Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 A B C Homework

Solutions for Session 4 Homework

See solutions for Problems: H1 | H2 | H3 | H4 | H5 | H6 | H7

Problem H1

 a. Four triangles are formed. b. The total sum of the angles in the four triangles is 4 • 180° = 720°. c. The sum of the angles at the center is 360 degrees (a full circle). d. The interior angles sum to 360 degrees (or 720° - 360°). This shows that a quadrilateral has 360 degrees in its angles. e. A five-sided polygon: Five triangles are formed. The sum of the angles is 5 • 180°. The sum of the angles at the center point is still 360 degrees. The sum of the interior angles is (5 • 180°) - 360° = 180°(5 - 2) = 540°. An eight-sided polygon: Eight triangles are formed. The sum of the angles is 8 • 180°. The sum of the angles at the center point is 360 degrees. The sum of the interior angles is (8 • 180°) - 360° = 180°(8 - 2) = 1,080°. An n-sided polygon: n triangles are formed. The sum of the angles is n • 180°. The sum of the angles at the center point is 360 degrees. The sum of the interior angles, then, is (n • 180°) - 360°, or 180°(n - 2).

 Problem H2 A good estimate is 72 degrees, since there are five angles roughly equally spaced around a circle (72 = 360 5). Depending on which angle you actually measure, it may be slightly larger or smaller than 72 degrees.

 Problem H3 Angles a and e are congruent only when the two lines are parallel. Angles d and h are congruent to a and e when the lines are parallel, as are b, c, f, and g to one another. (In Figure 3, all the angles are congruent.) If the lines are not parallel, a and d are still congruent, as are e and h, but they are not congruent to one another.

Problem H4

a.

Two rays form one angle. Three rays form three angles. Four rays form six angles. Five rays form 10 angles. Six rays form 15 angles.

b.

With each new ray, the number of angles seems to grow by one less than the total number of rays, so seven rays should form 21 angles (15 + [7 - 1]). Using this information we can make the following table:

 Rays 2 3 4 5 6 7 8 9 10 Angles 1 3 6 10 15 21 28 36 45

A formula is harder to find. One way to do it is to recognize that the first ray in an angle can be picked out of any of the n rays; the second ray can be picked out of (n - 1) rays. Multiplying the two will give us double the count of all the angles that can be formed (try some of the numbers from the table above to test this). So there are a total of n(n - 1)/2 angles that can be formed.

 Problem H5 Here is one way to do it: Repeat 5; Go Forward 4; Rotate Right 36; End Repeat; Repeat 5; Go Forward 4; Rotate Right 36; End Repeat. Notice that we've done this as a two-step process, following the same sequence twice. Alternatively, the sequence can also be written as the following: Repeat 2; Repeat 5; Go Forward 4; Rotate Right 36; End Repeat; End Repeat. The total turn was 360 degrees.

Problem H6

 a. The sled turns 30 degrees to the right, three times. b. Since it takes 360 degrees to turn all the way around, there would have to be 12 total turns. c. The resulting angle is the angle between the turns -- the interior angle. d. The pilot's description focused on interior angles rather than exterior angles. e. The resulting angle for a 40-degree turn would be 140 degrees. If the sled track forms a 130-degree resulting angle, the turn was 50 degrees.

Problem H7

 a. No. The acute angle must be exactly 60 degrees if six of them are to fit together at the center of the quilt. b. Yes. Angle B (and therefore A) must be 120 degrees. We know this because the angle shown is 30 degrees, meaning that the angle on the other side of B is also 30 degrees, leaving 120 degrees for B (since they form a straight line). Adding A and B gives us 240 degrees, so C (and its opposite angle) is 60 degrees, which is the required angle for the quilt.