## Learning Math: Measurement

# Angle Measurement Homework

## Session 4, Homework

**Problem H1**

Draw any quadrilateral. Then draw a point anywhere inside the quadrilateral, and connect that point to each of the vertices, as shown below:

Now answer the following questions:

- How many triangles have been formed?
- What is the total sum of the angle measures of all the triangles?
- How much of the total sum from part (b) is represented by the angles around the center point (i.e., what is their sum)?
- How much of the total sum from part (b) is represented by the interior angles of the quadrilateral?
- Repeat the activity with a five-sided polygon and an eight-sided polygon, and then attempt to generalize your result to an n-sided polygon.
**Note 8**

**Problem H2**

Estimate the number of degrees between two adjacent legs of the starfish below. Then, using a protractor, measure one of the angles. (You may want to print the PDF image in order to do this.) How close was your estimate?

**Problem H3**

Print the figures below from the PDF document. For each figure, cut out a and e. When are a and e congruent? What other angles have the same measure as ∠a? As ∠e?

**Problem H4**

- How many angles can be formed with the rays below?Look at all the possible combinations. For example, three rays can form three distinct angles. Do you see them?

- Predict the number of angles formed with seven rays and with 10 rays. Can you generalize your prediction to n rays?
**Note 9**

**Problem H5**

Write a series of commands in the style used in the Interactive Activity (or Geo-Logo commands) to draw a regular decagon (a 10-sided figure). You can test your commands using the Non-Interactive Activity. How many total degrees did you turn to make the decagon?

**Problem H6**

A sled got lost in the darkness of a polar night. Mayday emergency calls were received all night, but the darkness prohibited a search. The next morning, planes searched the area, and the pilots saw these tracks made by the sled:

- Use turns to describe the route of the sled as if you had been in it.
- If the sled continued in the same way, it might have returned to its starting point. How many turns would the sled have had to make to return to its starting point?The pilot described the track as follows: “It looks like the sled made three equal turns to the right. The four parts of the track seem to be equally long, and the resulting angle between each part measures about 150 degrees.”
- What do you think the pilot means by “the resulting angle”?
- How does the pilot’s description differ from your own?
- If you were to make a 40-degree turn on the sled, what would the resulting angle be? If the sled track forms a 130-degree resulting angle, what is the size of the turn?

**Problem H7**

You are interested in making a quilt like the one shown below. In the center, a star is made from six pieces of material: **Note 10**

- Is it possible to make the star with the piece below? Why or why not?

Print and cut out several copies of this image from the PDF document. - What about the highlighted piece below — could it be used to make the star in the quilt? Without measuring, determine the measures of angles A, B, and C, and explain how you arrived at your solution.

### Notes

**Note 8**

In the activities in the text, you determined the rule for finding the sum of the interior angles of a polygon, 180 • (n – 2), where n is the number of sides of the polygon. Problem H1 presents an alternative approach for finding the sum of the interior angles in a polygon. Think about the two rules and explain why they are the same.

**Note 9**

This is similar to the well-known handshake problem: If each person shakes hands with another person, how many handshakes will there be for a total of n people?

**Note 10**

There is a great deal of mathematics involved in designing quilts. If you’re working in a group, you may want to discuss Problem H7.

### Solutions

**Problem H1**

- Four triangles are formed.
- The total sum of the angles in the four triangles is 4 • 180° = 720°.
- The sum of the angles at the center is 360 degrees (a full circle).
- The interior angles sum to 360 degrees (or 720° – 360°). This shows that a quadrilateral has 360 degrees in its angles.
- 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**

- 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**

- The sled turns 30 degrees to the right, three times.
- Since it takes 360 degrees to turn all the way around, there would have to be 12 total turns.
- The resulting angle is the angle between the turns — the interior angle.
- The pilot’s description focused on interior angles rather than exterior angles.
- 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**

- No. The acute angle must be exactly 60 degrees if six of them are to fit together at the center of the quilt.
- 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.