Section C Practice Problems

Sample response:

1. 

   It is acute because it is less than 90 degrees.

2.  

   

   My acute angle is about $30^\circ$. The acute angle and the new angle make a line or an angle that is $180^\circ$. That means the new angle measures about $150^\circ$, so it is obtuse. I can also see that it is more than $90^\circ$.

1. $30^\circ$. Sample response: Because 12 of them make $360^\circ$.
2. Yes. Eighteen angles are needed. $18 \times 20 = 360$.

1. $45^\circ$. Sample responses:
   • There are 8 equal angles in one full turn, so each angle is $360 \div 8$, which is 45.
   • Angle M is half of a right angle and $90 \div 2 = 45$.
2. $72^\circ$. Sample response: If 5 equal angles make $360^\circ$, then each angle is $360 \div 5$ or 72.

1. • $90^\circ$. Sample response: The two hands make a right angle. (Students also may say $270^\circ$, and that there is a right angle and a big angle that together make $360^\circ$.)
   • $150^\circ$ (and $210^\circ$). Sample response: At 1 o’clock, the hour hand makes a $30^\circ$ angle with the minute hand at 12 because it’s $\frac{1}{12}$ of the circle or $\frac{1}{12}$ of $360^\circ$. There are 5 of the $30^\circ$ angles that make up the angle that the hands make at 5 o’clock. So their angle is $5 \times 30$ or $150^\circ$.
   • $180^\circ$. Sample response: The two hands make a straight line, or half of a $360^\circ$ angle.
2. $120^\circ$. Sample response: There are $360^\circ$ in the full circle and 3 groups of 4 hours, so one group of 4 hours must be $360 \div 3$, which is 120.

35 minutes. Sample response: Every time the long hand moves from one number to the next, the long hand turns $30^\circ$. If it has turned $210^\circ$, the long hand has gone past 7 numbers ($210 \div 30 = 7$), which means 7 times 5 minutes.

Angle C measures $36^\circ$, and Angles D and E each measure $72^\circ$. Sample response: The angles meeting at the center point add up to $360^\circ$ and there are 10 of them. Because they are all equal, each is $36^\circ$. Two of the outer angles plus the $216^\circ$ angle add up to $360^\circ$, so they are $144^\circ$ together and $72^\circ$ each.

Sample responses:

• At 12:00 noon or 12:00 midnight, the hour hand and the minute hand both point toward 12. At 6:00 a.m. or p.m., they are pointing in opposite directions. I think that there are some other times when they point in the same direction but could not find the exact time.
• Between 1:00 and 2:00, the hour hand starts pointing at the 1 and ends pointing at 2. The minute hand goes all the way around, so there is a time when the minute hand catches up with the hour hand. I think that time is a little after 1:05, but I can't find the exact time.

(Though students are not expected to find the exact time, it happens 11 times every 12 hours, at regular intervals, which means it happens every $\frac{12}{11}$ hours. The first time after 12:00 would be at 1 and $\frac{1}{11}$ hours, which is a little more than 27 seconds after 1:05.)

Sample response:

1. First, I made a $50^\circ$ angle, using my protractor. I made sure that the two sides of the angle were 1 inch long. Then I drew a line connecting the endpoints of the sides and folded the paper over. That gives me the other two sides of my rhombus.

   

2. This time I made the sides of the angle longer, 2 inches each. My shape looks like it's the same but it's bigger. I can fit 4 of the smaller shapes inside the big one.

1. $3^\circ$, because 30 seconds is half a minute, so it turns half of $6^\circ$.
2. $1^\circ$, because 10 seconds is a third of 30 seconds.
3. $480^\circ$. It is 20 minutes more than 1 hour, so it is $120^\circ$ more than $360^\circ$.
4. $900^\circ$. The minute hand turns $360^\circ$ twice in 2 hours, plus $180^\circ$ in half an hour. $(2 \times 360) + 180 = 720 + 180 = 900$

1. • a triangle: 3 angles
   • a trapezoid: 4 angles
   • a rhombus: 4 angles
   • a hexagon: 6 angles
      
2. Angle A is $120^\circ$ because $360 \div 3 = 120$.
   Angle B is $60^\circ$ because $360 \div 6 = 60$, or $180 \div 3 = 60$.
   Angle C is $60^\circ$ because $360 \div 6 = 60$, or $180 \div 3 = 60$.
   Angle D is $120^\circ$ because $360 \div 3 = 120$.
   Angle E is $60^\circ$ because $360 \div 6 = 60$, or $180 \div 3 = 60$.
   Angle F is $120^\circ$ because $360 \div 3 = 120$.