred because the included the measures of angles C, D, G, J, N ,
and R which are not angles of the polygon. Since these angles form a
circle, the student can get the correct answer of 720 by subtracting 360
from the answer that they got. Another approach would be to divide the
hexagon into 4 triangles instead of 6 in a similar way to their answer of
part 4. I would give the student 5 points for their work.
(4) The answer is correct, so the grade should be 10 points.
(7) (a) The polygon has 5 sides, so using the formula that the sum of the interior
angles of an n-gon is 180(n − 2), we get that the sum of the angles is 540
degrees.
(b) The polygon has 7 sides, so using the formula that the sum of the interior
angles of an n-gon is 180(n − 2), we get that the sum of the angles is 900
degrees.
(9) (a) The sides are congruent and the angles are also congruent, so this is a
regular polygon.
(b) The sides are congruent, but the angles are not congruent, so this is not
a regular polygon
(c) The angles are congruent, but the sides are not congruent, so this is not
a regular polygon
(d) Since the shape is a quadrilateral we know that the sum of the degrees
of the angles is 360. Also the 2 angles with unknown measurements are
congruent. So if x is the unknown measurement, then x + x + 91 + 91 =
360. Hence 2x + 182 = 360. So 2x = 178. Hence x = 89. Since the angles
aren’t all congruent, the polygon is not regular.
(11) (a) The formula for the measure of an interior angle of a regular n-gon is
180(n−2)
. So the measure of an interior angle of an equilateral triangle is
n
180(3−2)
= 180
3
3 = 60. The formula for the measure of a central angle of a
regular n-gon is 360
n . So the measure of a central angle of an equilateral
360)
triangle is 3 = 120.
(b) The formula for the measure of an interior angle of a regular n-gon is
180(n−2)
. So the measure of an interior angle of a regular octogan is
n
1
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180(8−2)
8
= 1080
= 135. The formula for the measure of a central angle
8
of a regular n-gon is 360
n . So the measure of a central angle of a regular
360)
octagon is 8 = 45.
(13) (a) The sum of the measures of the angles of an n-gon is 180(n − 2), so the
sum of the measures of the angles of a hexagon is 180(6−2) = 720. Hence
x + 121 + 128 + 124 + 109 + 120 = 720. So x + 602 = 720. Hence x = 118.
(b) The sum of the measures of the angles of an n-gon is 180(n − 2), so
the sum of the measures of the angles of a pentagon is 180(5 − 2) =
540. We know that the 2 right angles each measure 90 degrees. S0
x + 158 + 122 + 90 + 90 = 540. So x + 460 = 540. Hence x = 80.
(22) The student used the wrong formula for the central angle measure. The formula for the measure of a central angle of a regular n-gon is 360
n . So the
360)
measure of a central angle of a regular octagon is 8 = 45. I would encourage the student to review the section on angle measurements of regular
n-gons.
(23) (a) The angle formed between the new ray and the leftmost orginal ray is an
obtuse angle. He angle formed between the new ray and the rightmost
original ray is a straight angle. See the picture in Homework 6 solution
pictures.
(b) The measure of the obtuse angle is 110 degrees. Adding 180 to this gives
us that the measure of the reflex angle is 290 degrees.
(c) Adding the measures of the original angle and the reflex angle gives us
70 + 290 = 360 degrees.
(d) One method for finding the measure of a reflex angle is to subtract the
measure of the original angle from 360 degrees to get the measure of the
reflex angle.
(24) (a) Subtracting 90 from 360 degrees gives an answer of 270 degrees. So by
the result of problem 23 part d, the measure of the reflex angle of a right
angle is 270 degrees.
(b) The figure has 12 sides. Starting at the angle in the upper left corner and
going clockwise, the measures of the angles are 90, 90, 90, 270, 270, 90, 90, 90, 270, 270, 90, 90, 90.
See the picture in Homework 6 solution pictures.
(c) The sum of all the interior angle measurements is 1800 degrees.
(d) By the formula on page 387, the sum of the interior angle measurements
of a 12-gon is 180(12 − 2) = 1800. So the result agrees with the sum that
we found in part c.
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Page 3 of 3
(e) An activity that I would use so that students can inductively discover
that the formula for finding the sum of the interior angles of a polygon
can be used for both convex and concave polygons is as follows. I would
have students draw and cut out a convex and a concave polygon with
the same number of sides. Then using a protractor they would measure
the angles of each polygon and write down the sum of the angles of each
polygon. Then they can see that the sum of the angles is the same for
both polygons and is the same answer they get by using the formula for
finding the sum of the interior angles of a polygon.