Angles (II): Comparing angles
FoundationBigger angle = more open. Side length doesn't matter.
After this lesson, you can:
- Angle size = how far the two sides open
- Side length has nothing to do with angle size
- Compare angles by overlaying a paper corner
1Summary 重點摘要
Bigger angle = more open. Side length doesn't matter.
2Vocabulary cards Vocabulary
a
angle角
The opening formed by two lines meeting at a point.
c
compare angles比較角
Seeing which angle opens wider.
o
opening張開度
How far the two sides of an angle spread apart; it decides the angle size.
v
vertex頂點
The point where the two sides of an angle meet.
s
side (of angle)角的邊
The two lines that make the angle.
b
bigger angle較大的角
The one with the wider opening, no matter the side lengths.
p
paper corner紙角
A corner folded from paper, used to overlay and compare angles.
c
clock hands時鐘指針
Clock hands make angles, e.g. a right angle at 3 o'clock.
3Step by step Steps
- Part 1: What angle size meansThe size of an angle is how far its two sides open. At 3 o'clock the hour hand and minute hand open into a right angle; at 1 o'clock they open less, so the angle is smaller. The wider the opening, the bigger the angle.
- Part 2: Side length does not matterAngle size has nothing to do with side length. Draw a 90° angle with a short pencil, then draw one with a long ruler: the two angles are the same size. However long the sides are drawn, only the opening matters.
- Part 3: Compare with a paper cornerTo compare two angles, fold a paper corner and lay it over the other angle: vertex on vertex, one side lined up. If the paper's other side lies outside, the paper angle is bigger; if it lies inside, the paper angle is smaller.
- Part 4: Practise with a clockA clock is good practice for comparing angles. At 3 o'clock the hands make a right angle; at 6 o'clock they form a straight line, the widest opening; at 2 o'clock the angle is smaller than at 3. So the 6 o'clock angle is bigger than the 3 o'clock angle, which is bigger than the 2 o'clock angle.
- Part 5: Trimming the sides changes nothingCut a big angle from paper, then cut both sides down to half their length. The opening does not change at all, so the angle is still the same size. Remember: trimming the sides never makes an angle smaller; the opening decides everything.
- Part 6: Everyday useOpening a door shows angles in daily life. The wider the door opens, the bigger the angle between the door and the wall: fully open is about 180°, and half open is about 90°. Comparing two angles is like comparing how far two doors are open.
4Diagrams Diagrams
5Worked example Example
Example: angle A has two 8 cm sides and a narrow opening; angle B has two 3 cm sides and a wide opening. Which angle is bigger?
- Look at the side lengths first: one is 8 cm and the other 3 cm, but angle size does not depend on side length, so ignore them.
- Look at the openings next: angle A opens narrowly, while angle B opens widely.
- The angle with the wider opening is the bigger angle; this is the only rule for comparing angles.
- Conclusion: angle B is bigger than angle A.
- Answer: Answer: angle B is bigger, even though its sides are shorter. The size of an angle only compares how far the two sides open, and has nothing to do with how long the sides are. This is the key idea of this lesson.
6Common mistakes Mistakes
✗ Thinking longer sides make a bigger angle
✓ Angle size depends only on the opening, not side length
✗ Guessing by eye instead of overlaying
✓ When two angles look close, compare with a paper corner
✗ Overlaying without matching vertices
✓ Match the vertices first, then align one side
✗ Saying a trimmed angle gets smaller
✓ Cutting the sides does not change the opening, so the angle stays the same
7Quick check Quick check
Q1What decides the size of an angle?
Answer: How wide the sides open
Did you get it right?
Q2How does side length affect angle size?
Answer: No effect at all
Did you get it right?
Q3Which is bigger: the angle at 3 o'clock or at 2 o'clock?
Answer: 3 o'clock (right angle)
Did you get it right?
Q4When comparing with a paper corner, what must line up?
Answer: Vertex to vertex, then align one side
Did you get it right?
Q5If you draw an angle's sides longer, what happens?
Answer: The angle stays the same
Did you get it right?
8Recap Recap
- Angle size = how far the two sides open
- Side length has nothing to do with angle size
- Compare angles by overlaying a paper corner
- Match vertices and one side when overlaying
- Clock hands make ready-made angles: a right angle at 3 o'clock