Estimating angles
🎯 In this topic you will
- Estimate the size of an angle accurately by comparing it with known reference angles.
🧠 Key Words
- estimate
Show Definitions
- estimate: To make a sensible and approximate judgment of a value without measuring it exactly, often by comparing it with known reference values.
Using Angles to Give Directions
We can estimate an angle to help explain how far to turn and which direction to walk in, even without measuring exactly.
❓ EXERCISES
1. One right angle is $90$ degrees.
a. How many degrees are there in two right angles?
b. How many degrees are there in three right angles?
c. How many degrees are there in four right angles?
👀 Show answer
a. Two right angles are $2 \times 90 = 180$ degrees.
b. Three right angles are $3 \times 90 = 270$ degrees.
c. Four right angles are $4 \times 90 = 360$ degrees.
2. Stand up. Turn four right angles in the same direction. Describe what happens to the direction you are facing after turning four right angles.
👀 Show answer
Turning four right angles means turning through $360$ degrees, so you end up facing the same direction as when you started.
❓ EXERCISES
3. Estimate the size of these angles in degrees using the decision tree and diagram.
👀 Show answer
a. A sensible estimate is about $30$ degrees.
b. A sensible estimate is about $75$ degrees.
4. Estimate the size of the angle in degrees using the decision tree and diagram.
👀 Show answer
a. A sensible estimate is about $110$ degrees.
b. A sensible estimate is about $160$ degrees.
5. What is the best estimate for this angle? Explain why it is the best estimate.
Estimate $95$ degrees
Estimate $60$ degrees
Estimate $20$ degrees
Estimate $38$ degrees
Estimate $10$ degrees
Compare your answer and explanation with your partner.
Use the decision tree and diagram to decide who has the best explanation.
👀 Show answer
The best estimate is $38$ degrees because the angle is clearly less than $45$ degrees but much larger than $20$ degrees.
6. Carly says that she estimates that this angle is $175$ degrees. This is not a good estimate.
Explain how Carly could improve her estimate.
👀 Show answer
She should first decide whether the angle is closer to $180$ degrees or $135$ degrees using the decision tree, then compare it with known reference angles to narrow the estimate.
🧠 Think like a Mathematician
Task: Estimate the size of angles and justify why your estimate is the closest.
Method:
- Write down an estimate for the size of the given angle on a piece of paper.
- List all the estimates you have made in order, from smallest to largest.
- Choose one estimate and explain why you think it is closest to the actual size of the angle.
- Repeat the activity using a different angle and try to improve your estimate.
Goal: Improve the accuracy of your estimates and become better at explaining why one estimate is more reasonable than another.
Show Guidance
- Use reference angles such as $45^\circ$, $90^\circ$, and $180^\circ$ to guide your estimate.
- Decide first whether the angle is acute, right, or obtuse.
- Explain your reasoning clearly by comparing the angle with known angles.
💡 Quick Math Tip
Use reference angles first: Before guessing a number, decide whether an angle is less than, equal to, or greater than a right angle, then compare it with familiar angles like $45^\circ$, $90^\circ$, and $180^\circ$ to make a more accurate estimate.