Comparing angles
🎯 In this topic you will
- Compare the sizes of angles.
🧠 Key Words
- angle
- compare
- degrees
Show Definitions
- angle: The space between two straight lines that meet at a point, called the vertex.
- compare: To look at two or more things to decide which is bigger, smaller, or if they are the same.
- degrees: The unit used to measure the size of an angle, written with the symbol °.
Why Angle Size Matters 🎶
T alking about different sizes of angles is important. To make their dance look good, these dancers need their feet to all be at the same angle.
❓ EXERCISES
1. Which angle is greater?
👀 Show answer
a. Angle $C$ is greater than angle $D$.
b. Angle $F$ is greater than angle $E$.
c. Angle $H$ is greater than angle $G$.
2. Are these statements true or false?
a. Angle $W$ is greater than angle $X$.
b. Angle $X$ is less than angle $Z$.
c. Angle $Y$ is less than angle $W$.
d. Angle $Z$ is greater than angle $W$.
👀 Show answer
a. False.
b. True.
c. True.
d. False.
3. Order these angles from smallest to largest.
👀 Show answer
$J$, $L$, $K$
4. Order these angles from smallest to largest.
👀 Show answer
$p$, $r$, $s$, $t$, $q$
5. Arun has made a mistake.
Critique Arun’s statement. What is Arun’s mistake?
Why do you think Arun made that mistake?
How could you convince Arun that he is wrong?
👀 Show answer
Arun compared the lengths of the lines instead of the size of the angles. The angle depends only on how wide it opens, not on how long the lines are. Tracing one angle and placing it on the other would show that angle $B$ is actually greater.
🧠 Think like a Mathematician
Clock hands meet at an angle.
At 3 o’clock the hour and minute hands on this clock make a right angle.
Task:
- Write the time for the smallest angle you can find between the clock hands.
- Write the time for the largest angle you can find between the clock hands.
- Check your answers carefully. Think about whether it is possible to make an even smaller or larger angle by slightly changing the time, and improve your answers if needed.
👀 show answer
Smallest angle:
The smallest possible angle is 0°. This happens when the hour and minute hands are exactly on top of each other, for example at 12:00 (also at about 1:05, 2:11, etc.).
Largest angle:
The largest possible angle is 180°. This happens when the hands are directly opposite each other, for example at 6:00.
Improving the answers:
The smallest angle cannot be less than $0^\circ$ and the largest angle cannot be more than $180^\circ$. However, the exact times can vary slightly because the hour hand moves continuously. By choosing times just before or after exact hours, you can make angles very close to these values.